3.8.50 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^{13/2}} \, dx\) [750]

Optimal. Leaf size=267 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{5/2}} \]

[Out]

2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)/(e*x+d)^(5/2)/(g*x+f)^(11/2)+4/33*c*d*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)^2/(e*x+d)^(5/2)/(g*x+f)^(9/2)+16/231*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(5/2)/(g*x+f)^(7/2)+32/1155*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(5/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(5/2)/(g*x+f)^(5/2)

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Rubi [A]
time = 0.22, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^4}+\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 )^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^{5/2} (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 )^{5/2}}{11 (d+e x)^{5/2} (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)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(13/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(11/2)) + (4*c
*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(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)^(5/2))/(231*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(7/2)
) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1155*(c*d*f - a*e*g)^4*(d + e*x)^(5/2)*(f + g*
x)^(5/2))

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 )^{3/2}}{(d+e x)^{3/2} (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 )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {(6 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (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 )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (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 )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx}{33 (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 )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (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 )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {\left (16 c^3 d^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx}{231 (c d f-a e g)^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (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 )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 141, normalized size = 0.53 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{3/2} \left (-105 g^3+\frac {385 c d g^2 (f+g x)}{a e+c d x}-\frac {495 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {231 c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{1155 (c d f-a e g)^4 (d+e x)^{3/2} (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)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(13/2)),x]

[Out]

(2*(a*e + c*d*x)^4*((a*e + c*d*x)*(d + e*x))^(3/2)*(-105*g^3 + (385*c*d*g^2*(f + g*x))/(a*e + c*d*x) - (495*c^
2*d^2*g*(f + g*x)^2)/(a*e + c*d*x)^2 + (231*c^3*d^3*(f + g*x)^3)/(a*e + c*d*x)^3))/(1155*(c*d*f - a*e*g)^4*(d
+ e*x)^(3/2)*(f + g*x)^(11/2))

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Maple [A]
time = 0.14, size = 267, normalized size = 1.00

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+40 a \,c^{2} d^{2} e \,g^{3} x^{2}-88 c^{3} d^{3} f \,g^{2} x^{2}-70 a^{2} c d \,e^{2} g^{3} x +220 a \,c^{2} d^{2} e f \,g^{2} x -198 c^{3} d^{3} f^{2} g x +105 a^{3} e^{3} g^{3}-385 a^{2} c d \,e^{2} f \,g^{2}+495 a \,c^{2} d^{2} e \,f^{2} g -231 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (g x +f \right )^{\frac {11}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}\)	\(260\)
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{4} c^{4} d^{4}+24 a \,c^{3} d^{3} e \,g^{3} x^{3}-88 c^{4} d^{4} f \,g^{2} x^{3}-30 a^{2} c^{2} d^{2} e^{2} g^{3} x^{2}+132 a \,c^{3} d^{3} e f \,g^{2} x^{2}-198 c^{4} d^{4} f^{2} g \,x^{2}+35 a^{3} c d \,e^{3} g^{3} x -165 a^{2} c^{2} d^{2} e^{2} f \,g^{2} x +297 a \,c^{3} d^{3} e \,f^{2} g x -231 c^{4} d^{4} f^{3} x +105 g^{3} e^{4} a^{4}-385 a^{3} c d \,e^{3} f \,g^{2}+495 a^{2} c^{2} d^{2} e^{2} f^{2} g -231 a \,c^{3} d^{3} e \,f^{3}\right ) \left (c d x +a e \right )}{1155 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {11}{2}} \left (a e g -c d f \right )^{4}}\)	\(267\)

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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/1155*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(11/2)*(-16*c^4*d^4*g^3*x^4+24*a*c^3*d^3*e*g^3*x^3-8
8*c^4*d^4*f*g^2*x^3-30*a^2*c^2*d^2*e^2*g^3*x^2+132*a*c^3*d^3*e*f*g^2*x^2-198*c^4*d^4*f^2*g*x^2+35*a^3*c*d*e^3*
g^3*x-165*a^2*c^2*d^2*e^2*f*g^2*x+297*a*c^3*d^3*e*f^2*g*x-231*c^4*d^4*f^3*x+105*a^4*e^4*g^3-385*a^3*c*d*e^3*f*
g^2+495*a^2*c^2*d^2*e^2*f^2*g-231*a*c^3*d^3*e*f^3)*(c*d*x+a*e)/(a*e*g-c*d*f)^4

