Integrand size = 30, antiderivative size = 436 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=-\frac {\left (3 a^3 d^3 f^3+8 a^2 b d^2 f^2 (d e-3 c f)+64 b^3 (d e-c f)^3-80 a b^2 d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{128 b^2 f^4}+\frac {d \left (3 a^2 d^2 f^2-56 a b d f (d e-3 c f)+48 b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2}}{192 b f^3}-\frac {d^2 (8 b d e-24 b c f-9 a d f) x^5 \sqrt {a+b x^2}}{48 f^2}+\frac {b d^3 x^7 \sqrt {a+b x^2}}{8 f}+\frac {\left (3 a^4 d^3 f^4+8 a^3 b d^2 f^3 (d e-3 c f)+128 b^4 e (d e-c f)^3-192 a b^3 f (d e-c f)^3+48 a^2 b^2 d f^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2} f^5}-\frac {(b e-a f)^{3/2} (d e-c f)^3 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^5} \] Output:
-1/128*(3*a^3*d^3*f^3+8*a^2*b*d^2*f^2*(-3*c*f+d*e)+64*b^3*(-c*f+d*e)^3-80* a*b^2*d*f*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/b^2/f^4+1/192*d *(3*a^2*d^2*f^2-56*a*b*d*f*(-3*c*f+d*e)+48*b^2*(3*c^2*f^2-3*c*d*e*f+d^2*e^ 2))*x^3*(b*x^2+a)^(1/2)/b/f^3-1/48*d^2*(-9*a*d*f-24*b*c*f+8*b*d*e)*x^5*(b* x^2+a)^(1/2)/f^2+1/8*b*d^3*x^7*(b*x^2+a)^(1/2)/f+1/128*(3*a^4*d^3*f^4+8*a^ 3*b*d^2*f^3*(-3*c*f+d*e)+128*b^4*e*(-c*f+d*e)^3-192*a*b^3*f*(-c*f+d*e)^3+4 8*a^2*b^2*d*f^2*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a) ^(1/2))/b^(5/2)/f^5-(-a*f+b*e)^(3/2)*(-c*f+d*e)^3*arctanh((-a*f+b*e)^(1/2) *x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/f^5
Time = 1.98 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {\frac {f x \sqrt {a+b x^2} \left (-9 a^3 d^3 f^3+6 a^2 b d^2 f^2 \left (-4 d e+12 c f+d f x^2\right )+8 a b^2 d f \left (90 c^2 f^2+6 c d f \left (-15 e+7 f x^2\right )+d^2 \left (30 e^2-14 e f x^2+9 f^2 x^4\right )\right )-16 b^3 \left (-12 c^3 f^3-18 c^2 d f^2 \left (-2 e+f x^2\right )-6 c d^2 f \left (6 e^2-3 e f x^2+2 f^2 x^4\right )+d^3 \left (12 e^3-6 e^2 f x^2+4 e f^2 x^4-3 f^3 x^6\right )\right )\right )}{b^2}+\frac {384 (-b e+a f)^{3/2} (d e-c f)^3 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e}}-\frac {3 \left (3 a^4 d^3 f^4+8 a^3 b d^2 f^3 (d e-3 c f)+128 b^4 e (d e-c f)^3+192 a b^3 f (-d e+c f)^3+48 a^2 b^2 d f^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}}}{384 f^5} \] Input:
Integrate[((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2),x]
Output:
((f*x*Sqrt[a + b*x^2]*(-9*a^3*d^3*f^3 + 6*a^2*b*d^2*f^2*(-4*d*e + 12*c*f + d*f*x^2) + 8*a*b^2*d*f*(90*c^2*f^2 + 6*c*d*f*(-15*e + 7*f*x^2) + d^2*(30* e^2 - 14*e*f*x^2 + 9*f^2*x^4)) - 16*b^3*(-12*c^3*f^3 - 18*c^2*d*f^2*(-2*e + f*x^2) - 6*c*d^2*f*(6*e^2 - 3*e*f*x^2 + 2*f^2*x^4) + d^3*(12*e^3 - 6*e^2 *f*x^2 + 4*e*f^2*x^4 - 3*f^3*x^6))))/b^2 + (384*(-(b*e) + a*f)^(3/2)*(d*e - c*f)^3*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]*Sq rt[-(b*e) + a*f])])/Sqrt[e] - (3*(3*a^4*d^3*f^4 + 8*a^3*b*d^2*f^3*(d*e - 3 *c*f) + 128*b^4*e*(d*e - c*f)^3 + 192*a*b^3*f*(-(d*e) + c*f)^3 + 48*a^2*b^ 2*d*f^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x ^2]])/b^(5/2))/(384*f^5)
Time = 0.99 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.44, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {420, 318, 403, 299, 211, 224, 219, 420, 318, 299, 211, 224, 219, 420, 299, 211, 224, 219, 403, 25, 398, 224, 219, 291, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \int \sqrt {b x^2+a} \left (d x^2+c\right )^3dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {\int \sqrt {b x^2+a} \left (d x^2+c\right ) \left (d (12 b c-5 a d) x^2+c (8 b c-a d)\right )dx}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {\frac {\int \sqrt {b x^2+a} \left (d \left (72 b^2 c^2-52 a b d c+15 a^2 d^2\right ) x^2+c \left (48 b^2 c^2-18 a b d c+5 a^2 d^2\right )\right )dx}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {3 \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right ) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {3 \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {3 \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )^2dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \sqrt {b x^2+a} \left (d (8 b c-3 a d) x^2+c (6 b c-a d)\right )dx}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {\int -\frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}+\frac {d x \sqrt {a+b x^2}}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\int \frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2} (8 b c-3 a d)}{4 b}}{6 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{6 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{f}\right )}{f}\) |
