\(\int \frac {x^3 (a+b x+c x^2)^{3/2}}{d+e x} \, dx\) [20]

Optimal result

Integrand size = 25, antiderivative size = 635 \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (512 c^5 d^5+7 b^5 e^5+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+8 b c^2 e^3 \left (3 b^2 d^2-6 a b d e+2 a^2 e^2\right )-2 c e \left (128 c^4 d^4-7 b^4 e^4-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-8 c^2 e^2 \left (3 b^2 d^2-6 a b d e+2 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{512 c^4 e^6}-\frac {\left (64 c^3 d^3-24 b c^2 d^2 e-7 b^3 e^3-4 b c e^2 (3 b d-a e)-2 c e \left (24 c^2 d^2+7 b^2 e^2+4 c e (3 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}-\frac {(22 c d+7 b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac {\left (1024 c^6 d^6+7 b^6 e^6+12 b^4 c e^5 (b d-5 a e)-1536 c^5 d^4 e (b d-a e)+384 c^4 d^2 e^2 (b d-a e)^2+64 c^3 e^3 (b d-a e)^3+24 b^2 c^2 e^4 \left (b^2 d^2-4 a b d e+6 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2} e^7}-\frac {d^3 \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^7} \] Output:

-1/512*(512*c^5*d^5+7*b^5*e^5+4*b^3*c*e^4*(-8*a*e+3*b*d)-128*c^4*d^3*e*(-4 
*a*e+5*b*d)+32*b*c^3*d^2*e^2*(-3*a*e+2*b*d)+8*b*c^2*e^3*(2*a^2*e^2-6*a*b*d 
*e+3*b^2*d^2)-2*c*e*(128*c^4*d^4-7*b^4*e^4-4*b^2*c*e^3*(-8*a*e+3*b*d)-32*c 
^3*d^2*e*(-3*a*e+2*b*d)-8*c^2*e^2*(2*a^2*e^2-6*a*b*d*e+3*b^2*d^2))*x)*(c*x 
^2+b*x+a)^(1/2)/c^4/e^6-1/192*(64*c^3*d^3-24*b*c^2*d^2*e-7*b^3*e^3-4*b*c*e 
^2*(-a*e+3*b*d)-2*c*e*(24*c^2*d^2+7*b^2*e^2+4*c*e*(-a*e+3*b*d))*x)*(c*x^2+ 
b*x+a)^(3/2)/c^3/e^4-1/60*(7*b*e+22*c*d)*(c*x^2+b*x+a)^(5/2)/c^2/e^2+1/6*( 
e*x+d)*(c*x^2+b*x+a)^(5/2)/c/e^2+1/1024*(1024*c^6*d^6+7*b^6*e^6+12*b^4*c*e 
^5*(-5*a*e+b*d)-1536*c^5*d^4*e*(-a*e+b*d)+384*c^4*d^2*e^2*(-a*e+b*d)^2+64* 
c^3*e^3*(-a*e+b*d)^3+24*b^2*c^2*e^4*(6*a^2*e^2-4*a*b*d*e+b^2*d^2))*arctanh 
(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)/e^7-d^3*(a*e^2-b*d*e+c 
*d^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/ 
2)/(c*x^2+b*x+a)^(1/2))/e^7
 

Mathematica [A] (verified)

Time = 11.47 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.04 \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {-5120 d^3 (a+x (b+c x))^{3/2}+\frac {1920 d^2 e (b+2 c x) (a+x (b+c x))^{3/2}}{c}-\frac {3072 d e^2 (a+x (b+c x))^{5/2}}{c}+\frac {2560 e^3 x (a+x (b+c x))^{5/2}}{c}+\frac {360 \left (b^2-4 a c\right ) d^2 e \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}+\frac {60 b d e^2 \left (\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3 \left (b^2-4 a c\right ) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}\right )}{c}+\frac {e^3 \left (-1792 b (a+x (b+c x))^{5/2}+5 \left (7 b^2-4 a c\right ) \left (\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3 \left (b^2-4 a c\right ) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}\right )\right )}{c^2}+\frac {960 d^3 \left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (-b^2 e^2+4 c^2 d (-2 d+e x)-2 c e (-5 b d+4 a e+b e x)\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )\right )}{c^{3/2} e^3}}{15360 e^4} \] Input:

