\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx\) [759]

Optimal result

Integrand size = 48, antiderivative size = 198 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{7/2}} \]

[Out]

2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)/(e*x+d)^(7/2)/(g*x+f)^(11/2)+8/99*c*d*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)^2/(e*x+d)^(7/2)/(g*x+f)^(9/2)+16/693*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(7/2)/(g*x+f)^(7/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(13/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(11/2)) + (8*c
*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(9/2)) + (16
*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*(c*d*f - a*e*g)^3*(d + e*x)^(7/2)*(f + g*x)^(7/2)
)

Rule 874

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

Rule 886

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {(4 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx}{11 (c d f-a e g)} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx}{99 (c d f-a e g)^2} \\ & = \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (63 a^2 e^2 g^2-14 a c d e g (11 f+2 g x)+c^2 d^2 \left (99 f^2+44 f g x+8 g^2 x^2\right )\right )}{693 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{11/2}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(13/2)),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^2*g^2 - 14*a*c*d*e*g*(11*f + 2*g*x) + c^2*d^2*(99*f
^2 + 44*f*g*x + 8*g^2*x^2)))/(693*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(11/2))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-28 a c d e \,g^{2} x +44 c^{2} d^{2} f g x +63 a^{2} e^{2} g^{2}-154 a c d e f g +99 c^{2} d^{2} f^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (g x +f \right )^{\frac {11}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}\)	\(169\)
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (8 c^{4} d^{4} g^{2} x^{4}-12 a \,c^{3} d^{3} e \,g^{2} x^{3}+44 c^{4} d^{4} f g \,x^{3}+15 a^{2} c^{2} d^{2} e^{2} g^{2} x^{2}-66 a \,c^{3} d^{3} e f g \,x^{2}+99 c^{4} d^{4} f^{2} x^{2}+98 a^{3} c d \,e^{3} g^{2} x -264 a^{2} c^{2} d^{2} e^{2} f g x +198 a \,c^{3} d^{3} e \,f^{2} x +63 a^{4} e^{4} g^{2}-154 a^{3} c d \,e^{3} f g +99 a^{2} c^{2} d^{2} e^{2} f^{2}\right ) \left (c d x +a e \right )}{693 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {11}{2}} \left (a e g -c d f \right )^{3}}\)	\(231\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/693*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-28*a*c*d*e*g^2*x+44*c^2*d^2*f*g*x+63*a^2*e^2*g^2-154*a*c*d*e*f*g+99*c^2*
d^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(g*x+f)^(11/2)/(a^3*e^3*g^3-3*a^2*c*d*e^2*f*g^2+3*a*c^2*d^2*e
*f^2*g-c^3*d^3*f^3)/(e*x+d)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1101 vs. \(2 (174) = 348\).

