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

Optimal result

Integrand size = 48, antiderivative size = 267 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}} \]

[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)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)} \]

[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*} \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}}{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*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-231 a^3 e^3 g^3+63 a^2 c d e^2 g^2 (13 f+2 g x)-7 a c^2 d^2 e g \left (143 f^2+52 f g x+8 g^2 x^2\right )+c^3 d^3 \left (429 f^3+286 f^2 g x+104 f g^2 x^2+16 g^3 x^3\right )\right )}{3003 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)^{13/2}} \]

[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)^(15/2)),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^3*g^3 + 63*a^2*c*d*e^2*g^2*(13*f + 2*g*x) - 7*a*c
^2*d^2*e*g*(143*f^2 + 52*f*g*x + 8*g^2*x^2) + c^3*d^3*(429*f^3 + 286*f^2*g*x + 104*f*g^2*x^2 + 16*g^3*x^3)))/(
3003*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f + g*x)^(13/2))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.97

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

[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)*(-16*c^3*d^3*g^3*x^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+36
4*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*c^3
*d^3*f^3)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(g*x+f)^(13/2)/(a^4*e^4*g^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+c^4*d^4*f^4)/(e*x+d)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1648 vs. \(2 (235) = 470\).

Time = 1.57 (sec) , antiderivative size = 1648, normalized size of antiderivative = 6.17 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\text {Too large to display} \]

[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)^(15/2),x, algorithm="fricas")

[Out]

2/3003*(16*c^6*d^6*g^3*x^6 + 429*a^3*c^3*d^3*e^3*f^3 - 1001*a^4*c^2*d^2*e^4*f^2*g + 819*a^5*c*d*e^5*f*g^2 - 23
1*a^6*e^6*g^3 + 8*(13*c^6*d^6*f*g^2 - a*c^5*d^5*e*g^3)*x^5 + 2*(143*c^6*d^6*f^2*g - 26*a*c^5*d^5*e*f*g^2 + 3*a
^2*c^4*d^4*e^2*g^3)*x^4 + (429*c^6*d^6*f^3 - 143*a*c^5*d^5*e*f^2*g + 39*a^2*c^4*d^4*e^2*f*g^2 - 5*a^3*c^3*d^3*
e^3*g^3)*x^3 + (1287*a*c^5*d^5*e*f^3 - 2145*a^2*c^4*d^4*e^2*f^2*g + 1469*a^3*c^3*d^3*e^3*f*g^2 - 371*a^4*c^2*d
^2*e^4*g^3)*x^2 + (1287*a^2*c^4*d^4*e^2*f^3 - 2717*a^3*c^3*d^3*e^3*f^2*g + 2093*a^4*c^2*d^2*e^4*f*g^2 - 567*a^
5*c*d*e^5*g^3)*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^4*d^5*f^11 - 4*a*
c^3*d^4*e*f^10*g + 6*a^2*c^2*d^3*e^2*f^9*g^2 - 4*a^3*c*d^2*e^3*f^8*g^3 + a^4*d*e^4*f^7*g^4 + (c^4*d^4*e*f^4*g^
7 - 4*a*c^3*d^3*e^2*f^3*g^8 + 6*a^2*c^2*d^2*e^3*f^2*g^9 - 4*a^3*c*d*e^4*f*g^10 + a^4*e^5*g^11)*x^8 + (7*c^4*d^
4*e*f^5*g^6 + a^4*d*e^4*g^11 + (c^4*d^5 - 28*a*c^3*d^3*e^2)*f^4*g^7 - 2*(2*a*c^3*d^4*e - 21*a^2*c^2*d^2*e^3)*f
^3*g^8 + 2*(3*a^2*c^2*d^3*e^2 - 14*a^3*c*d*e^4)*f^2*g^9 - (4*a^3*c*d^2*e^3 - 7*a^4*e^5)*f*g^10)*x^7 + 7*(3*c^4
*d^4*e*f^6*g^5 + a^4*d*e^4*f*g^10 + (c^4*d^5 - 12*a*c^3*d^3*e^2)*f^5*g^6 - 2*(2*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^
3)*f^4*g^7 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^3*g^8 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^2*g^9)*x^6 + 7*(5*c
^4*d^4*e*f^7*g^4 + 3*a^4*d*e^4*f^2*g^9 + (3*c^4*d^5 - 20*a*c^3*d^3*e^2)*f^6*g^5 - 6*(2*a*c^3*d^4*e - 5*a^2*c^2
*d^2*e^3)*f^5*g^6 + 2*(9*a^2*c^2*d^3*e^2 - 10*a^3*c*d*e^4)*f^4*g^7 - (12*a^3*c*d^2*e^3 - 5*a^4*e^5)*f^3*g^8)*x
^5 + 35*(c^4*d^4*e*f^8*g^3 + a^4*d*e^4*f^3*g^8 + (c^4*d^5 - 4*a*c^3*d^3*e^2)*f^7*g^4 - 2*(2*a*c^3*d^4*e - 3*a^
2*c^2*d^2*e^3)*f^6*g^5 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^5*g^6 - (4*a^3*c*d^2*e^3 - a^4*e^5)*f^4*g^7)*
x^4 + 7*(3*c^4*d^4*e*f^9*g^2 + 5*a^4*d*e^4*f^4*g^7 + (5*c^4*d^5 - 12*a*c^3*d^3*e^2)*f^8*g^3 - 2*(10*a*c^3*d^4*
e - 9*a^2*c^2*d^2*e^3)*f^7*g^4 + 6*(5*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^6*g^5 - (20*a^3*c*d^2*e^3 - 3*a^4*e^5
)*f^5*g^6)*x^3 + 7*(c^4*d^4*e*f^10*g + 3*a^4*d*e^4*f^5*g^6 + (3*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^9*g^2 - 6*(2*a*c^
3*d^4*e - a^2*c^2*d^2*e^3)*f^8*g^3 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^7*g^4 - (12*a^3*c*d^2*e^3 - a^4*e
^5)*f^6*g^5)*x^2 + (c^4*d^4*e*f^11 + 7*a^4*d*e^4*f^6*g^5 + (7*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^10*g - 2*(14*a*c^3*
d^4*e - 3*a^2*c^2*d^2*e^3)*f^9*g^2 + 2*(21*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^8*g^3 - (28*a^3*c*d^2*e^3 - a^4*
e^5)*f^7*g^4)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\text {Timed out} \]

