\(\int \frac {x^2 (a+b x+c x^2)^{3/2}}{d+e x+f x^2} \, dx\) [135]

Optimal result

Integrand size = 30, antiderivative size = 1057 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx =\text {Too large to display} \] Output:

-1/192*(9*b^3*f^3+12*b*c*f^2*(-5*a*f+2*b*e)+192*c^3*(-2*d*e*f+e^3)+16*c^2* 
f*(16*a*e*f-15*b*(-d*f+e^2)))*(c*x^2+b*x+a)^(1/2)/c^2/f^4+1/96*(3*b^2*f^2+ 
48*c^2*(-d*f+e^2)-c*(-60*a*f^2+56*b*e*f))*x*(c*x^2+b*x+a)^(1/2)/c/f^3-1/24 
*(-9*b*f+8*c*e)*x^2*(c*x^2+b*x+a)^(1/2)/f^2+1/4*c*x^3*(c*x^2+b*x+a)^(1/2)/ 
f+1/128*(3*b^4*f^4+8*b^2*c*f^3*(-3*a*f+b*e)+128*c^4*(d^2*f^2-3*d*e^2*f+e^4 
)-48*c^2*f^2*(2*a*b*e*f-a^2*f^2-b^2*(-d*f+e^2))+192*c^3*f*(a*f*(-d*f+e^2)- 
b*(-2*d*e*f+e^3)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5 
/2)/f^5-1/2*((e-(-4*d*f+e^2)^(1/2))*(f*(-a*f+b*e)-c*(-d*f+e^2))*(f*(-a*e*f 
-2*b*d*f+b*e^2)-c*(-3*d*e*f+e^3))-2*d*f*(c^2*(d^2*f^2-3*d*e^2*f+e^4)-f^2*( 
2*a*b*e*f-a^2*f^2-b^2*(-d*f+e^2))+2*c*f*(a*f*(-d*f+e^2)-b*(-2*d*e*f+e^3))) 
)*arctanh(1/4*(4*a*f-b*(e-(-4*d*f+e^2)^(1/2))+2*(b*f-c*(e-(-4*d*f+e^2)^(1/ 
2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2) 
)^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/f^5/(-4*d*f+e^2)^(1/2)/(c*e^2-2*c*d*f 
-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/2*((e+(-4*d*f+e^2)^( 
1/2))*(f*(-a*f+b*e)-c*(-d*f+e^2))*(f*(-a*e*f-2*b*d*f+b*e^2)-c*(-3*d*e*f+e^ 
3))-2*d*f*(c^2*(d^2*f^2-3*d*e^2*f+e^4)-f^2*(2*a*b*e*f-a^2*f^2-b^2*(-d*f+e^ 
2))+2*c*f*(a*f*(-d*f+e^2)-b*(-2*d*e*f+e^3))))*arctanh(1/4*(4*a*f-b*(e+(-4* 
d*f+e^2)^(1/2))+2*(b*f-c*(e+(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f 
-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2 
^(1/2)/f^5/(-4*d*f+e^2)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3262\) vs. \(2(1057)=2114\).

Time = 17.58 (sec) , antiderivative size = 3262, normalized size of antiderivative = 3.09 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/6*(e*(1 - e/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2))/f^2 - (e*(1 + e 
/Sqrt[e^2 - 4*d*f])*(a + x*(b + c*x))^(3/2))/(6*f^2) + ((b + 2*c*x)*(a + x 
*(b + c*x))^(3/2))/(8*c*f) - (3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2)*((2* 
(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/c - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/ 
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(3/2)))/(128*c*f*(a + b*x + c*x^2)^( 
3/2)) + (d*(a + x*(b + c*x))^(3/2)*(((4*c*f*(-4*a*f + b*(e - Sqrt[e^2 - 4* 
d*f])) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(-(b*f) + 2*c*(e - Sqrt[e^2 - 
 4*d*f])) - 4*c*f*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*Sqrt[a + b*x + c*x^ 
2])/(8*c*f^2) - ((-2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e 
^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f*(3*a*f - b*(e - Sqrt[e^2 - 4*d*f 
])))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) - 
 (2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] 
 + b*f*Sqrt[e^2 - 4*d*f]]*(4*(e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^ 
2 - 4*d*f]))*(b^2*f^2 - 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) - 4*c*f* 
(3*a*f - b*(e - Sqrt[e^2 - 4*d*f]))) + 4*f*(2*c*f*(4*a*f - b*(e - Sqrt[e^2 
 - 4*d*f]))^2 - (e - Sqrt[e^2 - 4*d*f])*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))* 
(b^2*f + 4*a*c*f - 2*b*c*(e - Sqrt[e^2 - 4*d*f]))))*ArcTanh[(4*a*f - b*(e 
- Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2 
]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqr 
t[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 - 8*b*f*(e - Sqrt...
 