Time = 1.08 (sec) , antiderivative size = 1101, normalized size of antiderivative = 5.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {2 \, {\left (8 \, c^{5} d^{5} g^{2} x^{5} + 99 \, a^{3} c^{2} d^{2} e^{3} f^{2} - 154 \, a^{4} c d e^{4} f g + 63 \, a^{5} e^{5} g^{2} + 4 \, {\left (11 \, c^{5} d^{5} f g - a c^{4} d^{4} e g^{2}\right )} x^{4} + {\left (99 \, c^{5} d^{5} f^{2} - 22 \, a c^{4} d^{4} e f g + 3 \, a^{2} c^{3} d^{3} e^{2} g^{2}\right )} x^{3} + {\left (297 \, a c^{4} d^{4} e f^{2} - 330 \, a^{2} c^{3} d^{3} e^{2} f g + 113 \, a^{3} c^{2} d^{2} e^{3} g^{2}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{3} e^{2} f^{2} - 418 \, a^{3} c^{2} d^{2} e^{3} f g + 161 \, a^{4} c d e^{4} g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{693 \, {\left (c^{3} d^{4} f^{9} - 3 \, a c^{2} d^{3} e f^{8} g + 3 \, a^{2} c d^{2} e^{2} f^{7} g^{2} - a^{3} d e^{3} f^{6} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{6} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{7} + 3 \, a^{2} c d e^{3} f g^{8} - a^{3} e^{4} g^{9}\right )} x^{7} + {\left (6 \, c^{3} d^{3} e f^{4} g^{5} - a^{3} d e^{3} g^{9} + {\left (c^{3} d^{4} - 18 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{6} - 3 \, {\left (a c^{2} d^{3} e - 6 \, a^{2} c d e^{3}\right )} f^{2} g^{7} + 3 \, {\left (a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f g^{8}\right )} x^{6} + 3 \, {\left (5 \, c^{3} d^{3} e f^{5} g^{4} - 2 \, a^{3} d e^{3} f g^{8} + {\left (2 \, c^{3} d^{4} - 15 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{5} - 3 \, {\left (2 \, a c^{2} d^{3} e - 5 \, a^{2} c d e^{3}\right )} f^{3} g^{6} + {\left (6 \, a^{2} c d^{2} e^{2} - 5 \, a^{3} e^{4}\right )} f^{2} g^{7}\right )} x^{5} + 5 \, {\left (4 \, c^{3} d^{3} e f^{6} g^{3} - 3 \, a^{3} d e^{3} f^{2} g^{7} + 3 \, {\left (c^{3} d^{4} - 4 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{4} - 3 \, {\left (3 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f^{4} g^{5} + {\left (9 \, a^{2} c d^{2} e^{2} - 4 \, a^{3} e^{4}\right )} f^{3} g^{6}\right )} x^{4} + 5 \, {\left (3 \, c^{3} d^{3} e f^{7} g^{2} - 4 \, a^{3} d e^{3} f^{3} g^{6} + {\left (4 \, c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{6} g^{3} - 3 \, {\left (4 \, a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{5} g^{4} + 3 \, {\left (4 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{3} e f^{8} g - 5 \, a^{3} d e^{3} f^{4} g^{5} + {\left (5 \, c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2}\right )} f^{7} g^{2} - 3 \, {\left (5 \, a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{6} g^{3} + {\left (15 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{5} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{9} - 6 \, a^{3} d e^{3} f^{5} g^{4} + 3 \, {\left (2 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{8} g - 3 \, {\left (6 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{7} g^{2} + {\left (18 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{6} g^{3}\right )} x\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(13/2),x, algorithm="fricas")

[Out]

2/693*(8*c^5*d^5*g^2*x^5 + 99*a^3*c^2*d^2*e^3*f^2 - 154*a^4*c*d*e^4*f*g + 63*a^5*e^5*g^2 + 4*(11*c^5*d^5*f*g -
a*c^4*d^4*e*g^2)*x^4 + (99*c^5*d^5*f^2 - 22*a*c^4*d^4*e*f*g + 3*a^2*c^3*d^3*e^2*g^2)*x^3 + (297*a*c^4*d^4*e*f
^2 - 330*a^2*c^3*d^3*e^2*f*g + 113*a^3*c^2*d^2*e^3*g^2)*x^2 + (297*a^2*c^3*d^3*e^2*f^2 - 418*a^3*c^2*d^2*e^3*f
*g + 161*a^4*c*d*e^4*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^3*d^4*
f^9 - 3*a*c^2*d^3*e*f^8*g + 3*a^2*c*d^2*e^2*f^7*g^2 - a^3*d*e^3*f^6*g^3 + (c^3*d^3*e*f^3*g^6 - 3*a*c^2*d^2*e^2
*f^2*g^7 + 3*a^2*c*d*e^3*f*g^8 - a^3*e^4*g^9)*x^7 + (6*c^3*d^3*e*f^4*g^5 - a^3*d*e^3*g^9 + (c^3*d^4 - 18*a*c^2
*d^2*e^2)*f^3*g^6 - 3*(a*c^2*d^3*e - 6*a^2*c*d*e^3)*f^2*g^7 + 3*(a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^8)*x^6 + 3*(5*
c^3*d^3*e*f^5*g^4 - 2*a^3*d*e^3*f*g^8 + (2*c^3*d^4 - 15*a*c^2*d^2*e^2)*f^4*g^5 - 3*(2*a*c^2*d^3*e - 5*a^2*c*d*
e^3)*f^3*g^6 + (6*a^2*c*d^2*e^2 - 5*a^3*e^4)*f^2*g^7)*x^5 + 5*(4*c^3*d^3*e*f^6*g^3 - 3*a^3*d*e^3*f^2*g^7 + 3*(
c^3*d^4 - 4*a*c^2*d^2*e^2)*f^5*g^4 - 3*(3*a*c^2*d^3*e - 4*a^2*c*d*e^3)*f^4*g^5 + (9*a^2*c*d^2*e^2 - 4*a^3*e^4)
*f^3*g^6)*x^4 + 5*(3*c^3*d^3*e*f^7*g^2 - 4*a^3*d*e^3*f^3*g^6 + (4*c^3*d^4 - 9*a*c^2*d^2*e^2)*f^6*g^3 - 3*(4*a*
c^2*d^3*e - 3*a^2*c*d*e^3)*f^5*g^4 + 3*(4*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^5)*x^3 + 3*(2*c^3*d^3*e*f^8*g - 5*a^3
*d*e^3*f^4*g^5 + (5*c^3*d^4 - 6*a*c^2*d^2*e^2)*f^7*g^2 - 3*(5*a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^6*g^3 + (15*a^2*c
*d^2*e^2 - 2*a^3*e^4)*f^5*g^4)*x^2 + (c^3*d^3*e*f^9 - 6*a^3*d*e^3*f^5*g^4 + 3*(2*c^3*d^4 - a*c^2*d^2*e^2)*f^8*
g - 3*(6*a*c^2*d^3*e - a^2*c*d*e^3)*f^7*g^2 + (18*a^2*c*d^2*e^2 - a^3*e^4)*f^6*g^3)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {13}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(13/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(13/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2167 vs. \(2 (174) = 348\).