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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)^(3/2)/(e*x+d)^(3/2)/(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)^(3/2)/((g*x + f)^(13/2)*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1496 vs. \(2 (247) = 494\).
time = 1.01, size = 1496, normalized size = 5.60 \begin {gather*} \frac {2 \, {\left (16 \, c^{5} d^{5} g^{3} x^{5} + 88 \, c^{5} d^{5} f g^{2} x^{4} + 198 \, c^{5} d^{5} f^{2} g x^{3} + 231 \, c^{5} d^{5} f^{3} x^{2} - 105 \, a^{5} g^{3} e^{5} - 35 \, {\left (4 \, a^{4} c d g^{3} x - 11 \, a^{4} c d f g^{2}\right )} e^{4} - 5 \, {\left (a^{3} c^{2} d^{2} g^{3} x^{2} - 110 \, a^{3} c^{2} d^{2} f g^{2} x + 99 \, a^{3} c^{2} d^{2} f^{2} g\right )} e^{3} + 3 \, {\left (2 \, a^{2} c^{3} d^{3} g^{3} x^{3} + 11 \, a^{2} c^{3} d^{3} f g^{2} x^{2} - 264 \, a^{2} c^{3} d^{3} f^{2} g x + 77 \, a^{2} c^{3} d^{3} f^{3}\right )} e^{2} - {\left (8 \, a c^{4} d^{4} g^{3} x^{4} + 44 \, a c^{4} d^{4} f g^{2} x^{3} + 99 \, a c^{4} d^{4} f^{2} g x^{2} - 462 \, a c^{4} d^{4} f^{3} x\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}}{1155 \, {\left (c^{4} d^{5} f^{4} g^{6} x^{6} + 6 \, c^{4} d^{5} f^{5} g^{5} x^{5} + 15 \, c^{4} d^{5} f^{6} g^{4} x^{4} + 20 \, c^{4} d^{5} f^{7} g^{3} x^{3} + 15 \, c^{4} d^{5} f^{8} g^{2} x^{2} + 6 \, c^{4} d^{5} f^{9} g x + c^{4} d^{5} f^{10} + {\left (a^{4} g^{10} x^{7} + 6 \, a^{4} f g^{9} x^{6} + 15 \, a^{4} f^{2} g^{8} x^{5} + 20 \, a^{4} f^{3} g^{7} x^{4} + 15 \, a^{4} f^{4} g^{6} x^{3} + 6 \, a^{4} f^{5} g^{5} x^{2} + a^{4} f^{6} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{9} x^{7} - a^{4} d f^{6} g^{4} + {\left (24 \, a^{3} c d f^{2} g^{8} - a^{4} d g^{10}\right )} x^{6} + 6 \, {\left (10 \, a^{3} c d f^{3} g^{7} - a^{4} d f g^{9}\right )} x^{5} + 5 \, {\left (16 \, a^{3} c d f^{4} g^{6} - 3 \, a^{4} d f^{2} g^{8}\right )} x^{4} + 20 \, {\left (3 \, a^{3} c d f^{5} g^{5} - a^{4} d f^{3} g^{7}\right )} x^{3} + 3 \, {\left (8 \, a^{3} c d f^{6} g^{4} - 5 \, a^{4} d f^{4} g^{6}\right )} x^{2} + 2 \, {\left (2 \, a^{3} c d f^{7} g^{3} - 3 \, a^{4} d f^{5} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{8} x^{7} - 2 \, a^{3} c d^{2} f^{7} g^{3} + 2 \, {\left (9 \, a^{2} c^{2} d^{2} f^{3} g^{7} - a^{3} c d^{2} f g^{9}\right )} x^{6} + 3 \, {\left (15 \, a^{2} c^{2} d^{2} f^{4} g^{6} - 4 \, a^{3} c d^{2} f^{2} g^{8}\right )} x^{5} + 30 \, {\left (2 \, a^{2} c^{2} d^{2} f^{5} g^{5} - a^{3} c d^{2} f^{3} g^{7}\right )} x^{4} + 5 \, {\left (9 \, a^{2} c^{2} d^{2} f^{6} g^{4} - 8 \, a^{3} c d^{2} f^{4} g^{6}\right )} x^{3} + 6 \, {\left (3 \, a^{2} c^{2} d^{2} f^{7} g^{3} - 5 \, a^{3} c d^{2} f^{5} g^{5}\right )} x^{2} + 3 \, {\left (a^{2} c^{2} d^{2} f^{8} g^{2} - 4 \, a^{3} c