Rubi [A] (warning: unable to verify)

Time = 2.77 (sec) , antiderivative size = 1037, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2138, 27, 2138, 27, 2143, 27, 1092, 219, 1365, 1154, 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.

\(\displaystyle \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx\)

\(\Big \downarrow \) 2138

\(\displaystyle -\frac {\int -\frac {3 \sqrt {c x^2+b x+a} \left (-\left (\left (-16 \left (e^2-d f\right ) c^2+4 f (2 b e-3 a f) c+3 b^2 f^2\right ) x^2\right )+\left (-3 e f b^2+16 c^2 d e+4 c \left (2 b e^2-a f e-4 b d f\right )\right ) x+d \left (-3 f b^2+8 c e b-4 a c f\right )\right )}{4 \left (f x^2+e x+d\right )}dx}{12 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (-\left (\left (-16 \left (e^2-d f\right ) c^2+4 f (2 b e-3 a f) c+3 b^2 f^2\right ) x^2\right )+\left (-3 e f b^2+16 c^2 d e+4 c \left (2 b e^2-a f e-4 b d f\right )\right ) x+d \left (-3 f b^2+8 c e b-4 a c f\right )\right )}{f x^2+e x+d}dx}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 2138

\(\displaystyle \frac {-\frac {\int -\frac {\left (128 \left (e^4-3 d f e^2+d^2 f^2\right ) c^4+192 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c^3-48 f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right ) c^2+8 b^2 f^3 (b e-3 a f) c+3 b^4 f^4\right ) x^2+\left (3 e f^3 b^4+8 c e f^2 (b e-3 a f) b^2+128 c^4 d e \left (e^2-2 d f\right )-16 c^2 f \left (\left (5 e^3-13 d e f\right ) b^2-2 a f \left (5 e^2-8 d f\right ) b+5 a^2 e f^2\right )-64 c^3 \left (a e f \left (e^2-5 d f\right )-b \left (e^4-6 d f e^2+4 d^2 f^2\right )\right )\right ) x+d \left (3 f^3 b^4+8 c e f^2 b^3-8 c f \left (3 a f^2+10 c \left (e^2-d f\right )\right ) b^2+32 c^2 e \left (5 a f^2+2 c \left (e^2-2 d f\right )\right ) b-16 a c^2 f \left (5 a f^2+4 c \left (e^2-d f\right )\right )\right )}{4 \sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{2 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (128 \left (e^4-3 d f e^2+d^2 f^2\right ) c^4+192 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c^3-48 f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right ) c^2+8 b^2 f^3 (b e-3 a f) c+3 b^4 f^4\right ) x^2+\left (3 e f^3 b^4+8 c e f^2 (b e-3 a f) b^2+128 c^4 d e \left (e^2-2 d f\right )-16 c^2 f \left (\left (5 e^3-13 d e f\right ) b^2-2 a f \left (5 e^2-8 d f\right ) b+5 a^2 e f^2\right )-64 c^3 \left (a e f \left (e^2-5 d f\right )-b \left (e^4-6 d f e^2+4 d^2 f^2\right )\right )\right ) x+d \left (3 f^3 b^4+8 c e f^2 b^3-8 c f \left (3 a f^2+10 c \left (e^2-d f\right )\right ) b^2+32 c^2 e \left (5 a f^2+2 c \left (e^2-2 d f\right )\right ) b-16 a c^2 f \left (5 a f^2+4 c \left (e^2-d f\right )\right )\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 2143