Time = 1.77 (sec) , antiderivative size = 2167, normalized size of antiderivative = 10.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(13/2),x, algorithm="giac")

[Out]

2/693*(99*sqrt(-c*d^2*e + a*e^3)*c^5*d^8*e^2*f^2*abs(c)*abs(d) - 297*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^6*e^4*f^2*
abs(c)*abs(d) + 297*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^4*e^6*f^2*abs(c)*abs(d) - 99*sqrt(-c*d^2*e + a*e^3)*a^3*c
^2*d^2*e^8*f^2*abs(c)*abs(d) - 44*sqrt(-c*d^2*e + a*e^3)*c^5*d^9*e*f*g*abs(c)*abs(d) - 22*sqrt(-c*d^2*e + a*e^
3)*a*c^4*d^7*e^3*f*g*abs(c)*abs(d) + 330*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^5*e^5*f*g*abs(c)*abs(d) - 418*sqrt(-
c*d^2*e + a*e^3)*a^3*c^2*d^3*e^7*f*g*abs(c)*abs(d) + 154*sqrt(-c*d^2*e + a*e^3)*a^4*c*d*e^9*f*g*abs(c)*abs(d)
+ 8*sqrt(-c*d^2*e + a*e^3)*c^5*d^10*g^2*abs(c)*abs(d) + 4*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2*g^2*abs(c)*abs(
d) + 3*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4*g^2*abs(c)*abs(d) - 113*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*e^6*g
^2*abs(c)*abs(d) + 161*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8*g^2*abs(c)*abs(d) - 63*sqrt(-c*d^2*e + a*e^3)*a^5*
e^10*g^2*abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^3*e^5*f^8 - 5*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*
g)*c^3*d^4*e^4*f^7*g - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^2*e^6*f^7*g + 10*sqrt(c^2*d^2*e^2*f - c^2*d
^3*e*g)*c^3*d^5*e^3*f^6*g^2 + 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^3*e^5*f^6*g^2 + 3*sqrt(c^2*d^2*e^2*
f - c^2*d^3*e*g)*a^2*c*d*e^7*f^6*g^2 - 10*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^6*e^2*f^5*g^3 - 30*sqrt(c^2*
d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^4*e^4*f^5*g^3 - 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^2*e^6*f^5*g^3 -
sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*e^8*f^5*g^3 + 5*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^7*e*f^4*g^4 + 30
*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^5*e^3*f^4*g^4 + 30*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^3*e^5*
f^4*g^4 + 5*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*d*e^7*f^4*g^4 - sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^3*d^8*f^
3*g^5 - 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^2*d^6*e^2*f^3*g^5 - 30*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*
c*d^4*e^4*f^3*g^5 - 10*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*d^2*e^6*f^3*g^5 + 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*
e*g)*a*c^2*d^7*e*f^2*g^6 + 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^5*e^3*f^2*g^6 + 10*sqrt(c^2*d^2*e^2*f
- c^2*d^3*e*g)*a^3*d^3*e^5*f^2*g^6 - 3*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c*d^6*e^2*f*g^7 - 5*sqrt(c^2*d^2*
e^2*f - c^2*d^3*e*g)*a^3*d^4*e^4*f*g^7 + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*d^5*e^3*g^8) + 2/693*((e*x + d)
*c*d*e - c*d^2*e + a*e^3)^(7/2)*(4*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*(2*(c^12*d^12*e^10*f^2*g^7*abs(c)*abs(d
) - 2*a*c^11*d^11*e^11*f*g^8*abs(c)*abs(d) + a^2*c^10*d^10*e^12*g^9*abs(c)*abs(d))*((e*x + d)*c*d*e - c*d^2*e
+ a*e^3)/(c^5*d^5*e^10*f^5*g^5 - 5*a*c^4*d^4*e^11*f^4*g^6 + 10*a^2*c^3*d^3*e^12*f^3*g^7 - 10*a^3*c^2*d^2*e^13*
f^2*g^8 + 5*a^4*c*d*e^14*f*g^9 - a^5*e^15*g^10) + 11*(c^13*d^13*e^12*f^3*g^6*abs(c)*abs(d) - 3*a*c^12*d^12*e^1
3*f^2*g^7*abs(c)*abs(d) + 3*a^2*c^11*d^11*e^14*f*g^8*abs(c)*abs(d) - a^3*c^10*d^10*e^15*g^9*abs(c)*abs(d))/(c^
5*d^5*e^10*f^5*g^5 - 5*a*c^4*d^4*e^11*f^4*g^6 + 10*a^2*c^3*d^3*e^12*f^3*g^7 - 10*a^3*c^2*d^2*e^13*f^2*g^8 + 5*
a^4*c*d*e^14*f*g^9 - a^5*e^15*g^10)) + 99*(c^14*d^14*e^14*f^4*g^5*abs(c)*abs(d) - 4*a*c^13*d^13*e^15*f^3*g^6*a
bs(c)*abs(d) + 6*a^2*c^12*d^12*e^16*f^2*g^7*abs(c)*abs(d) - 4*a^3*c^11*d^11*e^17*f*g^8*abs(c)*abs(d) + a^4*c^1
0*d^10*e^18*g^9*abs(c)*abs(d))/(c^5*d^5*e^10*f^5*g^5 - 5*a*c^4*d^4*e^11*f^4*g^6 + 10*a^2*c^3*d^3*e^12*f^3*g^7
- 10*a^3*c^2*d^2*e^13*f^2*g^8 + 5*a^4*c*d*e^14*f*g^9 - a^5*e^15*g^10))/(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e*x +
d)*c*d*e - c*d^2*e + a*e^3)*c*d*g)^(11/2)

