Integrand size = 27, antiderivative size = 795 \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} f^5}-\frac {\left (a^2 f^4 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt {e^2-4 d f}+4 d e^3 f \sqrt {e^2-4 d f}-3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (a^2 f^4 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt {e^2-4 d f}-4 d e^3 f \sqrt {e^2-4 d f}+3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]
-1/12*(-3*f*x+4*e)*(c*x^2+a)^(3/2)/f^2+1/8*(3*a^2*f^4+12*a*c*f^2*(-d*f+e^2 )+8*c^2*(d^2*f^2-3*d*e^2*f+e^4))*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/f^5/c^ (1/2)-1/8*(8*e*(a*f^2+c*(-2*d*f+e^2))-f*(3*a*f^2+4*c*(-d*f+e^2))*x)*(c*x^2 +a)^(1/2)/f^4-1/2*arctanh(1/2*(2*a*f-c*x*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/( c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2))*(a^2*f^ 4*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2))+2*a*c*f^2*(e^4-4*d*e^2*f+2*d^2*f^2-e^3* (-4*d*f+e^2)^(1/2)+2*d*e*f*(-4*d*f+e^2)^(1/2))+c^2*(e^6-6*d*e^4*f+9*d^2*e^ 2*f^2-2*d^3*f^3-e^5*(-4*d*f+e^2)^(1/2)+4*d*e^3*f*(-4*d*f+e^2)^(1/2)-3*d^2* e*f^2*(-4*d*f+e^2)^(1/2)))/f^5*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2- 2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/2*arctanh(1/2*(2*a*f-c*x*(e+(-4*d*f+e ^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^( 1/2)))^(1/2))*(a^2*f^4*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2))+2*a*c*f^2*(e^4-4*d *e^2*f+2*d^2*f^2+e^3*(-4*d*f+e^2)^(1/2)-2*d*e*f*(-4*d*f+e^2)^(1/2))+c^2*(e ^6-6*d*e^4*f+9*d^2*e^2*f^2-2*d^3*f^3+e^5*(-4*d*f+e^2)^(1/2)-4*d*e^3*f*(-4* d*f+e^2)^(1/2)+3*d^2*e*f^2*(-4*d*f+e^2)^(1/2)))/f^5*2^(1/2)/(-4*d*f+e^2)^( 1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.82 (sec) , antiderivative size = 1239, normalized size of antiderivative = 1.56 \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\frac {f \sqrt {a+c x^2} \left (a f^2 (-32 e+15 f x)-2 c \left (12 e^3-6 e^2 f x+4 e f \left (-6 d+f x^2\right )-3 f^2 x \left (-2 d+f x^2\right )\right )\right )+\frac {6 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+c x^2}}\right )}{\sqrt {c}}+24 \text {RootSum}\left [c^2 d+2 \sqrt {a} c e \text {$\#$1}-2 c d \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-c^3 d e^4 \log (x)+3 c^3 d^2 e^2 f \log (x)-c^3 d^3 f^2 \log (x)-2 a c^2 d e^2 f^2 \log (x)+2 a c^2 d^2 f^3 \log (x)-a^2 c d f^4 \log (x)+c^3 d e^4 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-3 c^3 d^2 e^2 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+c^3 d^3 f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+2 a c^2 d e^2 f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-2 a c^2 d^2 f^3 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )+a^2 c d f^4 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right )-2 \sqrt {a} c^2 e^5 \log (x) \text {$\#$1}+8 \sqrt {a} c^2 d e^3 f \log (x) \text {$\#$1}-6 \sqrt {a} c^2 d^2 e f^2 \log (x) \text {$\#$1}-4 a^{3/2} c e^3 f^2 \log (x) \text {$\#$1}+8 a^{3/2} c d e f^3 \log (x) \text {$\#$1}-2 a^{5/2} e f^4 \log (x) \text {$\#$1}+2 \sqrt {a} c^2 e^5 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-8 \sqrt {a} c^2 d e^3 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+6 \sqrt {a} c^2 d^2 e f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+4 a^{3/2} c e^3 f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-8 a^{3/2} c d e f^3 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 a^{5/2} e f^4 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c^2 d e^4 \log (x) \text {$\#$1}^2-3 c^2 d^2 e^2 f \log (x) \text {$\#$1}^2+c^2 d^3 f^2 \log (x) \text {$\#$1}^2+2 a c d e^2 f^2 \log (x) \text {$\#$1}^2-2 a c d^2 f^3 \log (x) \text {$\#$1}^2+a^2 d f^4 \log (x) \text {$\#$1}^2-c^2 d e^4 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+3 c^2 d^2 e^2 f \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-c^2 d^3 f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a c d e^2 f^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+2 a c d^2 f^3 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a^2 d f^4 \log \left (-\sqrt {a}+\sqrt {a+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 c d \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{24 f^5} \]
