Integrand size = 30, antiderivative size = 177 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^3 x}{a b^2 (b e-a f) \sqrt {a+b x^2}}+\frac {d^3 x \sqrt {a+b x^2}}{2 b^2 f}-\frac {d^2 (2 b d e-6 b c f+3 a d f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2} f^2}+\frac {(d e-c f)^3 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f^2 (b e-a f)^{3/2}} \] Output:
(-a*d+b*c)^3*x/a/b^2/(-a*f+b*e)/(b*x^2+a)^(1/2)+1/2*d^3*x*(b*x^2+a)^(1/2)/ b^2/f-1/2*d^2*(3*a*d*f-6*b*c*f+2*b*d*e)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2)) /b^(5/2)/f^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^2/(-a*f+b*e)^(3/2)
Time = 1.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.27 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\frac {f x \left (-2 b^3 c^3 f+3 a^3 d^3 f+a b^2 d \left (6 c^2 f-d^2 e x^2\right )+a^2 b d^2 \left (-d e-6 c f+d f x^2\right )\right )}{a b^2 (-b e+a f) \sqrt {a+b x^2}}+\frac {2 (d e-c f)^3 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} (-b e+a f)^{3/2}}+\frac {d^2 (2 b d e-6 b c f+3 a d f) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}}}{2 f^2} \] Input:
Integrate[(c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)),x]
Output:
((f*x*(-2*b^3*c^3*f + 3*a^3*d^3*f + a*b^2*d*(6*c^2*f - d^2*e*x^2) + a^2*b* d^2*(-(d*e) - 6*c*f + d*f*x^2)))/(a*b^2*(-(b*e) + a*f)*Sqrt[a + b*x^2]) + (2*(d*e - c*f)^3*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sq rt[e]*Sqrt[-(b*e) + a*f])])/(Sqrt[e]*(-(b*e) + a*f)^(3/2)) + (d^2*(2*b*d*e - 6*b*c*f + 3*a*d*f)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(5/2))/(2*f^2 )
Leaf count is larger than twice the leaf count of optimal. \(538\) vs. \(2(177)=354\).
Time = 1.01 (sec) , antiderivative size = 538, normalized size of antiderivative = 3.04, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {419, 25, 401, 25, 27, 403, 299, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle -\frac {\int -\frac {\left (d x^2+c\right )^2 \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{3/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d x^2+c\right )^2 \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{3/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}-\frac {\int -\frac {b \left (d x^2+c\right ) \left (a b c (d e-c f)-d \left (4 d f a^2-5 b d e a-3 b c f a+4 b^2 c e\right ) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b \left (d x^2+c\right ) \left (a b c (d e-c f)-d \left (4 d f a^2-b (5 d e+3 c f) a+4 b^2 c e\right ) x^2\right )}{\sqrt {b x^2+a}}dx}{a b}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d x^2+c\right ) \left (a b c (d e-c f)-d \left (4 d f a^2-b (5 d e+3 c f) a+4 b^2 c e\right ) x^2\right )}{\sqrt {b x^2+a}}dx}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a c \left (4 c (2 d e-c f) b^2-a d (5 d e+3 c f) b+4 a^2 d^2 f\right )-d \left (-12 d^2 f a^3+b d (15 d e+17 c f) a^2-2 b^2 c (13 d e+c f) a+8 b^3 c^2 e\right ) x^2}{\sqrt {b x^2+a}}dx}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^2}{f x^2+e}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \int \sqrt {b x^2+a}dx}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {(4 b c-a d) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )}{f x^2+e}dx}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {\int -\frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}+\frac {d x \sqrt {a+b x^2}}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\int \frac {(2 b d e-2 b c f-a d f) x^2+a (d e-2 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {(-a d f-2 b c f+2 b d e) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 (b e-a f) (d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (12 a^3 d^3 f-5 a^2 b d^2 (5 c f+3 d e)+4 a b^2 c d (2 c f+9 d e)-8 b^3 c^2 (3 d e-c f)\right )}{2 b^{3/2}}-\frac {d x \sqrt {a+b x^2} \left (-12 a^3 d^2 f+a^2 b d (17 c f+15 d e)-2 a b^2 c (c f+13 d e)+8 b^3 c^2 e\right )}{2 b}}{4 b}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (4 a^2 d f-3 a b c f-5 a b d e+4 b^2 c e\right )}{4 b}}{a}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (b e-a f)}{a \sqrt {a+b x^2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \left (\frac {\left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) (4 b c-a d)}{4 b}+\frac {d x \left (a+b x^2\right )^{3/2}}{4 b}\right )}{f}-\frac {(d e-c f) \left (\frac {d x \sqrt {a+b x^2}}{2 f}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (-a d f-2 b c f+2 b d e)}{\sqrt {b} f}-\frac {2 \sqrt {b e-a f} (d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f}}{2 f}\right )}{f}\right )}{(b e-a f)^2}\) |
Input:
Int[(c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)),x]
Output:
(((b*c - a*d)*(b*e - a*f)*x*(c + d*x^2)^2)/(a*Sqrt[a + b*x^2]) + (-1/4*(d* (4*b^2*c*e - 5*a*b*d*e - 3*a*b*c*f + 4*a^2*d*f)*x*Sqrt[a + b*x^2]*(c + d*x ^2))/b + (-1/2*(d*(8*b^3*c^2*e - 12*a^3*d^2*f - 2*a*b^2*c*(13*d*e + c*f) + a^2*b*d*(15*d*e + 17*c*f))*x*Sqrt[a + b*x^2])/b - (a*(12*a^3*d^3*f - 8*b^ 3*c^2*(3*d*e - c*f) + 4*a*b^2*c*d*(9*d*e + 2*c*f) - 5*a^2*b*d^2*(3*d*e + 5 *c*f))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(4*b))/a)/(b*e - a*f)^2 - (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*Sq rt[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*Sqrt[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))/(b*e - a*f)^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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[b*((b*e - a*f)/(b*c - a*d)^2) Int[(c + d*x^2)^( q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2 I nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt Q[r, 1]
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 = 1.07 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.25
| method | result | size |
| pseudoelliptic | \(\frac {\frac {a \,d^{2} \left (\sqrt {b \,x^{2}+a}\, d f x -\frac {\left (3 a d f -6 b c f +2 b d e \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\sqrt {b}}\right )}{2 f^{2} b^{2}}-\frac {a \left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )}{\left (a f -b e \right ) f^{2} \sqrt {\left (a f -b e \right ) e}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{2} \left (a f -b e \right ) \sqrt {b \,x^{2}+a}}}{a}\) | \(221\) |
| risch | \(\frac {d^{3} x \sqrt {b \,x^{2}+a}}{2 b^{2} f}-\frac {\frac {d^{2} \left (3 a d f -6 b c f +2 b d e \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{f \sqrt {b}}-\frac {b^{3} \left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, \left (\sqrt {-a b}\, f +b \sqrt {-e f}\right ) \left (\sqrt {-a b}\, f -b \sqrt {-e f}\right ) \sqrt {\frac {a f -b e}{f}}}+\frac {f^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, f +b \sqrt {-e f}\right ) \left (\sqrt {-a b}\, f -b \sqrt {-e f}\right ) a \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{3} \left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\sqrt {-e f}\, \left (\sqrt {-a b}\, f +b \sqrt {-e f}\right ) \left (\sqrt {-a b}\, f -b \sqrt {-e f}\right ) \sqrt {\frac {a f -b e}{f}}}+\frac {f^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, f +b \sqrt {-e f}\right ) \left (\sqrt {-a b}\, f -b \sqrt {-e f}\right ) a \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 f \,b^{2}}\) | \(800\) |
| default | \(\frac {d \left (\frac {d^{2} e^{2} x}{a \sqrt {b \,x^{2}+a}}+d f \left (3 c f -d e \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+d^{2} f^{2} \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+\frac {3 c^{2} f^{2} x}{a \sqrt {b \,x^{2}+a}}-\frac {3 c d e f x}{a \sqrt {b \,x^{2}+a}}\right )}{f^{3}}-\frac {\left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \left (\frac {f}{\left (a f -b e \right ) \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}+\frac {2 b \sqrt {-e f}\, \left (2 b \left (x +\frac {\sqrt {-e f}}{f}\right )-\frac {2 b \sqrt {-e f}}{f}\right )}{\left (a f -b e \right ) \left (\frac {4 b \left (a f -b e \right )}{f}+\frac {4 b^{2} e}{f}\right ) \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}-\frac {f \ln \left (\frac {\frac {2 a f -2 b e}{f}-\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x +\frac {\sqrt {-e f}}{f}\right )^{2} b -\frac {2 b \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{2 f^{3} \sqrt {-e f}}+\frac {\left (c^{3} f^{3}-3 c^{2} e \,f^{2} d +3 c \,d^{2} e^{2} f -e^{3} d^{3}\right ) \left (\frac {f}{\left (a f -b e \right ) \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}-\frac {2 b \sqrt {-e f}\, \left (2 b \left (x -\frac {\sqrt {-e f}}{f}\right )+\frac {2 b \sqrt {-e f}}{f}\right )}{\left (a f -b e \right ) \left (\frac {4 b \left (a f -b e \right )}{f}+\frac {4 b^{2} e}{f}\right ) \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}-\frac {f \ln \left (\frac {\frac {2 a f -2 b e}{f}+\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {\left (x -\frac {\sqrt {-e f}}{f}\right )^{2} b +\frac {2 b \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {a f -b e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\left (a f -b e \right ) \sqrt {\frac {a f -b e}{f}}}\right )}{2 f^{3} \sqrt {-e f}}\) | \(982\) |
Input:
int((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
(1/2*a*d^2*((b*x^2+a)^(1/2)*d*f*x-(3*a*d*f-6*b*c*f+2*b*d*e)/b^(1/2)*arctan h((b*x^2+a)^(1/2)/x/b^(1/2)))/f^2/b^2-1/(a*f-b*e)*a*(c^3*f^3-3*c^2*d*e*f^2 +3*c*d^2*e^2*f-d^3*e^3)/f^2/((a*f-b*e)*e)^(1/2)*arctan(e*(b*x^2+a)^(1/2)/x /((a*f-b*e)*e)^(1/2))+(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^2/(a *f-b*e)*x/(b*x^2+a)^(1/2))/a
Leaf count of result is larger than twice the leaf count of optimal. 696 vs. \(2 (153) = 306\).