d^{2} f^{6} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{7} x^{7} - 3 \, a^{2} c^{2} d^{3} f^{8} g^{2} + 3 \, {\left (4 \, a c^{3} d^{3} f^{4} g^{6} - a^{2} c^{2} d^{3} f^{2} g^{8}\right )} x^{6} + 6 \, {\left (5 \, a c^{3} d^{3} f^{5} g^{5} - 3 \, a^{2} c^{2} d^{3} f^{3} g^{7}\right )} x^{5} + 5 \, {\left (8 \, a c^{3} d^{3} f^{6} g^{4} - 9 \, a^{2} c^{2} d^{3} f^{4} g^{6}\right )} x^{4} + 30 \, {\left (a c^{3} d^{3} f^{7} g^{3} - 2 \, a^{2} c^{2} d^{3} f^{5} g^{5}\right )} x^{3} + 3 \, {\left (4 \, a c^{3} d^{3} f^{8} g^{2} - 15 \, a^{2} c^{2} d^{3} f^{6} g^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{3} f^{9} g - 9 \, a^{2} c^{2} d^{3} f^{7} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{6} x^{7} - 4 \, a c^{3} d^{4} f^{9} g + 2 \, {\left (3 \, c^{4} d^{4} f^{5} g^{5} - 2 \, a c^{3} d^{4} f^{3} g^{7}\right )} x^{6} + 3 \, {\left (5 \, c^{4} d^{4} f^{6} g^{4} - 8 \, a c^{3} d^{4} f^{4} g^{6}\right )} x^{5} + 20 \, {\left (c^{4} d^{4} f^{7} g^{3} - 3 \, a c^{3} d^{4} f^{5} g^{5}\right )} x^{4} + 5 \, {\left (3 \, c^{4} d^{4} f^{8} g^{2} - 16 \, a c^{3} d^{4} f^{6} g^{4}\right )} x^{3} + 6 \, {\left (c^{4} d^{4} f^{9} g - 10 \, a c^{3} d^{4} f^{7} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{10} - 24 \, a c^{3} d^{4} f^{8} 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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(16*c^5*d^5*g^3*x^5 + 88*c^5*d^5*f*g^2*x^4 + 198*c^5*d^5*f^2*g*x^3 + 231*c^5*d^5*f^3*x^2 - 105*a^5*g^3*
e^5 - 35*(4*a^4*c*d*g^3*x - 11*a^4*c*d*f*g^2)*e^4 - 5*(a^3*c^2*d^2*g^3*x^2 - 110*a^3*c^2*d^2*f*g^2*x + 99*a^3*
c^2*d^2*f^2*g)*e^3 + 3*(2*a^2*c^3*d^3*g^3*x^3 + 11*a^2*c^3*d^3*f*g^2*x^2 - 264*a^2*c^3*d^3*f^2*g*x + 77*a^2*c^
3*d^3*f^3)*e^2 - (8*a*c^4*d^4*g^3*x^4 + 44*a*c^4*d^4*f*g^2*x^3 + 99*a*c^4*d^4*f^2*g*x^2 - 462*a*c^4*d^4*f^3*x)
*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^4*d^5*f^4*g^6*x^6 + 6*c^4*d^5*f
^5*g^5*x^5 + 15*c^4*d^5*f^6*g^4*x^4 + 20*c^4*d^5*f^7*g^3*x^3 + 15*c^4*d^5*f^8*g^2*x^2 + 6*c^4*d^5*f^9*g*x + c^
4*d^5*f^10 + (a^4*g^10*x^7 + 6*a^4*f*g^9*x^6 + 15*a^4*f^2*g^8*x^5 + 20*a^4*f^3*g^7*x^4 + 15*a^4*f^4*g^6*x^3 +
6*a^4*f^5*g^5*x^2 + a^4*f^6*g^4*x)*e^5 - (4*a^3*c*d*f*g^9*x^7 - a^4*d*f^6*g^4 + (24*a^3*c*d*f^2*g^8 - a^4*d*g^
10)*x^6 + 6*(10*a^3*c*d*f^3*g^7 - a^4*d*f*g^9)*x^5 + 5*(16*a^3*c*d*f^4*g^6 - 3*a^4*d*f^2*g^8)*x^4 + 20*(3*a^3*
c*d*f^5*g^5 - a^4*d*f^3*g^7)*x^3 + 3*(8*a^3*c*d*f^6*g^4 - 5*a^4*d*f^4*g^6)*x^2 + 2*(2*a^3*c*d*f^7*g^3 - 3*a^4*
d*f^5*g^5)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^8*x^7 - 2*a^3*c*d^2*f^7*g^3 + 2*(9*a^2*c^2*d^2*f^3*g^7 - a^3*c*d^2*
f*g^9)*x^6 + 3*(15*a^2*c^2*d^2*f^4*g^6 - 4*a^3*c*d^2*f^2*g^8)*x^5 + 30*(2*a^2*c^2*d^2*f^5*g^5 - a^3*c*d^2*f^3*
g^7)*x^4 + 5*(9*a^2*c^2*d^2*f^6*g^4 - 8*a^3*c*d^2*f^4*g^6)*x^3 + 6*(3*a^2*c^2*d^2*f^7*g^3 - 5*a^3*c*d^2*f^5*g^
