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

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

[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)

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Rubi [A]
time = 0.16, 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 = {886, 874} \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 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*} \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 = 0.22, size = 113, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^3 ((a e+c d x) (d+e x))^{5/2} \left (63 g^2-\frac {154 c d g (f+g x)}{a e+c d x}+\frac {99 c^2 d^2 (f+g x)^2}{(a e+c d x)^2}\right )}{693 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{11/2}} \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*((a*e + c*d*x)*(d + e*x))^(5/2)*(63*g^2 - (154*c*d*g*(f + g*x))/(a*e + c*d*x) + (99*c^2*d^2
*(f + g*x)^2)/(a*e + c*d*x)^2))/(693*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(11/2))

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Maple [A]
time = 0.15, size = 231, normalized size = 1.17

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

Verification of antiderivative is not currently implemented for this CAS.

[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)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(11/2)*(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)*(c*d*x+a
*e)/(a*e*g-c*d*f)^3

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((g*x + f)^(13/2)*(x*e + d)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (183) = 366\).
time = 0.93, size = 1159, normalized size = 5.85 \begin {gather*} \frac {2 \, {\left (8 \, c^{5} d^{5} g^{2} x^{5} + 44 \, c^{5} d^{5} f g x^{4} + 99 \, c^{5} d^{5} f^{2} x^{3} + 63 \, a^{5} g^{2} e^{5} + 7 \, {\left (23 \, a^{4} c d g^{2} x - 22 \, a^{4} c d f g\right )} e^{4} + {\left (113 \, a^{3} c^{2} d^{2} g^{2} x^{2} - 418 \, a^{3} c^{2} d^{2} f g x + 99 \, a^{3} c^{2} d^{2} f^{2}\right )} e^{3} + 3 \, {\left (a^{2} c^{3} d^{3} g^{2} x^{3} - 110 \, a^{2} c^{3} d^{3} f g x^{2} + 99 \, a^{2} c^{3} d^{3} f^{2} x\right )} e^{2} - {\left (4 \, a c^{4} d^{4} g^{2} x^{4} + 22 \, a c^{4} d^{4} f g x^{3} - 297 \, a c^{4} d^{4} f^{2} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{693 \, {\left (c^{3} d^{4} f^{3} g^{6} x^{6} + 6 \, c^{3} d^{4} f^{4} g^{5} x^{5} + 15 \, c^{3} d^{4} f^{5} g^{4} x^{4} + 20 \, c^{3} d^{4} f^{6} g^{3} x^{3} + 15 \, c^{3} d^{4} f^{7} g^{2} x^{2} + 6 \, c^{3} d^{4} f^{8} g x + c^{3} d^{4} f^{9} - {\left (a^{3} g^{9} x^{7} + 6 \, a^{3} f g^{8} x^{6} + 15 \, a^{3} f^{2} g^{7} x^{5} + 20 \, a^{3} f^{3} g^{6} x^{4} + 15 \, a^{3} f^{4} g^{5} x^{3} + 6 \, a^{3} f^{5} g^{4} x^{2} + a^{3} f^{6} g^{3} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{8} x^{7} - a^{3} d f^{6} g^{3} + {\left (18 \, a^{2} c d f^{2} g^{7} - a^{3} d g^{9}\right )} x^{6} + 3 \, {\left (15 \, a^{2} c d f^{3} g^{6} - 2 \, a^{3} d f g^{8}\right )} x^{5} + 15 \, {\left (4 \, a^{2} c d f^{4} g^{5} - a^{3} d f^{2} g^{7}\right )} x^{4} + 5 \, {\left (9 \, a^{2} c d f^{5} g^{4} - 4 \, a^{3} d f^{3} g^{6}\right )} x^{3} + 3 \, {\left (6 \, a^{2} c d f^{6} g^{3} - 5 \, a^{3} d f^{4} g^{5}\right )} x^{2} + 3 \, {\left (a^{2} c d f^{7} g^{2} - 2 \, a^{3} d f^{5} g^{4}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{7} x^{7} - a^{2} c