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

Output:

(-5120*d^3*(a + x*(b + c*x))^(3/2) + (1920*d^2*e*(b + 2*c*x)*(a + x*(b + c 
*x))^(3/2))/c - (3072*d*e^2*(a + x*(b + c*x))^(5/2))/c + (2560*e^3*x*(a + 
x*(b + c*x))^(5/2))/c + (360*(b^2 - 4*a*c)*d^2*e*(-2*Sqrt[c]*(b + 2*c*x)*S 
qrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a 
 + x*(b + c*x)])]))/c^(5/2) + (60*b*d*e^2*((16*(b + 2*c*x)*(a + x*(b + c*x 
))^(3/2))/c + (3*(b^2 - 4*a*c)*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x 
)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) 
)/c^(5/2)))/c + (e^3*(-1792*b*(a + x*(b + c*x))^(5/2) + 5*(7*b^2 - 4*a*c)* 
((16*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3*(b^2 - 4*a*c)*(-2*Sqrt[c] 
*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2* 
Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2))))/c^2 + (960*d^3*((2*c*d - b*e) 
*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqr 
t[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(-(b^2*e 
^2) + 4*c^2*d*(-2*d + e*x) - 2*c*e*(-5*b*d + 4*a*e + b*e*x)) + 8*c*(c*d^2 
+ e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sq 
rt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])))/(c^(3/2)*e^3))/(15 
360*e^4)
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {1267, 27, 2184, 27, 1231, 27, 1231, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1267