Input:
Int[((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2),x]
Output:
(b*((d*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + ((d*(12*b*c - 5*a*d)*x*( a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + ((d*(72*b^2*c^2 - 52*a*b*c*d + 15*a^ 2*d^2)*x*(a + b*x^2)^(3/2))/(4*b) + (3*(64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a ^2*b*c*d^2 - 5*a^3*d^3)*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sq rt[a + b*x^2]])/(2*Sqrt[b])))/(4*b))/(6*b))/(8*b)))/f - ((b*e - a*f)*((d*( (d*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + ((d*(8*b*c - 3*a*d)*x*(a + b*x ^2)^(3/2))/(4*b) + (3*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*((x*Sqrt[a + b*x^2 ])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(4*b))/(6*b) ))/f - ((d*e - c*f)*((d*((d*x*(a + b*x^2)^(3/2))/(4*b) + ((4*b*c - a*d)*(( x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b] )))/(4*b)))/f - ((d*e - c*f)*((d*x*Sqrt[a + b*x^2])/(2*f) - (((2*b*d*e - 2 *b*c*f - a*d*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - (2*Sqr t[b*e - a*f]*(d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x ^2])])/(Sqrt[e]*f))/(2*f)))/f))/f))/f
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Time = 1.08 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(\frac {-2 \left (-a f +b e \right )^{2} \left (c f -d e \right )^{3} b^{\frac {9}{2}} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+\sqrt {\left (a f -b e \right ) e}\, \left (\frac {3 \left (\frac {128 b^{4} d^{3} e^{4}}{3}-64 b^{3} d^{2} f \left (a d +2 b c \right ) e^{3}+16 b^{2} d \,f^{2} \left (a^{2} d^{2}+12 a b c d +8 b^{2} c^{2}\right ) e^{2}+\frac {8 b \,f^{3} \left (a^{3} d^{3}-18 a^{2} b c \,d^{2}-72 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) e}{3}+a \,f^{4} \left (a^{3} d^{3}-8 a^{2} b c \,d^{2}+48 a \,b^{2} c^{2} d +64 b^{3} c^{3}\right )\right ) b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{64}+\sqrt {b \,x^{2}+a}\, x f \,b^{\frac {5}{2}} \left (-b^{3} d^{3} e^{3}+\frac {5 d^{2} \left (\left (\frac {2 x^{2} d}{5}+\frac {12 c}{5}\right ) b +a d \right ) b^{2} f \,e^{2}}{4}-\frac {d b \,f^{2} \left (\left (\frac {8}{3} d^{2} x^{4}+12 c d \,x^{2}+24 c^{2}\right ) b^{2}+30 a d \left (\frac {7 x^{2} d}{45}+c \right ) b +a^{2} d^{2}\right ) e}{8}-\frac {3 \left (\left (-\frac {16}{3} d^{3} x^{6}-\frac {64}{3} c \,d^{2} x^{4}-32 c^{2} d \,x^{2}-\frac {64}{3} c^{3}\right ) b^{3}-80 a d \left (\frac {1}{10} d^{2} x^{4}+\frac {7}{15} c d \,x^{2}+c^{2}\right ) b^{2}-8 a^{2} \left (\frac {x^{2} d}{12}+c \right ) d^{2} b +a^{3} d^{3}\right ) f^{3}}{64}\right )\right )}{2 \sqrt {\left (a f -b e \right ) e}\, b^{\frac {9}{2}} f^{5}}\) | \(461\) |
| risch | \(\text {Expression too large to display}\) | \(1142\) |
| default | \(\text {Expression too large to display}\) | \(1674\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/2*(-2*(-a*f+b*e)^2*(c*f-d*e)^3*b^(9/2)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f- b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(3/64*(128/3*b^4*d^3*e^4-64*b^3*d^2*f*( a*d+2*b*c)*e^3+16*b^2*d*f^2*(a^2*d^2+12*a*b*c*d+8*b^2*c^2)*e^2+8/3*b*f^3*( a^3*d^3-18*a^2*b*c*d^2-72*a*b^2*c^2*d-16*b^3*c^3)*e+a*f^4*(a^3*d^3-8*a^2*b *c*d^2+48*a*b^2*c^2*d+64*b^3*c^3))*b^2*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+ (b*x^2+a)^(1/2)*x*f*b^(5/2)*(-b^3*d^3*e^3+5/4*d^2*((2/5*x^2*d+12/5*c)*b+a* d)*b^2*f*e^2-1/8*d*b*f^2*((8/3*d^2*x^4+12*c*d*x^2+24*c^2)*b^2+30*a*d*(7/45 *x^2*d+c)*b+a^2*d^2)*e-3/64*((-16/3*d^3*x^6-64/3*c*d^2*x^4-32*c^2*d*x^2-64 /3*c^3)*b^3-80*a*d*(1/10*d^2*x^4+7/15*c*d*x^2+c^2)*b^2-8*a^2*(1/12*x^2*d+c )*d^2*b+a^3*d^3)*f^3)))/((a*f-b*e)*e)^(1/2)/b^(9/2)/f^5
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{3}}{e + f x^{2}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**3/(f*x**2+e),x)
Output:
Integral((a + b*x**2)**(3/2)*(c + d*x**2)**3/(e + f*x**2), x)
Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^3}{f\,x^2+e} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2),x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^3)/(e + f*x^2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{e+f x^2} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{3}}{f \,x^{2}+e}d x \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e),x)
Output:
int((b*x^2+a)^(3/2)*(d*x^2+c)^3/(f*x^2+e),x)