3.8.60 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx\) [760]

Optimal. Leaf size=267 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}} \]

[Out]

2/13*(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)^(13/2)+12/143*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)^(11/2)+16/429*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)^(9/2)+32/3003*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(7/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(7/2)/(g*x+f)^(7/2)

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Rubi [A]
time = 0.21, 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 )^{7/2}}{3003 (d+e x)^{7/2} (f+g x)^{7/2} (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 )^{7/2}}{429 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^{7/2} (f+g x)^{11/2} (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}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (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)^(15/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(13/2)) + (12*
c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(11/2)) +
(16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(429*(c*d*f - a*e*g)^3*(d + e*x)^(7/2)*(f + g*x)^(9
/2)) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*(c*d*f - a*e*g)^4*(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)^{15/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\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 )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx}{13 (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}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {\left (24 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)^{11/2}} \, dx}{143 (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}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\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}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\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 )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx}{429 (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 )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\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}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\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 )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 141, normalized size = 0.53 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{5/2} \left (-231 g^3+\frac {819 c d g^2 (f+g x)}{a e+c d x}-\frac {1001 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {429 c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{3003 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{13/2}} \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)^(15/2)),x]

[Out]

(2*(a*e + c*d*x)^4*((a*e + c*d*x)*(d + e*x))^(5/2)*(-231*g^3 + (819*c*d*g^2*(f + g*x))/(a*e + c*d*x) - (1001*c
^2*d^2*g*(f + g*x)^2)/(a*e + c*d*x)^2 + (429*c^3*d^3*(f + g*x)^3)/(a*e + c*d*x)^3))/(3003*(c*d*f - a*e*g)^4*(d
+ e*x)^(5/2)*(f + g*x)^(13/2))

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Maple [A]
time = 0.15, size = 349, normalized size = 1.31

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+56 a \,c^{2} d^{2} e \,g^{3} x^{2}-104 c^{3} d^{3} f \,g^{2} x^{2}-126 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -286 c^{3} d^{3} f^{2} g x +231 a^{3} e^{3} g^{3}-819 a^{2} c d \,e^{2} f \,g^{2}+1001 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (g x +f \right )^{\frac {13}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}\)	\(260\)
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 c^{5} d^{5} g^{3} x^{5}+24 a \,c^{4} d^{4} e \,g^{3} x^{4}-104 c^{5} d^{5} f \,g^{2} x^{4}-30 a^{2} c^{3} d^{3} e^{2} g^{3} x^{3}+156 a \,c^{4} d^{4} e f \,g^{2} x^{3}-286 c^{5} d^{5} f^{2} g \,x^{3}+35 a^{3} c^{2} d^{2} e^{3} g^{3} x^{2}-195 a^{2} c^{3} d^{3} e^{2} f \,g^{2} x^{2}+429 a \,c^{4} d^{4} e \,f^{2} g \,x^{2}-429 c^{5} d^{5} f^{3} x^{2}+336 a^{4} c d \,e^{4} g^{3} x -1274 a^{3} c^{2} d^{2} e^{3} f \,g^{2} x +1716 a^{2} c^{3} d^{3} e^{2} f^{2} g x -858 a \,c^{4} d^{4} e \,f^{3} x +231 a^{5} e^{5} g^{3}-819 a^{4} c d \,e^{4} f \,g^{2}+1001 a^{3} c^{2} d^{2} e^{3} f^{2} g -429 a^{2} c^{3} d^{3} e^{2} f^{3}\right ) \left (c d x +a e \right )}{3003 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {13}{2}} \left (a e g -c d f \right )^{4}}\)	\(349\)

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)^(15/2),x,method=_RETURNVERBOSE)

[Out]