\(\displaystyle \frac {\int -\frac {\left (c x^2+b x+a\right )^{3/2} \left (e^2 (22 c d+7 b e) x^2+2 e \left (5 c d^2+e (6 b d+a e)\right ) x+d e (5 b d+2 a e)\right )}{2 (d+e x)}dx}{6 c e^3}+\frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\int \frac {\left (c x^2+b x+a\right )^{3/2} \left (e^2 (22 c d+7 b e) x^2+2 e \left (5 c d^2+e (6 b d+a e)\right ) x+d e (5 b d+2 a e)\right )}{d+e x}dx}{12 c e^3}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {\int -\frac {5 e^3 \left (d \left (7 e b^2+12 c d b-4 a c e\right )+\left (24 c^2 d^2+7 b^2 e^2+4 c e (3 b d-a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 (d+e x)}dx}{5 c e^2}+\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}}{12 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \int \frac {\left (d \left (7 e b^2+12 c d b-4 a c e\right )+\left (24 c^2 d^2+7 b^2 e^2+4 c e (3 b d-a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{d+e x}dx}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {\int \frac {3 \left (d \left (7 e^3 b^4+12 c d e^2 b^3+8 c e \left (3 c d^2-4 a e^2\right ) b^2-16 c^2 d \left (4 c d^2+3 a e^2\right ) b+16 a c^2 e \left (2 c d^2+a e^2\right )\right )-\left (128 c^4 d^4-32 c^3 e (2 b d-3 a e) d^2-7 b^4 e^4-4 b^2 c e^3 (3 b d-8 a e)-8 c^2 e^2 \left (3 b^2 d^2-6 a b e d+2 a^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \int \frac {\left (d \left (7 e^3 b^4+12 c d e^2 b^3+8 c e \left (3 c d^2-4 a e^2\right ) b^2-16 c^2 d \left (4 c d^2+3 a e^2\right ) b+16 a c^2 e \left (2 c d^2+a e^2\right )\right )-\left (128 c^4 d^4-32 c^3 e (2 b d-3 a e) d^2-7 b^4 e^4-4 b^2 c e^3 (3 b d-8 a e)-8 c^2 e^2 \left (3 b^2 d^2-6 a b e d+2 a^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\int \frac {d \left (7 e^5 b^6+12 c d e^4 b^5+12 c e^3 \left (2 c d^2-5 a e^2\right ) b^4+32 c^2 d e^2 \left (2 c d^2-3 a e^2\right ) b^3-16 c^2 e \left (40 c^2 d^4+12 a c e^2 d^2-9 a^2 e^4\right ) b^2+64 c^3 d \left (8 c^2 d^4+20 a c e^2 d^2+3 a^2 e^4\right ) b-64 a c^3 e \left (8 c^2 d^4+10 a c e^2 d^2+a^2 e^4\right )\right )+\left (1024 c^6 d^6-1536 c^5 e (b d-a e) d^4+384 c^4 e^2 (b d-a e)^2 d^2+7 b^6 e^6+64 c^3 e^3 (b d-a e)^3+12 b^4 c e^5 (b d-5 a e)+24 b^2 c^2 e^4 \left (b^2 d^2-4 a b e d+6 a^2 e^2\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\int \frac {d \left (7 e^5 b^6+12 c d e^4 b^5+12 c e^3 \left (2 c d^2-5 a e^2\right ) b^4+32 c^2 d e^2 \left (2 c d^2-3 a e^2\right ) b^3-16 c^2 e \left (40 c^2 d^4+12 a c e^2 d^2-9 a^2 e^4\right ) b^2+64 c^3 d \left (8 c^2 d^4+20 a c e^2 d^2+3 a^2 e^4\right ) b-64 a c^3 e \left (8 c^2 d^4+10 a c e^2 d^2+a^2 e^4\right )\right )+\left (1024 c^6 d^6-1536 c^5 e (b d-a e) d^4+384 c^4 e^2 (b d-a e)^2 d^2+7 b^6 e^6+64 c^3 e^3 (b d-a e)^3+12 b^4 c e^5 (b d-5 a e)+24 b^2 c^2 e^4 \left (b^2 d^2-4 a b e d+6 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\frac {\left (24 b^2 c^2 e^4 \left (6 a^2 e^2-4 a b d e+b^2 d^2\right )+12 b^4 c e^5 (b d-5 a e)-1536 c^5 d^4 e (b d-a e)+384 c^4 d^2 e^2 (b d-a e)^2+64 c^3 e^3 (b d-a e)^3+7 b^6 e^6+1024 c^6 d^6\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {1024 c^4 d^3 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\frac {2 \left (24 b^2 c^2 e^4 \left (6 a^2 e^2-4 a b d e+b^2 d^2\right )+12 b^4 c e^5 (b d-5 a e)-1536 c^5 d^4 e (b d-a e)+384 c^4 d^2 e^2 (b d-a e)^2+64 c^3 e^3 (b d-a e)^3+7 b^6 e^6+1024 c^6 d^6\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {1024 c^4 d^3 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 b^2 c^2 e^4 \left (6 a^2 e^2-4 a b d e+b^2 d^2\right )+12 b^4 c e^5 (b d-5 a e)-1536 c^5 d^4 e (b d-a e)+384 c^4 d^2 e^2 (b d-a e)^2+64 c^3 e^3 (b d-a e)^3+7 b^6 e^6+1024 c^6 d^6\right )}{\sqrt {c} e}-\frac {1024 c^4 d^3 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\frac {2048 c^4 d^3 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 b^2 c^2 e^4 \left (6 a^2 e^2-4 a b d e+b^2 d^2\right )+12 b^4 c e^5 (b d-5 a e)-1536 c^5 d^4 e (b d-a e)+384 c^4 d^2 e^2 (b d-a e)^2+64 c^3 e^3 (b d-a e)^3+7 b^6 e^6+1024 c^6 d^6\right )}{\sqrt {c} e}}{8 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac {\frac {e \left (a+b x+c x^2\right )^{5/2} (7 b e+22 c d)}{5 c}-\frac {e \left (-\frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (8 b c^2 e^3 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-2 c e x \left (-8 c^2 e^2 \left (2 a^2 e^2-6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (3 b d-8 a e)-32 c^3 d^2 e (2 b d-3 a e)-7 b^4 e^4+128 c^4 d^4\right )+4 b^3 c e^4 (3 b d-8 a e)-128 c^4 d^3 e (5 b d-4 a e)+32 b c^3 d^2 e^2 (2 b d-3 a e)+7 b^5 e^5+512 c^5 d^5\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 b^2 c^2 e^4 \left (6 a^2 e^2-4 a b d e+b^2 d^2\right )+12 b^4 c e^5 (b d-5 a e)-1536 c^5 d^4 e (b d-a e)+384 c^4 d^2 e^2 (b d-a e)^2+64 c^3 e^3 (b d-a e)^3+7 b^6 e^6+1024 c^6 d^6\right )}{\sqrt {c} e}-\frac {1024 c^4 d^3 \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}\right )}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (4 c e (3 b d-a e)+7 b^2 e^2+24 c^2 d^2\right )-4 b c e^2 (3 b d-a e)-7 b^3 e^3-24 b c^2 d^2 e+64 c^3 d^3\right )}{8 c e^2}\right )}{2 c}}{12 c e^3}\)