\(\displaystyle \frac {\frac {\frac {\int -\frac {128 c^2 \left (d \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )+\left (c e^2-b f e+a f^2-c d f\right ) \left (c e^3-b f e^2+a f^2 e-3 c d f e+2 b d f^2\right ) x\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}+\frac {\left (-48 c^2 f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+8 b^2 c f^3 (b e-3 a f)+192 c^3 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+3 b^4 f^4+128 c^4 \left (d^2 f^2-3 d e^2 f+e^4\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\left (-48 c^2 f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+8 b^2 c f^3 (b e-3 a f)+192 c^3 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+3 b^4 f^4+128 c^4 \left (d^2 f^2-3 d e^2 f+e^4\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}-\frac {128 c^2 \int \frac {d \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )+\left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {2 \left (-48 c^2 f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+8 b^2 c f^3 (b e-3 a f)+192 c^3 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+3 b^4 f^4+128 c^4 \left (d^2 f^2-3 d e^2 f+e^4\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{f}-\frac {128 c^2 \int \frac {d \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )+\left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-48 c^2 f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+8 b^2 c f^3 (b e-3 a f)+192 c^3 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+3 b^4 f^4+128 c^4 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{\sqrt {c} f}-\frac {128 c^2 \int \frac {d \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )+\left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-48 c^2 f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+8 b^2 c f^3 (b e-3 a f)+192 c^3 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+3 b^4 f^4+128 c^4 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )}{\sqrt {c} f}-\frac {128 c^2 \left (\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )-2 d f \left (-f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (-a e f-2 b d f+b e^2\right )-c \left (e^3-3 d e f\right )\right )-2 d f \left (-f^2 \left (-a^2 f^2+2 a b e f-\left (b^2 \left (e^2-d f\right )\right )\right )+2 c f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}\right )}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c f x \left (4 c f (2 b e-3 a f)+3 b^2 f^2-16 c^2 \left (e^2-d f\right )\right )-16 c^2 f \left (-4 a e f-5 b d f+5 b e^2\right )+4 b c f^2 (2 b e-3 a f)+3 b^3 f^3+64 c^3 \left (e^3-2 d e f\right )\right )}{4 c f^2}}{16 c f^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b f+8 c e-6 c f x)}{24 c f^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {\left (128 \left (e^4-3 d f e^2+d^2 f^2\right ) c^4+192 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c^3-48 f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right ) c^2+8 b^2 f^3 (b e-3 a f) c+3 b^4 f^4\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} f}-\frac {128 c^2 \left (\frac {2 \left (\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right )-2 d f \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {2 \left (\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right )-2 d f \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}\right )}{f}}{8 c f^2}-\frac {\left (64 \left (e^3-2 d e f\right ) c^3-16 f \left (5 b e^2-4 a f e-5 b d f\right ) c^2+4 b f^2 (2 b e-3 a f) c+2 f \left (-16 \left (e^2-d f\right ) c^2+4 f (2 b e-3 a f) c+3 b^2 f^2\right ) x c+3 b^3 f^3\right ) \sqrt {c x^2+b x+a}}{4 c f^2}}{16 c f^2}-\frac {(8 c e-3 b f-6 c f x) \left (c x^2+b x+a\right )^{3/2}}{24 c f^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (128 \left (e^4-3 d f e^2+d^2 f^2\right ) c^4+192 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c^3-48 f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right ) c^2+8 b^2 f^3 (b e-3 a f) c+3 b^4 f^4\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{\sqrt {c} f}-\frac {128 c^2 \left (\frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right )-2 d f \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right ) \left (f \left (b e^2-a f e-2 b d f\right )-c \left (e^3-3 d e f\right )\right )-2 d f \left (\left (e^4-3 d f e^2+d^2 f^2\right ) c^2+2 f \left (a f \left (e^2-d f\right )-b \left (e^3-2 d e f\right )\right ) c-f^2 \left (-\left (\left (e^2-d f\right ) b^2\right )+2 a e f b-a^2 f^2\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}}}\right )}{f}}{8 c f^2}-\frac {\left (64 \left (e^3-2 d e f\right ) c^3-16 f \left (5 b e^2-4 a f e-5 b d f\right ) c^2+4 b f^2 (2 b e-3 a f) c+2 f \left (-16 \left (e^2-d f\right ) c^2+4 f (2 b e-3 a f) c+3 b^2 f^2\right ) x c+3 b^3 f^3\right ) \sqrt {c x^2+b x+a}}{4 c f^2}}{16 c f^2}-\frac {(8 c e-3 b f-6 c f x) \left (c x^2+b x+a\right )^{3/2}}{24 c f^2}\)

Input:

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

Output:

-1/24*((8*c*e - 3*b*f - 6*c*f*x)*(a + b*x + c*x^2)^(3/2))/(c*f^2) + (-1/4* 
((3*b^3*f^3 + 4*b*c*f^2*(2*b*e - 3*a*f) - 16*c^2*f*(5*b*e^2 - 5*b*d*f - 4* 
a*e*f) + 64*c^3*(e^3 - 2*d*e*f) + 2*c*f*(3*b^2*f^2 + 4*c*f*(2*b*e - 3*a*f) 
 - 16*c^2*(e^2 - d*f))*x)*Sqrt[a + b*x + c*x^2])/(c*f^2) + (((3*b^4*f^4 + 
8*b^2*c*f^3*(b*e - 3*a*f) + 128*c^4*(e^4 - 3*d*e^2*f + d^2*f^2) - 48*c^2*f 
^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 192*c^3*f*(a*f*(e^2 - d*f) - 
b*(e^3 - 2*d*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])] 
)/(Sqrt[c]*f) - (128*c^2*((((e - Sqrt[e^2 - 4*d*f])*(f*(b*e - a*f) - c*(e^ 
2 - d*f))*(f*(b*e^2 - 2*b*d*f - a*e*f) - c*(e^3 - 3*d*e*f)) - 2*d*f*(c^2*( 
e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 
 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f))))*ArcTanh[(4*a*f - b*(e - Sqr 
t[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c 
*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + 
 b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 
 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - (((e + Sqrt[e^2 - 4*d*f])*(f* 
(b*e - a*f) - c*(e^2 - d*f))*(f*(b*e^2 - 2*b*d*f - a*e*f) - c*(e^3 - 3*d*e 
*f)) - 2*d*f*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - 
 b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f))))*ArcTanh[ 
(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x 
)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[...
 

