\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^5} \, dx\) [601]

Optimal result

Integrand size = 22, antiderivative size = 422 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {5 c (2 c d-b e) \sqrt {a+b x+c x^2}}{2 e^5 (d+e x)}+\frac {5 \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 e^4 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {5 (8 c d-b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3 (d+e x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac {5 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^6}+\frac {5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 e^6 \left (c d^2-b d e+a e^2\right )^{3/2}} \] Output:

5/2*c*(-b*e+2*c*d)*(c*x^2+b*x+a)^(1/2)/e^5/(e*x+d)+5/64*(16*c^2*d^2+b^2*e^ 
2-4*c*e*(-3*a*e+4*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e^4 
/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2+5/24*(6*c*e*x-b*e+8*c*d)*(c*x^2+b*x+a)^(3/2 
)/e^3/(e*x+d)^3-1/4*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^4-5/2*c^(3/2)*(-b*e+2*c* 
d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^6+5/128*(128*c^4*d 
^4-b^4*e^4-8*b^2*c*e^3*(-3*a*e+2*b*d)-64*c^3*d^2*e*(-3*a*e+4*b*d)+48*c^2*e 
^2*(a^2*e^2-4*a*b*d*e+3*b^2*d^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/(a*e^2-b*d*e+c*d^2)^(3/2 
)
 

Mathematica [A] (verified)

Time = 12.68 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (16 c^3 d^2 \left (60 d^4+210 d^3 e x+260 d^2 e^2 x^2+125 d e^3 x^3+12 e^4 x^4\right )+e^3 \left (-48 a^3 e^3+8 a^2 b e^2 (d-17 e x)+2 a b^2 e \left (5 d^2+18 d e x-59 e^2 x^2\right )+b^3 \left (15 d^3+55 d^2 e x+73 d e^2 x^2-15 e^3 x^3\right )\right )+2 c e^2 \left (-4 a^2 e^2 \left (11 d^2+20 d e x+27 e^2 x^2\right )-2 a b e \left (45 d^3+169 d^2 e x+191 d e^2 x^2+139 e^3 x^3\right )+b^2 d \left (120 d^3+435 d^2 e x+566 d e^2 x^2+323 e^3 x^3\right )\right )-8 c^2 e \left (-a e \left (100 d^4+355 d^3 e x+448 d^2 e^2 x^2+235 d e^3 x^3+24 e^4 x^4\right )+b d \left (150 d^4+530 d^3 e x+665 d^2 e^2 x^2+327 d e^3 x^3+24 e^4 x^4\right )\right )\right )}{\left (c d^2+e (-b d+a e)\right ) (d+e x)^4}-960 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {15 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2}}}{384 e^6} \] Input:

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

Output:

((2*e*Sqrt[a + x*(b + c*x)]*(16*c^3*d^2*(60*d^4 + 210*d^3*e*x + 260*d^2*e^ 
2*x^2 + 125*d*e^3*x^3 + 12*e^4*x^4) + e^3*(-48*a^3*e^3 + 8*a^2*b*e^2*(d - 
17*e*x) + 2*a*b^2*e*(5*d^2 + 18*d*e*x - 59*e^2*x^2) + b^3*(15*d^3 + 55*d^2 
*e*x + 73*d*e^2*x^2 - 15*e^3*x^3)) + 2*c*e^2*(-4*a^2*e^2*(11*d^2 + 20*d*e* 
x + 27*e^2*x^2) - 2*a*b*e*(45*d^3 + 169*d^2*e*x + 191*d*e^2*x^2 + 139*e^3* 
x^3) + b^2*d*(120*d^3 + 435*d^2*e*x + 566*d*e^2*x^2 + 323*e^3*x^3)) - 8*c^ 
2*e*(-(a*e*(100*d^4 + 355*d^3*e*x + 448*d^2*e^2*x^2 + 235*d*e^3*x^3 + 24*e 
^4*x^4)) + b*d*(150*d^4 + 530*d^3*e*x + 665*d^2*e^2*x^2 + 327*d*e^3*x^3 + 
24*e^4*x^4))))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^4) - 960*c^(3/2)*(2*c 
*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - (15*(12 
8*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 
3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^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)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(384*e^6)
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1161, 1229, 27, 1230, 25, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1229

