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

Optimal. Leaf size=198 \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)} \]

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Rubi [A]  time = 0.23, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[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 860

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 872

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*} \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 {(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*}

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Mathematica [A]  time = 0.11, size = 115, normalized size = 0.58 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d 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 \sqrt {d+e x} (f+g x)^{11/2} (c d f-a e g)^3} \end {gather*}

Antiderivative was successfully verified.

[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))

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IntegrateAlgebraic [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(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]

$Aborted

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fricas [B]  time = 0.48, size = 1101, normalized size = 5.56 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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maple [A]  time = 0.01, size = 169, normalized size = 0.85 \begin {gather*} -\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 f^{2} c^{2} d^{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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.82, size = 465, normalized size = 2.35 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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