Defintions of rubi rules used

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

rule 2138

Int[(Px_)*((a_) + (b_.)*(x_) + (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[Px, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + 
 q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + 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))   Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si 
mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + 
(p + q + 1)*(b^2*C*d*f*p + 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 - b*f)*(q + 1) - c*(C*e - 
 B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(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 + (p*( 
c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q 
 + 1)*(C*f^2*p*(b^2 - 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, b, 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 2143

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (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 + e*x + 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 + e*x + f*x 
^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2001\) vs. \(2(984)=1968\).

Time = 2.81 (sec) , antiderivative size = 2002, normalized size of antiderivative = 1.89

method	result	size
risch	\(\text {Expression too large to display}\)	\(2002\)
default	\(\text {Expression too large to display}\)	\(3016\)

Input:

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

Output:

1/192/c^2*(48*c^3*f^3*x^3+72*b*c^2*f^3*x^2-64*c^3*e*f^2*x^2+120*a*c^2*f^3* 
x+6*b^2*c*f^3*x-112*b*c^2*e*f^2*x-96*c^3*d*f^2*x+96*c^3*e^2*f*x+60*a*b*c*f 
^3-256*a*c^2*e*f^2-9*b^3*f^3-24*b^2*c*e*f^2-240*b*c^2*d*f^2+240*b*c^2*e^2* 
f+384*c^3*d*e*f-192*c^3*e^3)*(c*x^2+b*x+a)^(1/2)/f^4+1/128/f^4/c^2*(1/f*(4 
8*a^2*c^2*f^4-24*a*b^2*c*f^4-96*a*b*c^2*e*f^3-192*a*c^3*d*f^3+192*a*c^3*e^ 
2*f^2+3*b^4*f^4+8*b^3*c*e*f^3-48*b^2*c^2*d*f^3+48*b^2*c^2*e^2*f^2+384*b*c^ 
3*d*e*f^2-192*b*c^3*e^3*f+128*c^4*d^2*f^2-384*c^4*d*e^2*f+128*c^4*e^4)*ln( 
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+64/f^2*c^2*(b^2*e^3*f^2*( 
-4*d*f+e^2)^(1/2)-2*a*b*e^3*f^3-8*a*c*d*e^2*f^3-10*b*c*d^2*e*f^3+2*a*c*e^4 
*f^2+9*c^2*d^2*e^2*f^2-4*b^2*d*e^2*f^3+a^2*e*f^4*(-4*d*f+e^2)^(1/2)+2*a*b* 
d*f^4*(-4*d*f+e^2)^(1/2)-2*a*b*e^2*f^3*(-4*d*f+e^2)^(1/2)-2*b^2*d*e*f^3*(- 
4*d*f+e^2)^(1/2)-2*d^2*f^3*b*c*(-4*d*f+e^2)^(1/2)-2*b*c*e^4*f*(-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)+2*b^2*d 
^2*f^4+4*a*c*d^2*f^4-6*c^2*d*e^4*f+10*b*c*d*e^3*f^2+6*a*b*d*e*f^4+6*b*c*d* 
e^2*f^2*(-4*d*f+e^2)^(1/2)+b^2*e^4*f^2+a^2*e^2*f^4-2*c^2*d^3*f^3+c^2*e^6-2 
*a^2*d*f^5+e^5*c^2*(-4*d*f+e^2)^(1/2)-2*f*e^5*c*b)/(-4*d*f+e^2)^(1/2)*2^(1 
/2)/((-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c 
+c*e^2)/f^2)^(1/2)*ln(((-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a 
*f^2-b*e*f-2*d*f*c+c*e^2)/f^2+1/f*(-c*(-4*d*f+e^2)^(1/2)+f*b-c*e)*(x+1/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

integrate(x^2*(c*x^2+b*x+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:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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