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {5 \left (-\frac {-\frac {\int -\frac {e^3 b^4+16 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+64 c^2 d \left (c d^2+2 a e^2\right ) b-16 a c^2 e \left (4 c d^2+3 a e^2\right )+64 c^2 (2 c d-b e) \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 \left (-\frac {\frac {\int \frac {e^3 b^4+16 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+64 c^2 d \left (c d^2+2 a e^2\right ) b-16 a c^2 e \left (4 c d^2+3 a e^2\right )+64 c^2 (2 c d-b e) \left (c d^2-b e d+a e^2\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5 \left (-\frac {\frac {\frac {64 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {5 \left (-\frac {\frac {\frac {128 c^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (-\frac {\frac {\frac {64 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {5 \left (-\frac {\frac {\frac {2 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {64 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (-\frac {\frac {\frac {64 c^{3/2} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a e^2-b d e+c d^2\right )}{e}-\frac {\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{e^2 (d+e x)}}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{8 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}\)

Input:

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

Output:

-1/4*(a + b*x + c*x^2)^(5/2)/(e*(d + e*x)^4) + (5*(-1/12*((16*c^2*d^3 - b* 
e^2*(b*d - 4*a*e) - 4*c*d*e*(3*b*d - a*e) + 3*e*(8*c^2*d^2 + b^2*e^2 - 4*c 
*e*(2*b*d - a*e))*x)*(a + b*x + c*x^2)^(3/2))/(e^2*(c*d^2 - b*d*e + a*e^2) 
*(d + e*x)^3) - (-(((64*c^3*d^3 + b^3*e^3 + 4*b*c*e^2*(4*b*d - 5*a*e) - 16 
*c^2*d*e*(5*b*d - 4*a*e) + 2*c*e*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3* 
a*e))*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x))) + ((64*c^(3/2)*(2*c*d - b 
*e)*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + 
c*x^2])])/e - ((128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c 
^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*A 
rcTanh[(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*Sqrt[c*d^2 - b*d*e + a*e^2]))/(2*e^2))/(8*e^2*(c*d 
^2 - b*d*e + a*e^2))))/(8*e)
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 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 1161

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 + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4039\) vs. \(2(386)=772\).

Time = 1.58 (sec) , antiderivative size = 4040, normalized size of antiderivative = 9.57

method	result	size
risch	\(\text {Expression too large to display}\)	\(4040\)
default	\(\text {Expression too large to display}\)	\(7554\)

Input:

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

Output:

c^2/e^5*(c*x^2+b*x+a)^(1/2)+1/2/e^5*(2/e^3*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3 
*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)*(-1/(a*e^2-b*d*e+c*d^2)*e^2 
/(x+d/e)*(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) 
+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)))+6/e^4*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d 
*e^3+6*b^2*c*d^2*e^2-10*b*c^2*d^3*e+5*c^3*d^4)*(-1/2/(a*e^2-b*d*e+c*d^2)*e 
^2/(x+d/e)^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)-3/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+ 
d/e)*(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)+1/2 
*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)))+1/2*c/(a*e^2-b*d*e+c*d^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)))+6/e^5*(a^2*b*e^5-2*a^2*c*d*e^4-2*a*b^2*d*e^4+6*a*b*c*d^2*e^3-4* 
a*c^2*d^3*e^2+b^3*d^2*e^3-4*b^2*c*d^3*e^2+5*b*c^2*d^4*e-2*c^3*d^5)*(-1/3/( 
a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^5,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(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 26.20 (sec) , antiderivative size = 8880, normalized size of antiderivative = 21.04 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(720*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e** 
2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**4*e**4 + 
 2880*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e* 
*2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**3*e**5* 
x + 4320*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a 
*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**2*e* 
*6*x**2 + 2880*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)* 
sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d 
*e**7*x**3 + 720*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2 
)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2 
*e**8*x**4 + 360*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2 
)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b**2*c* 
d**4*e**4 + 1440*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2 
)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b**2*c* 
d**3*e**5*x + 2160*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x* 
*2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b**2* 
c*d**2*e**6*x**2 + 1440*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + 
 c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a* 
b**2*c*d*e**7*x**3 + 360*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x 
+ c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x...