d^{2} f^{7} g^{2} + {\left (6 \, a c^{2} d^{2} f^{3} g^{6} - a^{2} c d^{2} f g^{8}\right )} x^{6} + 3 \, {\left (5 \, a c^{2} d^{2} f^{4} g^{5} - 2 \, a^{2} c d^{2} f^{2} g^{7}\right )} x^{5} + 5 \, {\left (4 \, a c^{2} d^{2} f^{5} g^{4} - 3 \, a^{2} c d^{2} f^{3} g^{6}\right )} x^{4} + 5 \, {\left (3 \, a c^{2} d^{2} f^{6} g^{3} - 4 \, a^{2} c d^{2} f^{4} g^{5}\right )} x^{3} + 3 \, {\left (2 \, a c^{2} d^{2} f^{7} g^{2} - 5 \, a^{2} c d^{2} f^{5} g^{4}\right )} x^{2} + {\left (a c^{2} d^{2} f^{8} g - 6 \, a^{2} c d^{2} f^{6} g^{3}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{6} x^{7} - 3 \, a c^{2} d^{3} f^{8} g + 3 \, {\left (2 \, c^{3} d^{3} f^{4} g^{5} - a c^{2} d^{3} f^{2} g^{7}\right )} x^{6} + 3 \, {\left (5 \, c^{3} d^{3} f^{5} g^{4} - 6 \, a c^{2} d^{3} f^{3} g^{6}\right )} x^{5} + 5 \, {\left (4 \, c^{3} d^{3} f^{6} g^{3} - 9 \, a c^{2} d^{3} f^{4} g^{5}\right )} x^{4} + 15 \, {\left (c^{3} d^{3} f^{7} g^{2} - 4 \, a c^{2} d^{3} f^{5} g^{4}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{3} f^{8} g - 15 \, a c^{2} d^{3} f^{6} g^{3}\right )} x^{2} + {\left (c^{3} d^{3} f^{9} - 18 \, a c^{2} d^{3} f^{7} g^{2}\right )} x\right )} e\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 + 44*c^5*d^5*f*g*x^4 + 99*c^5*d^5*f^2*x^3 + 63*a^5*g^2*e^5 + 7*(23*a^4*c*d*g^2*x - 22
*a^4*c*d*f*g)*e^4 + (113*a^3*c^2*d^2*g^2*x^2 - 418*a^3*c^2*d^2*f*g*x + 99*a^3*c^2*d^2*f^2)*e^3 + 3*(a^2*c^3*d^
3*g^2*x^3 - 110*a^2*c^3*d^3*f*g*x^2 + 99*a^2*c^3*d^3*f^2*x)*e^2 - (4*a*c^4*d^4*g^2*x^4 + 22*a*c^4*d^4*f*g*x^3
- 297*a*c^4*d^4*f^2*x^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(c^3*d^4*f
^3*g^6*x^6 + 6*c^3*d^4*f^4*g^5*x^5 + 15*c^3*d^4*f^5*g^4*x^4 + 20*c^3*d^4*f^6*g^3*x^3 + 15*c^3*d^4*f^7*g^2*x^2
+ 6*c^3*d^4*f^8*g*x + c^3*d^4*f^9 - (a^3*g^9*x^7 + 6*a^3*f*g^8*x^6 + 15*a^3*f^2*g^7*x^5 + 20*a^3*f^3*g^6*x^4 +
15*a^3*f^4*g^5*x^3 + 6*a^3*f^5*g^4*x^2 + a^3*f^6*g^3*x)*e^4 + (3*a^2*c*d*f*g^8*x^7 - a^3*d*f^6*g^3 + (18*a^2*
c*d*f^2*g^7 - a^3*d*g^9)*x^6 + 3*(15*a^2*c*d*f^3*g^6 - 2*a^3*d*f*g^8)*x^5 + 15*(4*a^2*c*d*f^4*g^5 - a^3*d*f^2*
g^7)*x^4 + 5*(9*a^2*c*d*f^5*g^4 - 4*a^3*d*f^3*g^6)*x^3 + 3*(6*a^2*c*d*f^6*g^3 - 5*a^3*d*f^4*g^5)*x^2 + 3*(a^2*
c*d*f^7*g^2 - 2*a^3*d*f^5*g^4)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^7*x^7 - a^2*c*d^2*f^7*g^2 + (6*a*c^2*d^2*f^3*g^6 -
a^2*c*d^2*f*g^8)*x^6 + 3*(5*a*c^2*d^2*f^4*g^5 - 2*a^2*c*d^2*f^2*g^7)*x^5 + 5*(4*a*c^2*d^2*f^5*g^4 - 3*a^2*c*d^
2*f^3*g^6)*x^4 + 5*(3*a*c^2*d^2*f^6*g^3 - 4*a^2*c*d^2*f^4*g^5)*x^3 + 3*(2*a*c^2*d^2*f^7*g^2 - 5*a^2*c*d^2*f^5*
g^4)*x^2 + (a*c^2*d^2*f^8*g - 6*a^2*c*d^2*f^6*g^3)*x)*e^2 + (c^3*d^3*f^3*g^6*x^7 - 3*a*c^2*d^3*f^8*g + 3*(2*c^
3*d^3*f^4*g^5 - a*c^2*d^3*f^2*g^7)*x^6 + 3*(5*c^3*d^3*f^5*g^4 - 6*a*c^2*d^3*f^3*g^6)*x^5 + 5*(4*c^3*d^3*f^6*g^
3 - 9*a*c^2*d^3*f^4*g^5)*x^4 + 15*(c^3*d^3*f^7*g^2 - 4*a*c^2*d^3*f^5*g^4)*x^3 + 3*(2*c^3*d^3*f^8*g - 15*a*c^2*
d^3*f^6*g^3)*x^2 + (c^3*d^3*f^9 - 18*a*c^2*d^3*f^7*g^2)*x)*e)

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Sympy [F(-1)] Timed out
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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Giac [F(-1)] Timed out
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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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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