5)*x^2 + 3*(a^2*c^2*d^2*f^8*g^2 - 4*a^3*c*d^2*f^6*g^4)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^7*x^7 - 3*a^2*c^2*d^3*f^8
*g^2 + 3*(4*a*c^3*d^3*f^4*g^6 - a^2*c^2*d^3*f^2*g^8)*x^6 + 6*(5*a*c^3*d^3*f^5*g^5 - 3*a^2*c^2*d^3*f^3*g^7)*x^5
+ 5*(8*a*c^3*d^3*f^6*g^4 - 9*a^2*c^2*d^3*f^4*g^6)*x^4 + 30*(a*c^3*d^3*f^7*g^3 - 2*a^2*c^2*d^3*f^5*g^5)*x^3 +
3*(4*a*c^3*d^3*f^8*g^2 - 15*a^2*c^2*d^3*f^6*g^4)*x^2 + 2*(a*c^3*d^3*f^9*g - 9*a^2*c^2*d^3*f^7*g^3)*x)*e^2 + (c
^4*d^4*f^4*g^6*x^7 - 4*a*c^3*d^4*f^9*g + 2*(3*c^4*d^4*f^5*g^5 - 2*a*c^3*d^4*f^3*g^7)*x^6 + 3*(5*c^4*d^4*f^6*g^
4 - 8*a*c^3*d^4*f^4*g^6)*x^5 + 20*(c^4*d^4*f^7*g^3 - 3*a*c^3*d^4*f^5*g^5)*x^4 + 5*(3*c^4*d^4*f^8*g^2 - 16*a*c^
3*d^4*f^6*g^4)*x^3 + 6*(c^4*d^4*f^9*g - 10*a*c^3*d^4*f^7*g^3)*x^2 + (c^4*d^4*f^10 - 24*a*c^3*d^4*f^8*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)**(3/2)/(e*x+d)**(3/2)/(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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.83, size = 519, normalized size = 1.94 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {210\,a^5\,e^5\,g^3-770\,a^4\,c\,d\,e^4\,f\,g^2+990\,a^3\,c^2\,d^2\,e^3\,f^2\,g-462\,a^2\,c^3\,d^3\,e^2\,f^3}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^2\,\left (-10\,a^3\,c^2\,d^2\,e^3\,g^3+66\,a^2\,c^3\,d^3\,e^2\,f\,g^2-198\,a\,c^4\,d^4\,e\,f^2\,g+462\,c^5\,d^5\,f^3\right )}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^5\,d^5\,x^5}{1155\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^3\,d^3\,x^3\,\left (3\,a^2\,e^2\,g^2-22\,a\,c\,d\,e\,f\,g+99\,c^2\,d^2\,f^2\right )}{1155\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^4\,d^4\,x^4\,\left (a\,e\,g-11\,c\,d\,f\right )}{1155\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,a\,c\,d\,e\,x\,\left (70\,a^3\,e^3\,g^3-275\,a^2\,c\,d\,e^2\,f\,g^2+396\,a\,c^2\,d^2\,e\,f^2\,g-231\,c^3\,d^3\,f^3\right )}{1155\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}\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)^(3/2)/((f + g*x)^(13/2)*(d + e*x)^(3/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((210*a^5*e^5*g^3 - 462*a^2*c^3*d^3*e^2*f^3 + 990*a^3*c^2*d^2*
e^3*f^2*g - 770*a^4*c*d*e^4*f*g^2)/(1155*g^5*(a*e*g - c*d*f)^4) - (x^2*(462*c^5*d^5*f^3 - 10*a^3*c^2*d^2*e^3*g
^3 + 66*a^2*c^3*d^3*e^2*f*g^2 - 198*a*c^4*d^4*e*f^2*g))/(1155*g^5*(a*e*g - c*d*f)^4) - (32*c^5*d^5*x^5)/(1155*
g^2*(a*e*g - c*d*f)^4) - (4*c^3*d^3*x^3*(3*a^2*e^2*g^2 + 99*c^2*d^2*f^2 - 22*a*c*d*e*f*g))/(1155*g^4*(a*e*g -
c*d*f)^4) + (16*c^4*d^4*x^4*(a*e*g - 11*c*d*f))/(1155*g^3*(a*e*g - c*d*f)^4) + (4*a*c*d*e*x*(70*a^3*e^3*g^3 -
231*c^3*d^3*f^3 + 396*a*c^2*d^2*e*f^2*g - 275*a^2*c*d*e^2*f*g^2))/(1155*g^5*(a*e*g - c*d*f)^4)))/(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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