\(\int \frac {x^2 (a+c x^2)^{3/2}}{d+e x+f x^2} \, dx\) [25]

Optimal result

Integrand size = 27, antiderivative size = 657 \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=-\frac {e \left (a f^2+c \left (e^2-2 d f\right )\right ) \sqrt {a+c x^2}}{f^4}+\frac {\left (5 a f^2+4 c \left (e^2-d f\right )\right ) x \sqrt {a+c x^2}}{8 f^3}+\frac {c x^3 \sqrt {a+c x^2}}{4 f}-\frac {e \left (a+c x^2\right )^{3/2}}{3 f^2}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} f^5}-\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (e \left (e+\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \] Output:

-e*(a*f^2+c*(-2*d*f+e^2))*(c*x^2+a)^(1/2)/f^4+1/8*(5*a*f^2+4*c*(-d*f+e^2)) 
*x*(c*x^2+a)^(1/2)/f^3+1/4*c*x^3*(c*x^2+a)^(1/2)/f-1/3*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))*arc 
tanh(c^(1/2)*x/(c*x^2+a)^(1/2))/c^(1/2)/f^5-1/2*(e*(e-(-4*d*f+e^2)^(1/2))* 
(a*f^2+c*(-3*d*f+e^2))*(a*f^2+c*(-d*f+e^2))-2*d*f*(a^2*f^4+2*a*c*f^2*(-d*f 
+e^2)+c^2*(d^2*f^2-3*d*e^2*f+e^4)))*arctanh(1/2*(2*a*f-c*(e-(-4*d*f+e^2)^( 
1/2))*x)*2^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c*x^2 
+a)^(1/2))*2^(1/2)/f^5/(-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*(e*(e+(-4*d*f+e^2)^(1/2))*(a*f^2+c*(-3*d*f+e^2))*(a 
*f^2+c*(-d*f+e^2))-2*d*f*(a^2*f^4+2*a*c*f^2*(-d*f+e^2)+c^2*(d^2*f^2-3*d*e^ 
2*f+e^4)))*arctanh(1/2*(2*a*f-c*(e+(-4*d*f+e^2)^(1/2))*x)*2^(1/2)/(2*a*f^2 
+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c*x^2+a)^(1/2))*2^(1/2)/f^5/(- 
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)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 2.89 (sec) , antiderivative size = 1239, normalized size of antiderivative = 1.89 \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx =\text {Too large to display} \] Input:

Integrate[(x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x]
 

Output:

(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...
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.00, 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

\(\displaystyle -\frac {\frac {\frac {8 c^2 \left (\frac {\left (e \left (e-\sqrt {e^2-4 d f}\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (e \left (\sqrt {e^2-4 d f}+e\right ) \left (a f^2+c \left (e^2-3 d f\right )\right ) \left (a f^2+c \left (e^2-d f\right )\right )-2 d f \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{f}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{f}}{2 c f^2}+\frac {c \sqrt {a+c x^2} \left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f x \left (3 a f^2+4 c \left (e^2-d f\right )\right )\right )}{2 f^2}}{4 c f^2}-\frac {\left (a+c x^2\right )^{3/2} (4 e-3 f x)}{12 f^2}\)

Input:

Int[(x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x]
 

Output:

-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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 224

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]
 

rule 488

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]
 

rule 1367

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]
 

rule 2139

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]
 

rule 2145

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]
 

Maple [A] (verified)

Time = 2.26 (sec) , antiderivative size = 1177, normalized size of antiderivative = 1.79

method	result	size
risch	\(\text {Expression too large to display}\)	\(1177\)
default	\(\text {Expression too large to display}\)	\(2365\)

Input:

int(x^2*(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
 

Output:

-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(c^(1 
/2)*x+(c*x^2+a)^(1/2))/c^(1/2)+4/f^2*(a^2*e*f^4*(-4*d*f+e^2)^(1/2)-4*a*c*d 
*e*f^3*(-4*d*f+e^2)^(1/2)+2*a*c*e^3*f^2*(-4*d*f+e^2)^(1/2)+3*c^2*d^2*e*f^2 
*(-4*d*f+e^2)^(1/2)-4*c^2*d*e^3*f*(-4*d*f+e^2)^(1/2)+e^5*c^2*(-4*d*f+e^2)^ 
(1/2)+2*a^2*d*f^5-a^2*e^2*f^4-4*a*c*d^2*f^4+8*a*c*d*e^2*f^3-2*a*c*e^4*f^2+ 
2*c^2*d^3*f^3-9*c^2*d^2*e^2*f^2+6*c^2*d*e^4*f-c^2*e^6)/(-4*d*f+e^2)^(1/2)* 
2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*ln(((- 
(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2)) 
/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e 
+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^ 
2-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d 
*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*f+ 
e^2)^(1/2))))+4/f^2*(a^2*e*f^4*(-4*d*f+e^2)^(1/2)-4*a*c*d*e*f^3*(-4*d*f+e^ 
2)^(1/2)+2*a*c*e^3*f^2*(-4*d*f+e^2)^(1/2)+3*c^2*d^2*e*f^2*(-4*d*f+e^2)^(1/ 
2)-4*c^2*d*e^3*f*(-4*d*f+e^2)^(1/2)+e^5*c^2*(-4*d*f+e^2)^(1/2)-2*a^2*d*f^5 
+a^2*e^2*f^4+4*a*c*d^2*f^4-8*a*c*d*e^2*f^3+2*a*c*e^4*f^2-2*c^2*d^3*f^3+9*c 
^2*d^2*e^2*f^2-6*c^2*d*e^4*f+c^2*e^6)/(-4*d*f+e^2)^(1/2)*2^(1/2)/(((-4*d*f 
+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*ln((((-4*d*f+e^2)^(1/...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Timed out} \] Input:

