\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\) [649]

Optimal result

Integrand size = 24, antiderivative size = 696 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/63*(e*x+d)^(1/2)*(128*c^3*d^3-b^3*e^3+3*b*c*e^2*(-36*a*e+37*b*d)-12*c^2 
*d*e*(-11*a*e+20*b*d)-3*c*e*(32*c^2*d^2+b^2*e^2-4*c*e*(-7*a*e+8*b*d))*x)*( 
c*x^2+b*x+a)^(1/2)/c/e^5-10/63*(e*x+d)^(1/2)*(-14*c*e*x-15*b*e+16*c*d)*(c* 
x^2+b*x+a)^(3/2)/e^3-2*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(1/2)+2/63*2^(1/2)*(- 
4*a*c+b^2)^(1/2)*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-57*a*e+64*b*d)-b^2*c*e 
^3*(-15*a*e+7*b*d)+3*c^2*e^2*(28*a^2*e^2-76*a*b*d*e+45*b^2*d^2))*(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))/c^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)-2/63*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(a 
*e^2-b*d*e+c*d^2)*(128*c^2*d^2-b^2*e^2-4*c*e*(-33*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)*EllipticF(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^2/e^6/(e*x+d)^( 
1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 35.19 (sec) , antiderivative size = 7946, normalized size of antiderivative = 11.42 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 737, 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, 1231, 27, 1231, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5 \left (-\frac {2 \int \frac {c \left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e-\left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (-\frac {\int \frac {\left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e-\left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {2 \int \frac {\left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right )+5 c e (b d-2 a e) \left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e\right )+2 \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\int \frac {\left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right )+5 c e (b d-2 a e) \left (15 d e b^2-16 \left (c d^2+a e^2\right ) b+4 a c d e\right )+2 \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\frac {2 \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {5 \left (-\frac {2 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (c x^2+b x+a\right )^{3/2}}{63 e^2}-\frac {\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-12 c^2 e (20 b d-11 a e) d-b^3 e^3+3 b c e^2 (37 b d-36 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d-b^2 e^2+132 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{21 e^2}\right )}{e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {5 \left (-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 c e^2}}{21 e^2}-\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]
 

Output:

(-2*(a + b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (5*((-2*Sqrt[d + e*x]*(16 
*c*d - 15*b*e - 14*c*e*x)*(a + b*x + c*x^2)^(3/2))/(63*e^2) - ((2*Sqrt[d + 
 e*x]*(128*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(37*b*d - 36*a*e) - 12*c^2*d*e*(2 
0*b*d - 11*a*e) - 3*c*e*(32*c^2*d^2 + b^2*e^2 - 4*c*e*(8*b*d - 7*a*e))*x)* 
Sqrt[a + b*x + c*x^2])/(15*c*e^2) - ((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(128*c^4 
*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e 
) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*e^2))*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)])/(c*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)*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - 128*b*c*d*e - b^2 
*e^2 + 132*a*c*e^2)*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)/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)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b 
*x + c*x^2]))/(15*c*e^2))/(21*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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2071\) vs. \(2(632)=1264\).

Time = 9.79 (sec) , antiderivative size = 2072, normalized size of antiderivative = 2.98

method	result	size
risch	\(\text {Expression too large to display}\)	\(2072\)
elliptic	\(\text {Expression too large to display}\)	\(2528\)
default	\(\text {Expression too large to display}\)	\(9187\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/63/c*(7*c^3*e^3*x^3+19*b*c^2*e^3*x^2-17*c^3*d*e^2*x^2+28*a*c^2*e^3*x+15* 
b^2*c*e^3*x-50*b*c^2*d*e^2*x+33*c^3*d^2*e*x+57*a*b*c*e^3-86*a*c^2*d*e^2+b^ 
3*e^3-48*b^2*c*d*e^2+114*b*c^2*d^2*e-65*c^3*d^3)*(e*x+d)^(1/2)*(c*x^2+b*x+ 
a)^(1/2)/e^5+1/63/e^5/c*(2*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e 
^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^ 
3*d^3*e+193*c^4*d^4)*(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)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((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))+1/2/c*(-b+(-4*a*c+b^2)^(1/ 
2))*EllipticF(((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*(132*a^2*b*c*e^5-159*a^2*c^2*d*e^4-a*b^3*e^5-228*a*b^2*c*d*e^4+450*a*b*c 
^2*d^2*e^3-190*a*c^3*d^3*e^2-b^4*d*e^4+111*b^3*c*d^2*e^3-303*b^2*c^2*d^3*e 
^2+254*b*c^3*d^4*e-63*c^4*d^5)/e*(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...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")
 

Output:

-2/189*((256*c^5*d^6 - 640*b*c^4*d^5*e + 2*(239*b^2*c^3 + 324*a*c^4)*d^4*e 
^2 - (77*b^3*c^2 + 972*a*b*c^3)*d^3*e^3 - (13*b^4*c - 258*a*b^2*c^2 - 456* 
a^2*c^3)*d^2*e^4 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d*e^5 + (256*c^5*d 
^5*e - 640*b*c^4*d^4*e^2 + 2*(239*b^2*c^3 + 324*a*c^4)*d^3*e^3 - (77*b^3*c 
^2 + 972*a*b*c^3)*d^2*e^4 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d*e^5 
 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*e^6)*x)*sqrt(c*e)*weierstrassPInve 
rse(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^5*d^5*e - 256*b*c^4*d^ 
4*e^2 + 3*(45*b^2*c^3 + 76*a*c^4)*d^3*e^3 - (7*b^3*c^2 + 228*a*b*c^3)*d^2* 
e^4 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d*e^5 + (128*c^5*d^4*e^2 - 256*b 
*c^4*d^3*e^3 + 3*(45*b^2*c^3 + 76*a*c^4)*d^2*e^4 - (7*b^3*c^2 + 228*a*b*c^ 
3)*d*e^5 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*e^6)*x)*sqrt(c*e)*weierstra 
ssZeta(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), 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))) - 3*( 
7*c^5*e^6*x^4 - 128*c^5*d^4*e^2 + 240*b*c^4*d^3*e^3 - 63*a^2*c^3*e^6 - (11 
1*b^2*c^3 + 212*a*c^4)*d^2*e^4 + (b^3*c^2 + 183*a*b*c^3)*d*e^5 - (10*c^...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(3/2),x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{\left (e x +d \right )^{\frac {3}{2}}}d x \] Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x)
 

Output:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x)