(f*Sqrt[a + c*x^2]*(a*f^2*(-32*e + 15*f*x) - 2*c*(12*e^3 - 6*e^2*f*x + 4*e *f*(-6*d + f*x^2) - 3*f^2*x*(-2*d + f*x^2))) + (6*(3*a^2*f^4 + 12*a*c*f^2* (e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTanh[(Sqrt[c]*x)/(-Sqr t[a] + Sqrt[a + c*x^2])])/Sqrt[c] + 24*RootSum[c^2*d + 2*Sqrt[a]*c*e*#1 - 2*c*d*#1^2 + 4*a*f*#1^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (-(c^3*d*e^4*Log[x ]) + 3*c^3*d^2*e^2*f*Log[x] - c^3*d^3*f^2*Log[x] - 2*a*c^2*d*e^2*f^2*Log[x ] + 2*a*c^2*d^2*f^3*Log[x] - a^2*c*d*f^4*Log[x] + c^3*d*e^4*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] - 3*c^3*d^2*e^2*f*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] + c^3*d^3*f^2*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] + 2*a*c^2*d*e ^2*f^2*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] - 2*a*c^2*d^2*f^3*Log[-Sqrt[ a] + Sqrt[a + c*x^2] - x*#1] + a^2*c*d*f^4*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1] - 2*Sqrt[a]*c^2*e^5*Log[x]*#1 + 8*Sqrt[a]*c^2*d*e^3*f*Log[x]*#1 - 6*Sqrt[a]*c^2*d^2*e*f^2*Log[x]*#1 - 4*a^(3/2)*c*e^3*f^2*Log[x]*#1 + 8*a^(3 /2)*c*d*e*f^3*Log[x]*#1 - 2*a^(5/2)*e*f^4*Log[x]*#1 + 2*Sqrt[a]*c^2*e^5*Lo g[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1]*#1 - 8*Sqrt[a]*c^2*d*e^3*f*Log[-Sqrt[ a] + Sqrt[a + c*x^2] - x*#1]*#1 + 6*Sqrt[a]*c^2*d^2*e*f^2*Log[-Sqrt[a] + S qrt[a + c*x^2] - x*#1]*#1 + 4*a^(3/2)*c*e^3*f^2*Log[-Sqrt[a] + Sqrt[a + c* x^2] - x*#1]*#1 - 8*a^(3/2)*c*d*e*f^3*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*# 1]*#1 + 2*a^(5/2)*e*f^4*Log[-Sqrt[a] + Sqrt[a + c*x^2] - x*#1]*#1 + c^2*d* e^4*Log[x]*#1^2 - 3*c^2*d^2*e^2*f*Log[x]*#1^2 + c^2*d^3*f^2*Log[x]*#1^2...
Time = 1.56 (sec) , antiderivative size = 656, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2139, 27, 2139, 25, 2145, 27, 224, 219, 1367, 488, 219}
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.
| \(\displaystyle \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx\) |
| \(\Big \downarrow \) 2139 |
| \(\displaystyle -\frac {\int \frac {3 \sqrt {c x^2+a} \left (-c \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x^2-c e (4 c d-a f) x+a c d f\right )}{f x^2+e x+d}dx}{12 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {c x^2+a} \left (-c \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x^2-c e (4 c d-a f) x+a c d f\right )}{f x^2+e x+d}dx}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 2139 |
| \(\displaystyle -\frac {\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}-\frac {\int -\frac {-\left (\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) x^2 c^2\right )+a d f \left (5 a f^2+4 c \left (e^2-d f\right )\right ) c^2+e \left (5 a^2 f^3+4 a c \left (e^2-5 d f\right ) f-8 c^2 d \left (e^2-2 d f\right )\right ) x c^2}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{2 c f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {-\left (\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) x^2 c^2\right )+a d f \left (5 a f^2+4 c \left (e^2-d f\right )\right ) c^2+e \left (5 a^2 f^3+4 a c \left (e^2-5 d f\right ) f-8 c^2 d \left (e^2-2 d f\right )\right ) x c^2}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 2145 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {8 c^2 \left (d \left (a^2 f^4+2 a c \left (e^2-d f\right ) f^2+c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right )+e \left (c e^2+a f^2-3 c d f\right ) \left (c e^2+a f^2-c d f\right ) x\right )}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {c^2 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {8 c^2 \int \frac {d \left (a^2 f^4+2 a c \left (e^2-d f\right ) f^2+c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right )+e \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right ) x}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {c^2 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {\frac {\frac {8 c^2 \int \frac {d \left (a^2 f^4+2 a c \left (e^2-d f\right ) f^2+c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right )+e \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right ) x}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {c^2 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {8 c^2 \int \frac {d \left (a^2 f^4+2 a c \left (e^2-d f\right ) f^2+c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right )+e \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right ) x}{\sqrt {c x^2+a} \left (f x^2+e x+d\right )}dx}{f}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 1367 |
