\(\int \frac {(a+b x+c x^2)^{5/2}}{d+e x} \, dx\) [597]

Optimal result

Integrand size = 22, antiderivative size = 459 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{5/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6} \] Output:

1/128*(128*c^4*d^4-3*b^4*e^4-2*b^2*c*e^3*(-14*a*e+5*b*d)-32*c^3*d^2*e*(-8* 
a*e+9*b*d)+8*c^2*e^2*(16*a^2*e^2-39*a*b*d*e+22*b^2*d^2)-2*c*e*(-b*e+2*c*d) 
*(16*c^2*d^2-3*b^2*e^2-4*c*e*(-7*a*e+4*b*d))*x)*(c*x^2+b*x+a)^(1/2)/c^2/e^ 
5+1/48*(16*c^2*d^2+3*b^2*e^2-2*c*e*(-8*a*e+11*b*d)-6*c*e*(-b*e+2*c*d)*x)*( 
c*x^2+b*x+a)^(3/2)/c/e^3+1/5*(c*x^2+b*x+a)^(5/2)/e-1/256*(-b*e+2*c*d)*(128 
*c^4*d^4+3*b^4*e^4+8*b^2*c*e^3*(-5*a*e+2*b*d)-64*c^3*d^2*e*(-5*a*e+4*b*d)+ 
16*c^2*e^2*(15*a^2*e^2-20*a*b*d*e+7*b^2*d^2))*arctanh(1/2*(2*c*x+b)/c^(1/2 
)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/e^6+(a*e^2-b*d*e+c*d^2)^(5/2)*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 = 3.69 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {\frac {e \sqrt {a+x (b+c x)} \left (-45 b^4 e^4+30 b^2 c e^3 (-5 b d+18 a e+b e x)+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+4 c^2 e^2 \left (736 a^2 e^2+2 a b e (-695 d+311 e x)+b^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )\right )-16 c^3 e \left (a e \left (-280 d^2+135 d e x-88 e^2 x^2\right )+b \left (270 d^3-130 d^2 e x+85 d e^2 x^2-63 e^3 x^3\right )\right )\right )}{c^2}+3840 \sqrt {-c d^2+e (b d-a e)} \left (c d^2+e (-b d+a e)\right )^2 \arctan \left (\frac {\sqrt {-c d^2+e (b d-a e)} x}{\sqrt {a} (d+e x)-d \sqrt {a+x (b+c x)}}\right )-\frac {15 (2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{5/2}}}{1920 e^6} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x),x]
 

Output:

((e*Sqrt[a + x*(b + c*x)]*(-45*b^4*e^4 + 30*b^2*c*e^3*(-5*b*d + 18*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) + 4*c^2*e^2*(736*a^2*e^2 + 2*a*b*e*(-695*d + 311*e*x) + b^2*(660*d 
^2 - 295*d*e*x + 186*e^2*x^2)) - 16*c^3*e*(a*e*(-280*d^2 + 135*d*e*x - 88* 
e^2*x^2) + b*(270*d^3 - 130*d^2*e*x + 85*d*e^2*x^2 - 63*e^3*x^3))))/c^2 + 
3840*Sqrt[-(c*d^2) + e*(b*d - a*e)]*(c*d^2 + e*(-(b*d) + a*e))^2*ArcTan[(S 
qrt[-(c*d^2) + e*(b*d - a*e)]*x)/(Sqrt[a]*(d + e*x) - d*Sqrt[a + x*(b + c* 
x)])] - (15*(2*c*d - b*e)*(128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(2*b*d - 
5*a*e) - 64*c^3*d^2*e*(4*b*d - 5*a*e) + 16*c^2*e^2*(7*b^2*d^2 - 20*a*b*d*e 
 + 15*a^2*e^2))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/c 
^(5/2))/(1920*e^6)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1162, 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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\int \frac {\left (8 c e (b d-2 a e)^2-d (2 c d-b e) \left (-3 e b^2+8 c d b-4 a c e\right )-(2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\int \frac {\left (8 c e (b d-2 a e)^2-d (2 c d-b e) \left (-3 e b^2+8 c d b-4 a c e\right )-(2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\int \frac {d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right ) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right )+4 c e (b d-2 a e) \left (8 c e (b d-2 a e)^2-d (2 c d-b e) \left (-3 e b^2+8 c d b-4 a c e\right )\right )+(2 c d-b e) \left (128 c^4 d^4-64 c^3 e (4 b d-5 a e) d^2+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b e d+15 a^2 e^2\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\int \frac {d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right ) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right )+4 c e (b d-2 a e) \left (8 c e (b d-2 a e)^2-d (2 c d-b e) \left (-3 e b^2+8 c d b-4 a c e\right )\right )+(2 c d-b e) \left (128 c^4 d^4-64 c^3 e (4 b d-5 a e) d^2+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b e d+15 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {(2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {256 c^2 \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {2 (2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {256 c^2 \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {256 c^2 \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {512 c^2 \left (a e^2-b d e+c d^2\right )^3 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {256 c^2 \left (a e^2-b d e+c d^2\right )^{5/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{24 c e^2}}{2 e}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x),x]
 