Input:

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

Output:

((d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c*e^2) - ((e*(22*c*d + 7*b*e)*(a + 
b*x + c*x^2)^(5/2))/(5*c) - (e*(-1/8*((64*c^3*d^3 - 24*b*c^2*d^2*e - 7*b^3 
*e^3 - 4*b*c*e^2*(3*b*d - a*e) - 2*c*e*(24*c^2*d^2 + 7*b^2*e^2 + 4*c*e*(3* 
b*d - a*e))*x)*(a + b*x + c*x^2)^(3/2))/(c*e^2) - (3*(((512*c^5*d^5 + 7*b^ 
5*e^5 + 4*b^3*c*e^4*(3*b*d - 8*a*e) - 128*c^4*d^3*e*(5*b*d - 4*a*e) + 32*b 
*c^3*d^2*e^2*(2*b*d - 3*a*e) + 8*b*c^2*e^3*(3*b^2*d^2 - 6*a*b*d*e + 2*a^2* 
e^2) - 2*c*e*(128*c^4*d^4 - 7*b^4*e^4 - 4*b^2*c*e^3*(3*b*d - 8*a*e) - 32*c 
^3*d^2*e*(2*b*d - 3*a*e) - 8*c^2*e^2*(3*b^2*d^2 - 6*a*b*d*e + 2*a^2*e^2))* 
x)*Sqrt[a + b*x + c*x^2])/(4*c*e^2) - (((1024*c^6*d^6 + 7*b^6*e^6 + 12*b^4 
*c*e^5*(b*d - 5*a*e) - 1536*c^5*d^4*e*(b*d - a*e) + 384*c^4*d^2*e^2*(b*d - 
 a*e)^2 + 64*c^3*e^3*(b*d - a*e)^3 + 24*b^2*c^2*e^4*(b^2*d^2 - 4*a*b*d*e + 
 6*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt 
[c]*e) - (1024*c^4*d^3*(c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e 
+ (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])]) 
/e)/(8*c*e^2)))/(16*c*e^2)))/(2*c))/(12*c*e^3)
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1267