| \(\displaystyle -\frac {\frac {\frac {8 c^2 \left (\frac {\left (e \left (\sqrt {e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+a}}dx}{\sqrt {e^2-4 d f}}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+a}}dx}{\sqrt {e^2-4 d f}}\right )}{f}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle -\frac {\frac {\frac {8 c^2 \left (\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-\frac {\left (2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x\right )^2}{c x^2+a}}d\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {c x^2+a}}}{\sqrt {e^2-4 d f}}-\frac {\left (e \left (\sqrt {e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-\frac {\left (2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x\right )^2}{c x^2+a}}d\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {c x^2+a}}}{\sqrt {e^2-4 d f}}\right )}{f}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {8 c^2 \left (\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (e \left (\sqrt {e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{f}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\) |
-1/12*((4*e - 3*f*x)*(a + c*x^2)^(3/2))/f^2 - ((c*(8*e*(a*f^2 + c*(e^2 - 2 *d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x^2])/(2*f^2) + (-((c ^(3/2)*(3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2* f^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/f) + (8*c^2*(((e*(e - Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 3*d*f))*(a*f^2 + c*(e^2 - d*f)) - 2*d*f*(a^2* f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(2 *a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqr t[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((e*(e + Sqrt[e^2 - 4*d*f])*(a*f^2 + c*(e^2 - 3*d*f))*(a*f^2 + c*(e^2 - d*f)) - 2*d*f*(a^2*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*ArcTanh[(2*a* f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[2 *a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])))/f)/(2*c*f^2))/(4*c*f^2)
3.1.58.3.1 Defintions of rubi rules used
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)^(-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[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/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f _.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*c*g - h*( b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{ a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]
Int[(Px_)*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*((-c)*e*(2*p + q + 2)) + 2*c*C*f *(p + q + 1)*x)*(a + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) I nt[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*((-a)*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)* (C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2 *A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, c, d, e, f, q}, x] & & PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0 ] && !IGtQ[p, 0] && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + f*x^2], x], x] + Simp[1/c Int[(A*c - a* C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a , b, c, d, f}, x] && PolyQ[Px, x, 2]
Time = 0.66 (sec) , antiderivative size = 1183, normalized size of antiderivative = 1.49
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1183\) |
| default | \(\text {Expression too large to display}\) | \(2367\) |
-1/24*(-6*c*f^3*x^3+8*c*e*f^2*x^2-15*a*f^3*x+12*c*d*f^2*x-12*c*e^2*f*x+32* a*e*f^2-48*c*d*e*f+24*c*e^3)*(c*x^2+a)^(1/2)/f^4+1/8/f^4*(1/f*(3*a^2*f^4-1 2*a*c*d*f^3+12*a*c*e^2*f^2+8*c^2*d^2*f^2-24*c^2*d*e^2*f+8*c^2*e^4)*ln(x*c^ (1/2)+(c*x^2+a)^(1/2))/c^(1/2)-1/2*(-8*e*f^4*a^2*(-4*d*f+e^2)^(1/2)+32*a*c *d*e*f^3*(-4*d*f+e^2)^(1/2)-16*a*c*e^3*f^2*(-4*d*f+e^2)^(1/2)-24*c^2*d^2*e *f^2*(-4*d*f+e^2)^(1/2)+32*c^2*d*e^3*f*(-4*d*f+e^2)^(1/2)-8*c^2*e^5*(-4*d* f+e^2)^(1/2)+16*a^2*d*f^5-8*a^2*e^2*f^4-32*a*c*d^2*f^4+64*a*c*d*e^2*f^3-16 *a*c*e^4*f^2+16*c^2*d^3*f^3-72*c^2*d^2*e^2*f^2+48*c^2*d*e^4*f-8*c^2*e^6)/f ^2/(-4*d*f+e^2)^(1/2)*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e ^2)/f^2)^(1/2)*ln((((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e +(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4* d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+ e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/ 2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2 *(e+(-4*d*f+e^2)^(1/2))/f))-1/2*(-8*e*f^4*a^2*(-4*d*f+e^2)^(1/2)+32*a*c*d* e*f^3*(-4*d*f+e^2)^(1/2)-16*a*c*e^3*f^2*(-4*d*f+e^2)^(1/2)-24*c^2*d^2*e*f^ 2*(-4*d*f+e^2)^(1/2)+32*c^2*d*e^3*f*(-4*d*f+e^2)^(1/2)-8*c^2*e^5*(-4*d*f+e ^2)^(1/2)-16*a^2*d*f^5+8*a^2*e^2*f^4+32*a*c*d^2*f^4-64*a*c*d*e^2*f^3+16*a* c*e^4*f^2-16*c^2*d^3*f^3+72*c^2*d^2*e^2*f^2-48*c^2*d*e^4*f+8*c^2*e^6)/f^2/ (-4*d*f+e^2)^(1/2)*2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*...
Timed out. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` for more deta
Exception generated. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\int \frac {x^2\,{\left (c\,x^2+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \]