\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\) [651]

Optimal result

Integrand size = 24, antiderivative size = 583 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+15 b^2 e^2-4 c e (32 b d-17 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/15*(128*c^2*d^2+15*b^2*e^2-4*c*e*(-9*a*e+28*b*d)+16*c*e*(-b*e+2*c*d)*x) 
*(c*x^2+b*x+a)^(1/2)/e^5/(e*x+d)^(1/2)+2/15*(6*c*e*x-5*b*e+16*c*d)*(c*x^2+ 
b*x+a)^(3/2)/e^3/(e*x+d)^(3/2)-2/5*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(5/2)+2/1 
5*2^(1/2)*(-4*a*c+b^2)^(1/2)*(128*c^2*d^2+23*b^2*e^2-4*c*e*(-9*a*e+32*b*d) 
)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2* 
c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-( 
b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/e^6/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/ 
2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)-2/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2* 
c*d)*(128*c^2*d^2+15*b^2*e^2-4*c*e*(-17*a*e+32*b*d))*(c*(e*x+d)/(2*c*d-(b+ 
(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*Ellipt 
icF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1 
/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c/e^6/(e*x+d)^(1/2)/(c*x^2+ 
b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.21 (sec) , antiderivative size = 1283, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x]
 

Output:

(Sqrt[d + e*x]*(a + x*(b + c*x))^(5/2)*((2*c*(-19*c*d + 11*b*e))/(15*e^5) 
+ (2*c^2*x)/(5*e^4) - (2*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^5*(d + e*x)^3) - 
(22*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2))/(15*e^5*(d + e*x)^2) - (2*(128 
*c^2*d^2 - 128*b*c*d*e + 23*b^2*e^2 + 36*a*c*e^2))/(15*e^5*(d + e*x))))/(a 
 + b*x + c*x^2)^2 - ((d + e*x)^(3/2)*(a + x*(b + c*x))^(5/2)*(-4*Sqrt[(c*d 
^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(128*c^2* 
d^2 + 23*b^2*e^2 + 4*c*e*(-32*b*d + 9*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*( 
b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + (I*Sqrt[2]*(2*c*d - b 
*e + Sqrt[(b^2 - 4*a*c)*e^2])*(128*c^2*d^2 + 23*b^2*e^2 + 4*c*e*(-32*b*d + 
 9*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + 
 d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a 
*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 
 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4 
*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2 
*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + 
Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d 
 + e*x] - (I*Sqrt[2]*(-8*b^3*e^3 + b^2*e^2*(16*c*d + 23*Sqrt[(b^2 - 4*a*c) 
*e^2]) + 32*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(32*c*d^2* 
Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-16*c*d + 9*Sqrt[(b^2 - 4*a*c)*e^2])))*Sq 
rt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + ...
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1161, 1230, 27, 1230, 27, 1269, 1172, 321, 327}

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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {\left (-5 e b^2+16 c d b-12 a c e+16 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^{3/2}}dx}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\int \frac {\left (-5 e b^2+16 c d b-12 a c e+16 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{3/2}}dx}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {2 \int -\frac {-15 e^2 b^3+112 c d e b^2-4 c \left (32 c d^2+17 a e^2\right ) b+64 a c^2 d e-2 c \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\int \frac {-15 e^2 b^3+112 c d e b^2-4 c \left (32 c d^2+17 a e^2\right ) b+64 a c^2 d e-2 c \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {(2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 c \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x]
 

Output:

(-2*(a + b*x + c*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) + ((2*(16*c*d - 5*b*e + 
 6*c*e*x)*(a + b*x + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) - ((2*(128*c^2 
*d^2 + 15*b^2*e^2 - 4*c*e*(28*b*d - 9*a*e) + 16*c*e*(2*c*d - b*e)*x)*Sqrt[ 
a + b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) + ((-2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*( 
128*c^2*d^2 + 23*b^2*e^2 - 4*c*e*(32*b*d - 9*a*e))*Sqrt[d + e*x]*Sqrt[-((c 
*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 
4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c 
*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt 
[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*( 
2*c*d - b*e)*(128*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(32*b*d - 17*a*e))*Sqrt[(c* 
(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2 
))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/S 
qrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^ 
2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]))/(3*e^2))/(5*e^ 
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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

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 1172

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

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. \(1496\) vs. \(2(517)=1034\).

Time = 15.31 (sec) , antiderivative size = 1497, normalized size of antiderivative = 2.57

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1497\)
risch	\(\text {Expression too large to display}\)	\(4069\)
default	\(\text {Expression too large to display}\)	\(14453\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/5*(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)/e^8*(c*e*x 
^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^3-22/15*(a*b*e^3-2*a*c*d 
*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^7*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b 
*d*x+a*d)^(1/2)/(x+d/e)^2-2/15*(c*e*x^2+b*e*x+a*e)*(36*a*c*e^2+23*b^2*e^2- 
128*b*c*d*e+128*c^2*d^2)/e^6/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2/5*c^2/e 
^4*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(3*c^2/e^4*(b*e-c 
*d)-2/5*c^2/e^4*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a* 
d)^(1/2)+2*((6*a*b*c*e^3-9*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+18*b*c^2*d^2* 
e-10*c^3*d^3)/e^6-11/15*c*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2 
*d^3)/e^6-1/15*(36*a*c*e^2+23*b^2*e^2-128*b*c*d*e+128*c^2*d^2)/e^6*(b*e-c* 
d)+1/15*b/e^5*(36*a*c*e^2+23*b^2*e^2-128*b*c*d*e+128*c^2*d^2)-2/5*c^2/e^4* 
a*d-2/3*(3*c^2/e^4*(b*e-c*d)-2/5*c^2/e^4*(2*b*e+2*c*d))/c/e*(1/2*a*e+1/2*b 
*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^ 
(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a* 
c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a 
*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*E 
llipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+ 
(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(3*c 
/e^5*(a*c*e^2+b^2*e^2-3*b*c*d*e+2*c^2*d^2)+1/15*(36*a*c*e^2+23*b^2*e^2-...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 1018, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

-2/45*((256*c^3*d^6 - 384*b*c^2*d^5*e + 6*(21*b^2*c + 44*a*c^2)*d^4*e^2 + 
(b^3 - 132*a*b*c)*d^3*e^3 + (256*c^3*d^3*e^3 - 384*b*c^2*d^2*e^4 + 6*(21*b 
^2*c + 44*a*c^2)*d*e^5 + (b^3 - 132*a*b*c)*e^6)*x^3 + 3*(256*c^3*d^4*e^2 - 
 384*b*c^2*d^3*e^3 + 6*(21*b^2*c + 44*a*c^2)*d^2*e^4 + (b^3 - 132*a*b*c)*d 
*e^5)*x^2 + 3*(256*c^3*d^5*e - 384*b*c^2*d^4*e^2 + 6*(21*b^2*c + 44*a*c^2) 
*d^3*e^3 + (b^3 - 132*a*b*c)*d^2*e^4)*x)*sqrt(c*e)*weierstrassPInverse(4/3 
*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b 
*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 
 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 6*(128*c^3*d^5*e - 128*b*c^2*d^4*e^2 + 
 (23*b^2*c + 36*a*c^2)*d^3*e^3 + (128*c^3*d^2*e^4 - 128*b*c^2*d*e^5 + (23* 
b^2*c + 36*a*c^2)*e^6)*x^3 + 3*(128*c^3*d^3*e^3 - 128*b*c^2*d^2*e^4 + (23* 
b^2*c + 36*a*c^2)*d*e^5)*x^2 + 3*(128*c^3*d^4*e^2 - 128*b*c^2*d^3*e^3 + (2 
3*b^2*c + 36*a*c^2)*d^2*e^4)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b 
*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 
3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassP 
Inverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^ 
3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3) 
/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) - 3*(3*c^3*e^6*x^4 - 128*c^3 
*d^4*e^2 + 112*b*c^2*d^3*e^3 - 5*a*b*c*d*e^5 - 3*a^2*c*e^6 - 5*(3*b^2*c + 
4*a*c^2)*d^2*e^4 - (10*c^3*d*e^5 - 11*b*c^2*e^6)*x^3 - (176*c^3*d^2*e^4...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(7/2),x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{\left (e x +d \right )^{\frac {7}{2}}}d x \] Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(7/2),x)
 

Output:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(7/2),x)