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.40

method	result	size
risch	\(-\frac {\left (-1280 c^{5} e^{5} x^{5}-1664 b \,c^{4} e^{5} x^{4}+1536 c^{5} d \,e^{4} x^{4}-2240 a \,c^{4} e^{5} x^{3}-48 b^{2} c^{3} e^{5} x^{3}+2112 b \,c^{4} d \,e^{4} x^{3}-1920 c^{5} d^{2} e^{3} x^{3}-288 a b \,c^{3} e^{5} x^{2}+3072 a \,c^{4} d \,e^{4} x^{2}+56 b^{3} c^{2} e^{5} x^{2}+96 b^{2} c^{3} d \,e^{4} x^{2}-2880 b \,c^{4} d^{2} e^{3} x^{2}+2560 c^{5} d^{3} e^{2} x^{2}-480 a^{2} c^{3} e^{5} x +432 a \,b^{2} c^{2} e^{5} x +672 a b \,c^{3} d \,e^{4} x -4800 a \,c^{4} d^{2} e^{3} x -70 b^{4} c \,e^{5} x -120 b^{3} c^{2} d \,e^{4} x -240 b^{2} c^{3} d^{2} e^{3} x +4480 b \,c^{4} d^{3} e^{2} x -3840 c^{5} d^{4} e x +1296 a^{2} b \,c^{2} e^{5}+1536 a^{2} c^{3} d \,e^{4}-760 a \,b^{3} c \,e^{5}-1200 a \,b^{2} c^{2} d \,e^{4}-2400 a b \,c^{3} d^{2} e^{3}+10240 a \,c^{4} d^{3} e^{2}+105 b^{5} e^{5}+180 b^{4} c d \,e^{4}+360 b^{3} d^{2} e^{3} c^{2}+960 b^{2} c^{3} d^{3} e^{2}-9600 b \,c^{4} d^{4} e +7680 d^{5} c^{5}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4} e^{6}}-\frac {\frac {\left (64 a^{3} c^{3} e^{6}-144 a^{2} b^{2} c^{2} e^{6}-192 a^{2} b \,c^{3} d \,e^{5}-384 a^{2} c^{4} d^{2} e^{4}+60 a \,b^{4} c \,e^{6}+96 a \,b^{3} c^{2} d \,e^{5}+192 a \,b^{2} c^{3} d^{2} e^{4}+768 a b \,c^{4} d^{3} e^{3}-1536 a \,c^{5} d^{4} e^{2}-7 b^{6} e^{6}-12 b^{5} c d \,e^{5}-24 b^{4} c^{2} d^{2} e^{4}-64 b^{3} c^{3} d^{3} e^{3}-384 b^{2} c^{4} d^{4} e^{2}+1536 b \,c^{5} d^{5} e -1024 c^{6} d^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {1024 d^{3} \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) c^{4} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{1024 e^{6} c^{4}}\)	\(890\)
default	\(\text {Expression too large to display}\)	\(1138\)

Input:

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

Output:

-1/7680/c^4*(-1280*c^5*e^5*x^5-1664*b*c^4*e^5*x^4+1536*c^5*d*e^4*x^4-2240* 
a*c^4*e^5*x^3-48*b^2*c^3*e^5*x^3+2112*b*c^4*d*e^4*x^3-1920*c^5*d^2*e^3*x^3 
-288*a*b*c^3*e^5*x^2+3072*a*c^4*d*e^4*x^2+56*b^3*c^2*e^5*x^2+96*b^2*c^3*d* 
e^4*x^2-2880*b*c^4*d^2*e^3*x^2+2560*c^5*d^3*e^2*x^2-480*a^2*c^3*e^5*x+432* 
a*b^2*c^2*e^5*x+672*a*b*c^3*d*e^4*x-4800*a*c^4*d^2*e^3*x-70*b^4*c*e^5*x-12 
0*b^3*c^2*d*e^4*x-240*b^2*c^3*d^2*e^3*x+4480*b*c^4*d^3*e^2*x-3840*c^5*d^4* 
e*x+1296*a^2*b*c^2*e^5+1536*a^2*c^3*d*e^4-760*a*b^3*c*e^5-1200*a*b^2*c^2*d 
*e^4-2400*a*b*c^3*d^2*e^3+10240*a*c^4*d^3*e^2+105*b^5*e^5+180*b^4*c*d*e^4+ 
360*b^3*c^2*d^2*e^3+960*b^2*c^3*d^3*e^2-9600*b*c^4*d^4*e+7680*c^5*d^5)*(c* 
x^2+b*x+a)^(1/2)/e^6-1/1024/e^6/c^4*((64*a^3*c^3*e^6-144*a^2*b^2*c^2*e^6-1 
92*a^2*b*c^3*d*e^5-384*a^2*c^4*d^2*e^4+60*a*b^4*c*e^6+96*a*b^3*c^2*d*e^5+1 
92*a*b^2*c^3*d^2*e^4+768*a*b*c^4*d^3*e^3-1536*a*c^5*d^4*e^2-7*b^6*e^6-12*b 
^5*c*d*e^5-24*b^4*c^2*d^2*e^4-64*b^3*c^3*d^3*e^3-384*b^2*c^4*d^4*e^2+1536* 
b*c^5*d^5*e-1024*c^6*d^6)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^ 
(1/2)-1024*d^3*(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-2*b*c*d^3*e+ 
c^2*d^4)*c^4/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2) 
/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+ 
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(c*x^2+b*x+a)^(3/2)/(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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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