integrate(x^2*(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\int \frac {x^{2} \left (a + c x^{2}\right )^{\frac {3}{2}}}{d + e x + f x^{2}}\, dx \] Input:

integrate(x**2*(c*x**2+a)**(3/2)/(f*x**2+e*x+d),x)
 

Output:

Integral(x**2*(a + c*x**2)**(3/2)/(d + e*x + f*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(c*x^2+a)^(3/2)/(f*x^2+e*x+d),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(4*d*f-e^2>0)', see `assume?` for 
 more deta
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^2*(c*x^2+a)^(3/2)/(f*x^2+e*x+d),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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\int \frac {x^2\,{\left (c\,x^2+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \] Input:

int((x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x)
                                                                                    
                                                                                    
 

Output:

int((x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {too large to display} \] Input:

int(x^2*(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x)
 

Output:

( - 48*sqrt(a + c*x**2)*a**2*d*e*f**5 + 48*sqrt(a + c*x**2)*a**2*e**3*f**4 
 + 128*sqrt(a + c*x**2)*a*c*d**2*e*f**4 + 30*sqrt(a + c*x**2)*a*c*d**2*f** 
5*x - 288*sqrt(a + c*x**2)*a*c*d*e**3*f**3 + 96*sqrt(a + c*x**2)*a*c*e**5* 
f**2 - 48*sqrt(a + c*x**2)*c**2*d**3*e*f**3 - 24*sqrt(a + c*x**2)*c**2*d** 
3*f**4*x + 288*sqrt(a + c*x**2)*c**2*d**2*e**3*f**2 + 24*sqrt(a + c*x**2)* 
c**2*d**2*e**2*f**3*x - 16*sqrt(a + c*x**2)*c**2*d**2*e*f**4*x**2 + 12*sqr 
t(a + c*x**2)*c**2*d**2*f**5*x**3 - 240*sqrt(a + c*x**2)*c**2*d*e**5*f + 4 
8*sqrt(a + c*x**2)*c**2*e**7 - 9*sqrt(c)*log(sqrt(a + c*x**2) - sqrt(c)*x) 
*a**2*d**2*f**5 + 36*sqrt(c)*log(sqrt(a + c*x**2) - sqrt(c)*x)*a*c*d**3*f* 
*4 - 36*sqrt(c)*log(sqrt(a + c*x**2) - sqrt(c)*x)*a*c*d**2*e**2*f**3 - 24* 
sqrt(c)*log(sqrt(a + c*x**2) - sqrt(c)*x)*c**2*d**4*f**3 + 72*sqrt(c)*log( 
sqrt(a + c*x**2) - sqrt(c)*x)*c**2*d**3*e**2*f**2 - 24*sqrt(c)*log(sqrt(a 
+ c*x**2) - sqrt(c)*x)*c**2*d**2*e**4*f + 9*sqrt(c)*log(sqrt(a + c*x**2) + 
 sqrt(c)*x)*a**2*d**2*f**5 - 36*sqrt(c)*log(sqrt(a + c*x**2) + sqrt(c)*x)* 
a*c*d**3*f**4 + 36*sqrt(c)*log(sqrt(a + c*x**2) + sqrt(c)*x)*a*c*d**2*e**2 
*f**3 + 24*sqrt(c)*log(sqrt(a + c*x**2) + sqrt(c)*x)*c**2*d**4*f**3 - 72*s 
qrt(c)*log(sqrt(a + c*x**2) + sqrt(c)*x)*c**2*d**3*e**2*f**2 + 24*sqrt(c)* 
log(sqrt(a + c*x**2) + sqrt(c)*x)*c**2*d**2*e**4*f - 48*int(sqrt(a + c*x** 
2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*a**2*c*d** 
3*f**5 + 96*int(sqrt(a + c*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c...