Output:

(a + b*x + c*x^2)^(5/2)/(5*e) - (-1/24*((16*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(1 
1*b*d - 8*a*e) - 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 - 2*b^2*c*e^3*(5*b*d - 14*a*e) - 32*c^3*d^2*e* 
(9*b*d - 8*a*e) + 8*c^2*e^2*(22*b^2*d^2 - 39*a*b*d*e + 16*a^2*e^2) - 2*c*e 
*(2*c*d - b*e)*(16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(4*b*d - 7*a*e))*x)*Sqrt[a 
+ b*x + c*x^2])/(4*c*e^2) - (((2*c*d - b*e)*(128*c^4*d^4 + 3*b^4*e^4 + 8*b 
^2*c*e^3*(2*b*d - 5*a*e) - 64*c^3*d^2*e*(4*b*d - 5*a*e) + 16*c^2*e^2*(7*b^ 
2*d^2 - 20*a*b*d*e + 15*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + 
b*x + c*x^2])])/(Sqrt[c]*e) - (256*c^2*(c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTa 
nh[(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*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 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, 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 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 = 1.14 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.56

method	result	size
risch	\(\frac {\left (384 c^{4} e^{4} x^{4}+1008 b \,c^{3} e^{4} x^{3}-480 c^{4} d \,e^{3} x^{3}+1408 a \,c^{3} e^{4} x^{2}+744 b^{2} c^{2} e^{4} x^{2}-1360 b \,c^{3} d \,e^{3} x^{2}+640 c^{4} d^{2} e^{2} x^{2}+2488 a b \,c^{2} e^{4} x -2160 a \,c^{3} d \,e^{3} x +30 x \,b^{3} c \,e^{4}-1180 b^{2} c^{2} d \,e^{3} x +2080 b \,c^{3} d^{2} e^{2} x -960 c^{4} d^{3} e x +2944 e^{4} a^{2} c^{2}+540 a \,b^{2} c \,e^{4}-5560 a b \,c^{2} d \,e^{3}+4480 d^{2} e^{2} a \,c^{3}-45 b^{4} e^{4}-150 d \,e^{3} b^{3} c +2640 d^{2} e^{2} b^{2} c^{2}-4320 d^{3} e b \,c^{3}+1920 d^{4} c^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{2} e^{5}}+\frac {\frac {\left (240 a^{2} b \,c^{2} e^{5}-480 a^{2} c^{3} d \,e^{4}-40 a \,b^{3} c \,e^{5}-240 a \,b^{2} c^{2} d \,e^{4}+960 a b \,c^{3} d^{2} e^{3}-640 a \,c^{4} d^{3} e^{2}+3 b^{5} e^{5}+10 b^{4} c d \,e^{4}+80 b^{3} d^{2} e^{3} c^{2}-480 b^{2} c^{3} d^{3} e^{2}+640 b \,c^{4} d^{4} e -256 d^{5} c^{5}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {256 \left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right ) c^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{256 e^{5} c^{2}}\)	\(715\)
default	\(\text {Expression too large to display}\)	\(1046\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/1920/c^2*(384*c^4*e^4*x^4+1008*b*c^3*e^4*x^3-480*c^4*d*e^3*x^3+1408*a*c^ 
3*e^4*x^2+744*b^2*c^2*e^4*x^2-1360*b*c^3*d*e^3*x^2+640*c^4*d^2*e^2*x^2+248 
8*a*b*c^2*e^4*x-2160*a*c^3*d*e^3*x+30*b^3*c*e^4*x-1180*b^2*c^2*d*e^3*x+208 
0*b*c^3*d^2*e^2*x-960*c^4*d^3*e*x+2944*a^2*c^2*e^4+540*a*b^2*c*e^4-5560*a* 
b*c^2*d*e^3+4480*a*c^3*d^2*e^2-45*b^4*e^4-150*b^3*c*d*e^3+2640*b^2*c^2*d^2 
*e^2-4320*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^2* 
((240*a^2*b*c^2*e^5-480*a^2*c^3*d*e^4-40*a*b^3*c*e^5-240*a*b^2*c^2*d*e^4+9 
60*a*b*c^3*d^2*e^3-640*a*c^4*d^3*e^2+3*b^5*e^5+10*b^4*c*d*e^4+80*b^3*c^2*d 
^2*e^3-480*b^2*c^3*d^3*e^2+640*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*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2 
*e^4+3*a*b^2*d^2*e^4-6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d 
^4*e^2-3*b*c^2*d^5*e+c^3*d^6)*c^2/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. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d),x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(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. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{d+e\,x} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(d + e*x),x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{e x +d}d x \] Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d),x)
 

Output:

int((c*x^2+b*x+a)^(5/2)/(e*x+d),x)