\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [646]

Optimal result

Integrand size = 24, antiderivative size = 570 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {a+b x+c x^2}}{5 e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {2 (8 c d-b e+6 c e x) \sqrt {a+b x+c x^2}}{5 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {16 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.07 (sec) , antiderivative size = 1269, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x]
 

Output:

(Sqrt[d + e*x]*(a + x*(b + c*x))^(3/2)*((-2*(c*d^2 - b*d*e + a*e^2))/(5*e^ 
3*(d + e*x)^3) + (4*(2*c*d - b*e))/(5*e^3*(d + e*x)^2) - (2*(11*c^2*d^2 - 
11*b*c*d*e + b^2*e^2 + 7*a*c*e^2))/(5*e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x 
))))/(a + b*x + c*x^2) - ((d + e*x)^(3/2)*(a + x*(b + c*x))^(3/2)*(-4*Sqrt 
[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16* 
c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*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])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*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]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 
4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 
 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*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*(-1 
 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqr...
 

Rubi [A] (warning: unable to verify)

Time = 0.90 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1161, 1229, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {3 \left (-\frac {2 \int \frac {c \left (7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (-\frac {c \int \frac {7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {3 \left (-\frac {c \left (\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {3 \left (-\frac {c \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {3 \left (-\frac {c \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {3 \left (-\frac {c \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{5 e}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x]
 

Output:

(-2*(a + b*x + c*x^2)^(3/2))/(5*e*(d + e*x)^(5/2)) + (3*((-2*(8*c^2*d^3 + 
a*b*e^3 - c*d*e*(7*b*d - 4*a*e) + e*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 
 3*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x 
)^(3/2)) - (c*(-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*c^2*d^2 + b^2*e^2 - 4*c*e* 
(4*b*d - 3*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)])/ 
(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])) + (16*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + 
a*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])) 
)/(3*e^2*(c*d^2 - b*d*e + a*e^2))))/(5*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 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 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. \(1152\) vs. \(2(504)=1008\).

Time = 7.42 (sec) , antiderivative size = 1153, normalized size of antiderivative = 2.02

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1153\)
default	\(\text {Expression too large to display}\)	\(12944\)

Input:

int((c*x^2+b*x+a)^(3/2)/(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*e 
^2-b*d*e+c*d^2)/e^6*(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 
)^3-4/5*(b*e-2*c*d)/e^5*(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)^2-2/5*(c*e*x^2+b*e*x+a*e)/e^4/(a*e^2-b*d*e+c*d^2)*(7*a*c*e^2+b^2*e^2 
-11*b*c*d*e+11*c^2*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(c*(2*b*e-3* 
c*d)/e^4-2/5*c*(b*e-2*c*d)/e^4-1/5/e^4*(b*e-c*d)*(7*a*c*e^2+b^2*e^2-11*b*c 
*d*e+11*c^2*d^2)/(a*e^2-b*d*e+c*d^2)+1/5*b/e^3/(a*e^2-b*d*e+c*d^2)*(7*a*c* 
e^2+b^2*e^2-11*b*c*d*e+11*c^2*d^2))*(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)*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*(c^2/e^3+1/5*c/e^3*(7*a*c*e^2+b^2*e^2-11*b*c*d* 
e+11*c^2*d^2)/(a*e^2-b*d*e+c*d^2))*(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)))*Ellipti 
cE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1032 vs. \(2 (512) = 1024\).

Time = 0.11 (sec) , antiderivative size = 1032, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

-2/15*((16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(b^2*c + 4*a*c^2)*d^4*e^2 + (b^3 - 
 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*(b^2*c + 4*a*c 
^2)*d*e^5 + (b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e 
^3 + 6*(b^2*c + 4*a*c^2)*d^2*e^4 + (b^3 - 12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3 
*d^5*e - 24*b*c^2*d^4*e^2 + 6*(b^2*c + 4*a*c^2)*d^3*e^3 + (b^3 - 12*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)) + 3*(16*c^3*d^5*e - 16*b*c^2*d^4*e^2 + (b^2*c + 12*a*c^2)*d^3*e^3 + 
(16*c^3*d^2*e^4 - 16*b*c^2*d*e^5 + (b^2*c + 12*a*c^2)*e^6)*x^3 + 3*(16*c^3 
*d^3*e^3 - 16*b*c^2*d^2*e^4 + (b^2*c + 12*a*c^2)*d*e^5)*x^2 + 3*(16*c^3*d^ 
4*e^2 - 16*b*c^2*d^3*e^3 + (b^2*c + 12*a*c^2)*d^2*e^4)*x)*sqrt(c*e)*weiers 
trassZeta(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*(8*c^3*d^4*e^2 - 7*b*c^2*d^3*e^3 + 5*a*c^2*d^2*e^4 + a^2*c*e^6 + (11*c^3 
*d^2*e^4 - 11*b*c^2*d*e^5 + (b^2*c + 7*a*c^2)*e^6)*x^2 + 2*(9*c^3*d^3*e^3 
- 8*b*c^2*d^2*e^4 + 5*a*c^2*d*e^5 + a*b*c*e^6)*x)*sqrt(c*x^2 + b*x + a)...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**(7/2),x)
 

Output:

Integral((a + b*x + c*x**2)**(3/2)/(d + e*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{\left (e x +d \right )^{\frac {7}{2}}}d x \] Input:

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x)
 

Output:

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^(7/2),x)