\(\int \frac {x^2 (a+b x+c x^2)^{3/2}}{d+e x} \, dx\) [21]

Optimal result

Integrand size = 25, antiderivative size = 461 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {\left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2} e^6}+\frac {d^2 \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6} \] Output:

1/128*(128*c^4*d^4+3*b^4*e^4-32*c^3*d^2*e*(-4*a*e+5*b*d)+8*b*c^2*d*e^2*(-3 
*a*e+2*b*d)+6*b^2*c*e^3*(-2*a*e+b*d)-2*c*e*(32*c^3*d^3-3*b^3*e^3-8*c^2*d*e 
*(-3*a*e+2*b*d)-6*b*c*e^2*(-2*a*e+b*d))*x)*(c*x^2+b*x+a)^(1/2)/c^3/e^5+1/4 
8*(16*c^2*d^2-6*b*c*d*e-3*b^2*e^2-6*c*e*(b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2) 
/c^2/e^3+1/5*(c*x^2+b*x+a)^(5/2)/c/e-1/256*(256*c^5*d^5+3*b^5*e^5+6*b^3*c* 
e^4*(-4*a*e+b*d)-384*c^4*d^3*e*(-a*e+b*d)+96*c^3*d*e^2*(-a*e+b*d)^2+16*b*c 
^2*e^3*(3*a^2*e^2-3*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^(7/2)/e^6+d^2*(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^6
 

Mathematica [A] (verified)

Time = 11.01 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {-15 \left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (45 b^4 e^4-30 b^2 c e^3 (-3 b d+10 a e+b e x)+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+12 c^2 e^2 \left (32 a^2 e^2+2 a b e (-25 d+7 e x)+b^2 \left (20 d^2-5 d e x+2 e^2 x^2\right )\right )+16 c^3 e \left (a e \left (160 d^2-75 d e x+48 e^2 x^2\right )+b \left (-150 d^3+70 d^2 e x-45 d e^2 x^2+33 e^3 x^3\right )\right )\right )-1920 c^3 d^2 \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{3840 c^{7/2} e^6} \] Input:

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

Output:

(-15*(256*c^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*d - 4*a*e) - 384*c^4*d^3*e* 
(b*d - a*e) + 96*c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e^3*(b^2*d^2 - 3*a*b*d 
*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 
2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(45*b^4*e^4 - 30*b^2*c*e^3*(-3*b*d + 10 
*a*e + b*e*x) + 32*c^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^ 
3 + 12*e^4*x^4) + 12*c^2*e^2*(32*a^2*e^2 + 2*a*b*e*(-25*d + 7*e*x) + b^2*( 
20*d^2 - 5*d*e*x + 2*e^2*x^2)) + 16*c^3*e*(a*e*(160*d^2 - 75*d*e*x + 48*e^ 
2*x^2) + b*(-150*d^3 + 70*d^2*e*x - 45*d*e^2*x^2 + 33*e^3*x^3))) - 1920*c^ 
3*d^2*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(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)])]))/(3840* 
c^(7/2)*e^6)
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1267, 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.

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

\(\Big \downarrow \) 1267

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\int \frac {d \left (3 e^4 b^5+6 c d e^3 b^4+8 c e^2 \left (2 c d^2-3 a e^2\right ) b^3-16 c^2 d e \left (10 c d^2+3 a e^2\right ) b^2+16 c^2 \left (8 c^2 d^4+20 a c e^2 d^2+3 a^2 e^4\right ) b-32 a c^3 d e \left (4 c d^2+5 a e^2\right )\right )+\left (256 c^5 d^5-384 c^4 e (b d-a e) d^3+96 c^3 e^2 (b d-a e)^2 d+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)+16 b c^2 e^3 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\int \frac {d \left (3 e^4 b^5+6 c d e^3 b^4+8 c e^2 \left (2 c d^2-3 a e^2\right ) b^3-16 c^2 d e \left (10 c d^2+3 a e^2\right ) b^2+16 c^2 \left (8 c^2 d^4+20 a c e^2 d^2+3 a^2 e^4\right ) b-32 a c^3 d e \left (4 c d^2+5 a e^2\right )\right )+\left (256 c^5 d^5-384 c^4 e (b d-a e) d^3+96 c^3 e^2 (b d-a e)^2 d+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)+16 b c^2 e^3 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {\left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {2 \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right )}{\sqrt {c} e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {512 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right )}{\sqrt {c} e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right )}{\sqrt {c} e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-3 b^2 e^2-6 c e x (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\)

Input:

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

Output:

(a + b*x + c*x^2)^(5/2)/(5*c*e) - (-1/24*((16*c^2*d^2 - 6*b*c*d*e - 3*b^2* 
e^2 - 6*c*e*(2*c*d + b*e)*x)*(a + b*x + c*x^2)^(3/2))/(c*e^2) - (((128*c^4 
*d^4 + 3*b^4*e^4 - 32*c^3*d^2*e*(5*b*d - 4*a*e) + 8*b*c^2*d*e^2*(2*b*d - 3 
*a*e) + 6*b^2*c*e^3*(b*d - 2*a*e) - 2*c*e*(32*c^3*d^3 - 3*b^3*e^3 - 8*c^2* 
d*e*(2*b*d - 3*a*e) - 6*b*c*e^2*(b*d - 2*a*e))*x)*Sqrt[a + b*x + c*x^2])/( 
4*c*e^2) - (((256*c^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*d - 4*a*e) - 384*c^ 
4*d^3*e*(b*d - a*e) + 96*c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e^3*(b^2*d^2 - 
 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x 
^2])])/(Sqrt[c]*e) - (256*c^3*d^2*(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*e)
 

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]
 

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {\left (384 c^{4} e^{4} x^{4}+528 b \,c^{3} e^{4} x^{3}-480 c^{4} d \,e^{3} x^{3}+768 a \,c^{3} e^{4} x^{2}+24 b^{2} c^{2} e^{4} x^{2}-720 b \,c^{3} d \,e^{3} x^{2}+640 c^{4} d^{2} e^{2} x^{2}+168 a b \,c^{2} e^{4} x -1200 a \,c^{3} d \,e^{3} x -30 x \,b^{3} c \,e^{4}-60 b^{2} c^{2} d \,e^{3} x +1120 b \,c^{3} d^{2} e^{2} x -960 c^{4} d^{3} e x +384 e^{4} a^{2} c^{2}-300 a \,b^{2} c \,e^{4}-600 a b \,c^{2} d \,e^{3}+2560 d^{2} e^{2} a \,c^{3}+45 b^{4} e^{4}+90 d \,e^{3} b^{3} c +240 d^{2} e^{2} b^{2} c^{2}-2400 d^{3} e b \,c^{3}+1920 d^{4} c^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{3} e^{5}}-\frac {\frac {\left (48 a^{2} b \,c^{2} e^{5}+96 a^{2} c^{3} d \,e^{4}-24 a \,b^{3} c \,e^{5}-48 a \,b^{2} c^{2} d \,e^{4}-192 a b \,c^{3} d^{2} e^{3}+384 a \,c^{4} d^{3} e^{2}+3 b^{5} e^{5}+6 b^{4} c d \,e^{4}+16 b^{3} d^{2} e^{3} c^{2}+96 b^{2} c^{3} d^{3} e^{2}-384 b \,c^{4} d^{4} e +256 d^{5} c^{5}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}+\frac {256 d^{2} \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) c^{3} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{256 e^{5} c^{3}}\)	\(664\)
default	\(\frac {\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}}{e}+\frac {d^{2} \left (\frac {\left (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}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \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}}}}{4 c}+\frac {\left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\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}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \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}}}}\right )}{e^{2}}\right )}{e^{3}}-\frac {d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{e^{2}}\)	\(869\)

Input:

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

Output:

1/1920/c^3*(384*c^4*e^4*x^4+528*b*c^3*e^4*x^3-480*c^4*d*e^3*x^3+768*a*c^3* 
e^4*x^2+24*b^2*c^2*e^4*x^2-720*b*c^3*d*e^3*x^2+640*c^4*d^2*e^2*x^2+168*a*b 
*c^2*e^4*x-1200*a*c^3*d*e^3*x-30*b^3*c*e^4*x-60*b^2*c^2*d*e^3*x+1120*b*c^3 
*d^2*e^2*x-960*c^4*d^3*e*x+384*a^2*c^2*e^4-300*a*b^2*c*e^4-600*a*b*c^2*d*e 
^3+2560*a*c^3*d^2*e^2+45*b^4*e^4+90*b^3*c*d*e^3+240*b^2*c^2*d^2*e^2-2400*b 
*c^3*d^3*e+1920*c^4*d^4)*(c*x^2+b*x+a)^(1/2)/e^5-1/256/e^5/c^3*((48*a^2*b* 
c^2*e^5+96*a^2*c^3*d*e^4-24*a*b^3*c*e^5-48*a*b^2*c^2*d*e^4-192*a*b*c^3*d^2 
*e^3+384*a*c^4*d^3*e^2+3*b^5*e^5+6*b^4*c*d*e^4+16*b^3*c^2*d^2*e^3+96*b^2*c 
^3*d^3*e^2-384*b*c^4*d^4*e+256*c^5*d^5)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b* 
x+a)^(1/2))/c^(1/2)+256*d^2*(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^3/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. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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