-2/3003*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(13/2)*(-16*c^5*d^5*g^3*x^5+24*a*c^4*d^4*e*g^3*x^4-1
04*c^5*d^5*f*g^2*x^4-30*a^2*c^3*d^3*e^2*g^3*x^3+156*a*c^4*d^4*e*f*g^2*x^3-286*c^5*d^5*f^2*g*x^3+35*a^3*c^2*d^2
*e^3*g^3*x^2-195*a^2*c^3*d^3*e^2*f*g^2*x^2+429*a*c^4*d^4*e*f^2*g*x^2-429*c^5*d^5*f^3*x^2+336*a^4*c*d*e^4*g^3*x
-1274*a^3*c^2*d^2*e^3*f*g^2*x+1716*a^2*c^3*d^3*e^2*f^2*g*x-858*a*c^4*d^4*e*f^3*x+231*a^5*e^5*g^3-819*a^4*c*d*e
^4*f*g^2+1001*a^3*c^2*d^2*e^3*f^2*g-429*a^2*c^3*d^3*e^2*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)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(15/2),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)^(15/2)*(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. 1741 vs. \(2 (247) = 494\).
time = 0.95, size = 1741, normalized size = 6.52 \begin {gather*} \frac {2 \, {\left (16 \, c^{6} d^{6} g^{3} x^{6} + 104 \, c^{6} d^{6} f g^{2} x^{5} + 286 \, c^{6} d^{6} f^{2} g x^{4} + 429 \, c^{6} d^{6} f^{3} x^{3} - 231 \, a^{6} g^{3} e^{6} - 63 \, {\left (9 \, a^{5} c d g^{3} x - 13 \, a^{5} c d f g^{2}\right )} e^{5} - 7 \, {\left (53 \, a^{4} c^{2} d^{2} g^{3} x^{2} - 299 \, a^{4} c^{2} d^{2} f g^{2} x + 143 \, a^{4} c^{2} d^{2} f^{2} g\right )} e^{4} - {\left (5 \, a^{3} c^{3} d^{3} g^{3} x^{3} - 1469 \, a^{3} c^{3} d^{3} f g^{2} x^{2} + 2717 \, a^{3} c^{3} d^{3} f^{2} g x - 429 \, a^{3} c^{3} d^{3} f^{3}\right )} e^{3} + 3 \, {\left (2 \, a^{2} c^{4} d^{4} g^{3} x^{4} + 13 \, a^{2} c^{4} d^{4} f g^{2} x^{3} - 715 \, a^{2} c^{4} d^{4} f^{2} g x^{2} + 429 \, a^{2} c^{4} d^{4} f^{3} x\right )} e^{2} - {\left (8 \, a c^{5} d^{5} g^{3} x^{5} + 52 \, a c^{5} d^{5} f g^{2} x^{4} + 143 \, a c^{5} d^{5} f^{2} g x^{3} - 1287 \, a c^{5} d^{5} f^{3} 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}}{3003 \, {\left (c^{4} d^{5} f^{4} g^{7} x^{7} + 7 \, c^{4} d^{5} f^{5} g^{6} x^{6} + 21 \, c^{4} d^{5} f^{6} g^{5} x^{5} + 35 \, c^{4} d^{5} f^{7} g^{4} x^{4} + 35 \, c^{4} d^{5} f^{8} g^{3} x^{3} + 21 \, c^{4} d^{5} f^{9} g^{2} x^{2} + 7 \, c^{4} d^{5} f^{10} g x + c^{4} d^{5} f^{11} + {\left (a^{4} g^{11} x^{8} + 7 \, a^{4} f g^{10} x^{7} + 21 \, a^{4} f^{2} g^{9} x^{6} + 35 \, a^{4} f^{3} g^{8} x^{5} + 35 \, a^{4} f^{4} g^{7} x^{4} + 21 \, a^{4} f^{5} g^{6} x^{3} + 7 \, a^{4} f^{6} g^{5} x^{2} + a^{4} f^{7} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{10} x^{8} - a^{4} d f^{7} g^{4} + {\left (28 \, a^{3} c d f^{2} g^{9} - a^{4} d g^{11}\right )} x^{7} + 7 \, {\left (12 \, a^{3} c d f^{3} g^{8} - a^{4} d f g^{10}\right )} x^{6} + 7 \, {\left (20 \, a^{3} c d f^{4} g^{7} - 3 \, a^{4} d f^{2} g^{9}\right )} x^{5} + 35 \, {\left (4 \, a^{3} c d f^{5} g^{6} - a^{4} d f^{3} g^{8}\right )} x^{4} + 7 \, {\left (12 \, a^{3} c d f^{6} g^{5} - 5 \, a^{4} d f^{4} g^{7}\right )} x^{3} + 7 \, {\left (4 \, a^{3} c d f^{7} g^{4} - 3 \, a^{4} d f^{5} g^{6}\right )} x^{2} + {\left (4 \, a^{3} c d f^{8} g^{3} - 7 \, a^{4} d f^{6} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{9} x^{8} - 2 \, a^{3} c d^{2} f^{8} g^{3} + {\left (21 \, a^{2} c^{2} d^{2} f^{3} g^{8} - 2 \, a^{3} c d^{2} f g^{10}\right )} x^{7} + 7 \, {\left (9 \, a^{2} c^{2} d^{2} f^{4} g^{7} - 2 \, a^{3} c d^{2} f^{2} g^{9}\right )} x^{6} + 21 \, {\left (5 \, a^{2} c^{2} d^{2} f^{5} g^{6} - 2 \, a^{3} c d^{2} f^{3} g^{8}\right )} x^{5} + 35 \, {\left (3 \, a^{2} c^{2} d^{2} f^{6} g^{5} - 2 \, a^{3} c d^{2} f^{4} g^{7}\right )} x^{4} + 7 \, {\left (9 \, a^{2} c^{2} d^{2} f^{7} g^{4} - 10 \, a^{3} c d^{2} f^{5} g^{6}\right )} x^{3} + 21 \, {\left (a^{2} c^{2} d^{2} f^{8} g^{3} - 2 \, a^{3} c d^{2} f^{6} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{9} g^{2} - 14 \, a^{3} c d^{2} f^{7} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{8} x^{8} - 3 \, a^{2} c^{2} d^{3} f^{9} g^{2} + {\left (14 \, a c^{3} d^{3} f^{4} g^{7} - 3 \, a^{2} c^{2} d^{3} f^{2} g^{9}\right )} x^{7} + 21 \, {\left (2 \, a c^{3} d^{3} f^{5} g^{6} - a^{2} c^{2} d^{3} f^{3} g^{8}\right )} x^{6} + 7 \, {\left (10 \, a c^{3} d^{3} f^{6} g^{5} - 9 \, a^{2} c^{2} d^{3} f^{4} g^{7}\right )} x^{5} + 35 \, {\left (2 \, a c^{3} d^{3} f^{7} g^{4} - 3 \, a^{2} c^{2} d^{3} f^{5} g^{6}\right )} x^{4} + 21 \, {\left (2 \, a c^{3} d^{3} f^{8} g^{3} - 5 \, a^{2} c^{2} d^{3} f^{6} g^{5}\right )} x^{3} + 7 \, {\left (2 \, a c^{3} d^{3} f^{9} g^{2} - 9 \, a^{2} c^{2} d^{3} f^{7} g^{4}\right )} x^{2} + {\left (2 \, a c^{3} d^{3} f^{10} g - 21 \, a^{2} c^{2} d^{3} f^{8} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{7} x^{8} - 4 \, a c^{3} d^{4} f^{10} g + {\left (7 \, c^{4} d^{4} f^{5} g^{6} - 4 \, a c^{3} d^{4} f^{3} g^{8}\right )} x^{7} + 7 \, {\left (3 \, c^{4} d^{4} f^{6} g^{5} - 4 \, a c^{3} d^{4} f^{4} g^{7}\right )} x^{6} + 7 \, {\left (5 \, c^{4} d^{4} f^{7} g^{4} - 12 \, a c^{3} d^{4} f^{5} g^{6}\right )} x^{5} + 35 \, {\left (c^{4} d^{4} f^{8} g^{3} - 4 \, a c^{3} d^{4} f^{6} g^{5}\right )} x^{4} + 7 \, {\left (3 \, c^{4} d^{4} f^{9} g^{2} - 20 \, a c^{3} d^{4} f^{7} g^{4}\right )} x^{3} + 7 \, {\left (c^{4} d^{4} f^{10} g - 12 \, a c^{3} d^{4} f^{8} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{11} - 28 \, a c^{3} d^{4} f^{9} 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)^(15/2),x, algorithm="fricas")