Time = 9.83 (sec) , antiderivative size = 2877, normalized size of antiderivative = 16.25 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e),x, algorithm="fricas")
Output:
[1/4*((2*a^2*b^3*d^3*e^4 - (6*a^2*b^3*c*d^2 + a^3*b^2*d^3)*e^3*f + 4*(3*a^ 3*b^2*c*d^2 - a^4*b*d^3)*e^2*f^2 - 3*(2*a^4*b*c*d^2 - a^5*d^3)*e*f^3 + (2* a*b^4*d^3*e^4 - (6*a*b^4*c*d^2 + a^2*b^3*d^3)*e^3*f + 4*(3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*e^2*f^2 - 3*(2*a^3*b^2*c*d^2 - a^4*b*d^3)*e*f^3)*x^2)*sqrt(b )*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (a^2*b^3*d^3*e^3 - 3*a ^2*b^3*c*d^2*e^2*f + 3*a^2*b^3*c^2*d*e*f^2 - a^2*b^3*c^3*f^3 + (a*b^4*d^3* e^3 - 3*a*b^4*c*d^2*e^2*f + 3*a*b^4*c^2*d*e*f^2 - a*b^4*c^3*f^3)*x^2)*sqrt (b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2*f^2)*x^4 + a^2*e^2 + 2*( 4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 + a*e*x)*sqrt(b*e^2 - a* e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2)) + 2*((a*b^4*d^3*e^3*f - 2*a^2*b^3*d^3*e^2*f^2 + a^3*b^2*d^3*e*f^3)*x^3 + (a^2*b^3*d^3*e^3*f + 2*( b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*e^2*f^2 - (2*a* b^4*c^3 - 6*a^2*b^3*c^2*d + 6*a^3*b^2*c*d^2 - 3*a^4*b*d^3)*e*f^3)*x)*sqrt( b*x^2 + a))/(a^2*b^5*e^3*f^2 - 2*a^3*b^4*e^2*f^3 + a^4*b^3*e*f^4 + (a*b^6* e^3*f^2 - 2*a^2*b^5*e^2*f^3 + a^3*b^4*e*f^4)*x^2), 1/4*(2*(2*a^2*b^3*d^3*e ^4 - (6*a^2*b^3*c*d^2 + a^3*b^2*d^3)*e^3*f + 4*(3*a^3*b^2*c*d^2 - a^4*b*d^ 3)*e^2*f^2 - 3*(2*a^4*b*c*d^2 - a^5*d^3)*e*f^3 + (2*a*b^4*d^3*e^4 - (6*a*b ^4*c*d^2 + a^2*b^3*d^3)*e^3*f + 4*(3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*e^2*f^2 - 3*(2*a^3*b^2*c*d^2 - a^4*b*d^3)*e*f^3)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/s qrt(b*x^2 + a)) + (a^2*b^3*d^3*e^3 - 3*a^2*b^3*c*d^2*e^2*f + 3*a^2*b^3*...
\[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate((d*x**2+c)**3/(b*x**2+a)**(3/2)/(f*x**2+e),x)
Output:
Integral((c + d*x**2)**3/((a + b*x**2)**(3/2)*(e + f*x**2)), x)
Exception generated. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^(3/2)/(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. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x^2+c)^3/(b*x^2+a)^(3/2)/(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. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)),x)
Output:
int((c + d*x^2)^3/((a + b*x^2)^(3/2)*(e + f*x^2)), x)
Time = 0.23 (sec) , antiderivative size = 2263, normalized size of antiderivative = 12.79 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^3/(b*x^2+a)^(3/2)/(f*x^2+e),x)
Output:
( - 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x **2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3*c**3*f**3 + 24*sqrt (e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqr t(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3*c**2*d*e*f**2 - 24*sqrt(e)*sq rt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*s qrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3*c*d**2*e**2*f + 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)* x)/(sqrt(e)*sqrt(b)))*a**2*b**3*d**3*e**3 - 8*sqrt(e)*sqrt(a*f - b*e)*atan ((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e) *sqrt(b)))*a*b**4*c**3*f**3*x**2 + 24*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a *f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b) ))*a*b**4*c**2*d*e*f**2*x**2 - 24*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a *b**4*c*d**2*e**2*f*x**2 + 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a*b**4 *d**3*e**3*x**2 - 8*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f )*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3*c**3* f**3 + 24*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**3*c**2*d*e*f**2 - 24*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*...