Mupad [B] (verification not implemented)

Time = 13.49 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {126\,a^5\,e^5\,g^2-308\,a^4\,c\,d\,e^4\,f\,g+198\,a^3\,c^2\,d^2\,e^3\,f^2}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x^3\,\left (6\,a^2\,c^3\,d^3\,e^2\,g^2-44\,a\,c^4\,d^4\,e\,f\,g+198\,c^5\,d^5\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^5\,d^5\,x^5}{693\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^4\,d^4\,x^4\,\left (a\,e\,g-11\,c\,d\,f\right )}{693\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (161\,a^2\,e^2\,g^2-418\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (113\,a^2\,e^2\,g^2-330\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {5\,f\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {5\,f^4\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {10\,f^2\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {10\,f^3\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^(13/2)*(d + e*x)^(5/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((126*a^5*e^5*g^2 + 198*a^3*c^2*d^2*e^3*f^2 - 308*a^4*c*d*e^4*
f*g)/(693*g^5*(a*e*g - c*d*f)^3) + (x^3*(198*c^5*d^5*f^2 + 6*a^2*c^3*d^3*e^2*g^2 - 44*a*c^4*d^4*e*f*g))/(693*g
^5*(a*e*g - c*d*f)^3) + (16*c^5*d^5*x^5)/(693*g^3*(a*e*g - c*d*f)^3) - (8*c^4*d^4*x^4*(a*e*g - 11*c*d*f))/(693
*g^4*(a*e*g - c*d*f)^3) + (2*a^2*c*d*e^2*x*(161*a^2*e^2*g^2 + 297*c^2*d^2*f^2 - 418*a*c*d*e*f*g))/(693*g^5*(a*
e*g - c*d*f)^3) + (2*a*c^2*d^2*e*x^2*(113*a^2*e^2*g^2 + 297*c^2*d^2*f^2 - 330*a*c*d*e*f*g))/(693*g^5*(a*e*g -
c*d*f)^3)))/(x^5*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^5*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^5 + (5*f*x^4*(f + g
*x)^(1/2)*(d + e*x)^(1/2))/g + (5*f^4*x*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^4 + (10*f^2*x^3*(f + g*x)^(1/2)*(d
+ e*x)^(1/2))/g^2 + (10*f^3*x^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3)