[Out]

2/3003*(16*c^6*d^6*g^3*x^6 + 104*c^6*d^6*f*g^2*x^5 + 286*c^6*d^6*f^2*g*x^4 + 429*c^6*d^6*f^3*x^3 - 231*a^6*g^3
*e^6 - 63*(9*a^5*c*d*g^3*x - 13*a^5*c*d*f*g^2)*e^5 - 7*(53*a^4*c^2*d^2*g^3*x^2 - 299*a^4*c^2*d^2*f*g^2*x + 143
*a^4*c^2*d^2*f^2*g)*e^4 - (5*a^3*c^3*d^3*g^3*x^3 - 1469*a^3*c^3*d^3*f*g^2*x^2 + 2717*a^3*c^3*d^3*f^2*g*x - 429
*a^3*c^3*d^3*f^3)*e^3 + 3*(2*a^2*c^4*d^4*g^3*x^4 + 13*a^2*c^4*d^4*f*g^2*x^3 - 715*a^2*c^4*d^4*f^2*g*x^2 + 429*
a^2*c^4*d^4*f^3*x)*e^2 - (8*a*c^5*d^5*g^3*x^5 + 52*a*c^5*d^5*f*g^2*x^4 + 143*a*c^5*d^5*f^2*g*x^3 - 1287*a*c^5*
d^5*f^3*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^4*d^5*f^4*g^7*x^7 +
7*c^4*d^5*f^5*g^6*x^6 + 21*c^4*d^5*f^6*g^5*x^5 + 35*c^4*d^5*f^7*g^4*x^4 + 35*c^4*d^5*f^8*g^3*x^3 + 21*c^4*d^5
*f^9*g^2*x^2 + 7*c^4*d^5*f^10*g*x + c^4*d^5*f^11 + (a^4*g^11*x^8 + 7*a^4*f*g^10*x^7 + 21*a^4*f^2*g^9*x^6 + 35*
a^4*f^3*g^8*x^5 + 35*a^4*f^4*g^7*x^4 + 21*a^4*f^5*g^6*x^3 + 7*a^4*f^6*g^5*x^2 + a^4*f^7*g^4*x)*e^5 - (4*a^3*c*
d*f*g^10*x^8 - a^4*d*f^7*g^4 + (28*a^3*c*d*f^2*g^9 - a^4*d*g^11)*x^7 + 7*(12*a^3*c*d*f^3*g^8 - a^4*d*f*g^10)*x
^6 + 7*(20*a^3*c*d*f^4*g^7 - 3*a^4*d*f^2*g^9)*x^5 + 35*(4*a^3*c*d*f^5*g^6 - a^4*d*f^3*g^8)*x^4 + 7*(12*a^3*c*d
*f^6*g^5 - 5*a^4*d*f^4*g^7)*x^3 + 7*(4*a^3*c*d*f^7*g^4 - 3*a^4*d*f^5*g^6)*x^2 + (4*a^3*c*d*f^8*g^3 - 7*a^4*d*f
^6*g^5)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^9*x^8 - 2*a^3*c*d^2*f^8*g^3 + (21*a^2*c^2*d^2*f^3*g^8 - 2*a^3*c*d^2*f*
g^10)*x^7 + 7*(9*a^2*c^2*d^2*f^4*g^7 - 2*a^3*c*d^2*f^2*g^9)*x^6 + 21*(5*a^2*c^2*d^2*f^5*g^6 - 2*a^3*c*d^2*f^3*
g^8)*x^5 + 35*(3*a^2*c^2*d^2*f^6*g^5 - 2*a^3*c*d^2*f^4*g^7)*x^4 + 7*(9*a^2*c^2*d^2*f^7*g^4 - 10*a^3*c*d^2*f^5*
g^6)*x^3 + 21*(a^2*c^2*d^2*f^8*g^3 - 2*a^3*c*d^2*f^6*g^5)*x^2 + (3*a^2*c^2*d^2*f^9*g^2 - 14*a^3*c*d^2*f^7*g^4)
*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^8*x^8 - 3*a^2*c^2*d^3*f^9*g^2 + (14*a*c^3*d^3*f^4*g^7 - 3*a^2*c^2*d^3*f^2*g^9)*
x^7 + 21*(2*a*c^3*d^3*f^5*g^6 - a^2*c^2*d^3*f^3*g^8)*x^6 + 7*(10*a*c^3*d^3*f^6*g^5 - 9*a^2*c^2*d^3*f^4*g^7)*x^
5 + 35*(2*a*c^3*d^3*f^7*g^4 - 3*a^2*c^2*d^3*f^5*g^6)*x^4 + 21*(2*a*c^3*d^3*f^8*g^3 - 5*a^2*c^2*d^3*f^6*g^5)*x^
3 + 7*(2*a*c^3*d^3*f^9*g^2 - 9*a^2*c^2*d^3*f^7*g^4)*x^2 + (2*a*c^3*d^3*f^10*g - 21*a^2*c^2*d^3*f^8*g^3)*x)*e^2
+ (c^4*d^4*f^4*g^7*x^8 - 4*a*c^3*d^4*f^10*g + (7*c^4*d^4*f^5*g^6 - 4*a*c^3*d^4*f^3*g^8)*x^7 + 7*(3*c^4*d^4*f^
6*g^5 - 4*a*c^3*d^4*f^4*g^7)*x^6 + 7*(5*c^4*d^4*f^7*g^4 - 12*a*c^3*d^4*f^5*g^6)*x^5 + 35*(c^4*d^4*f^8*g^3 - 4*
a*c^3*d^4*f^6*g^5)*x^4 + 7*(3*c^4*d^4*f^9*g^2 - 20*a*c^3*d^4*f^7*g^4)*x^3 + 7*(c^4*d^4*f^10*g - 12*a*c^3*d^4*f
^8*g^3)*x^2 + (c^4*d^4*f^11 - 28*a*c^3*d^4*f^9*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)**(15/2),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)^(15/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.12, size = 627, 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 {462\,a^6\,e^6\,g^3-1638\,a^5\,c\,d\,e^5\,f\,g^2+2002\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^3\,\left (-10\,a^3\,c^3\,d^3\,e^3\,g^3+78\,a^2\,c^4\,d^4\,e^2\,f\,g^2-286\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^6\,d^6\,x^6}{3003\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^4\,d^4\,x^4\,\left (3\,a^2\,e^2\,g^2-26\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{3003\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^5\,d^5\,x^5\,\left (a\,e\,g-13\,c\,d\,f\right )}{3003\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (567\,a^3\,e^3\,g^3-2093\,a^2\,c\,d\,e^2\,f\,g^2+2717\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (371\,a^3\,e^3\,g^3-1469\,a^2\,c\,d\,e^2\,f\,g^2+2145\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^6}+\frac {6\,f\,x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {6\,f^5\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {15\,f^2\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {20\,f^3\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {15\,f^4\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}} \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)^(15/2)*(d + e*x)^(5/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((462*a^6*e^6*g^3 - 858*a^3*c^3*d^3*e^3*f^3 + 2002*a^4*c^2*d^2
*e^4*f^2*g - 1638*a^5*c*d*e^5*f*g^2)/(3003*g^6*(a*e*g - c*d*f)^4) - (x^3*(858*c^6*d^6*f^3 - 10*a^3*c^3*d^3*e^3
*g^3 + 78*a^2*c^4*d^4*e^2*f*g^2 - 286*a*c^5*d^5*e*f^2*g))/(3003*g^6*(a*e*g - c*d*f)^4) - (32*c^6*d^6*x^6)/(300
3*g^3*(a*e*g - c*d*f)^4) - (4*c^4*d^4*x^4*(3*a^2*e^2*g^2 + 143*c^2*d^2*f^2 - 26*a*c*d*e*f*g))/(3003*g^5*(a*e*g
- c*d*f)^4) + (16*c^5*d^5*x^5*(a*e*g - 13*c*d*f))/(3003*g^4*(a*e*g - c*d*f)^4) + (2*a^2*c*d*e^2*x*(567*a^3*e^
3*g^3 - 1287*c^3*d^3*f^3 + 2717*a*c^2*d^2*e*f^2*g - 2093*a^2*c*d*e^2*f*g^2))/(3003*g^6*(a*e*g - c*d*f)^4) + (2
*a*c^2*d^2*e*x^2*(371*a^3*e^3*g^3 - 1287*c^3*d^3*f^3 + 2145*a*c^2*d^2*e*f^2*g - 1469*a^2*c*d*e^2*f*g^2))/(3003
*g^6*(a*e*g - c*d*f)^4)))/(x^6*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^6*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^6 + (
6*f*x^5*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g + (6*f^5*x*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^5 + (15*f^2*x^4*(f +
g*x)^(1/2)*(d + e*x)^(1/2))/g^2 + (20*f^3*x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (15*f^4*x^2*(f + g*x)^(1/
2)*(d + e*x)^(1/2))/g^4)

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