[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)**(15/2),x)

[Out]

Timed out

Maxima [F]

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

[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)^(15/2),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)^(15/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3241 vs. \(2 (235) = 470\).

Time = 2.66 (sec) , antiderivative size = 3241, normalized size of antiderivative = 12.14 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\text {Too large to display} \]

[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)^(15/2),x, algorithm="giac")

[Out]

2/3003*(429*sqrt(-c*d^2*e + a*e^3)*c^6*d^9*e^3*f^3*abs(c)*abs(d) - 1287*sqrt(-c*d^2*e + a*e^3)*a*c^5*d^7*e^5*f
^3*abs(c)*abs(d) + 1287*sqrt(-c*d^2*e + a*e^3)*a^2*c^4*d^5*e^7*f^3*abs(c)*abs(d) - 429*sqrt(-c*d^2*e + a*e^3)*
a^3*c^3*d^3*e^9*f^3*abs(c)*abs(d) - 286*sqrt(-c*d^2*e + a*e^3)*c^6*d^10*e^2*f^2*g*abs(c)*abs(d) - 143*sqrt(-c*
d^2*e + a*e^3)*a*c^5*d^8*e^4*f^2*g*abs(c)*abs(d) + 2145*sqrt(-c*d^2*e + a*e^3)*a^2*c^4*d^6*e^6*f^2*g*abs(c)*ab
s(d) - 2717*sqrt(-c*d^2*e + a*e^3)*a^3*c^3*d^4*e^8*f^2*g*abs(c)*abs(d) + 1001*sqrt(-c*d^2*e + a*e^3)*a^4*c^2*d
^2*e^10*f^2*g*abs(c)*abs(d) + 104*sqrt(-c*d^2*e + a*e^3)*c^6*d^11*e*f*g^2*abs(c)*abs(d) + 52*sqrt(-c*d^2*e + a
*e^3)*a*c^5*d^9*e^3*f*g^2*abs(c)*abs(d) + 39*sqrt(-c*d^2*e + a*e^3)*a^2*c^4*d^7*e^5*f*g^2*abs(c)*abs(d) - 1469
*sqrt(-c*d^2*e + a*e^3)*a^3*c^3*d^5*e^7*f*g^2*abs(c)*abs(d) + 2093*sqrt(-c*d^2*e + a*e^3)*a^4*c^2*d^3*e^9*f*g^
2*abs(c)*abs(d) - 819*sqrt(-c*d^2*e + a*e^3)*a^5*c*d*e^11*f*g^2*abs(c)*abs(d) - 16*sqrt(-c*d^2*e + a*e^3)*c^6*
d^12*g^3*abs(c)*abs(d) - 8*sqrt(-c*d^2*e + a*e^3)*a*c^5*d^10*e^2*g^3*abs(c)*abs(d) - 6*sqrt(-c*d^2*e + a*e^3)*
a^2*c^4*d^8*e^4*g^3*abs(c)*abs(d) - 5*sqrt(-c*d^2*e + a*e^3)*a^3*c^3*d^6*e^6*g^3*abs(c)*abs(d) + 371*sqrt(-c*d
^2*e + a*e^3)*a^4*c^2*d^4*e^8*g^3*abs(c)*abs(d) - 567*sqrt(-c*d^2*e + a*e^3)*a^5*c*d^2*e^10*g^3*abs(c)*abs(d)
+ 231*sqrt(-c*d^2*e + a*e^3)*a^6*e^12*g^3*abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^4*e^6*f^10 -
6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^5*e^5*f^9*g - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^3*e^7*f^9
*g + 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^6*e^4*f^8*g^2 + 24*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^4
*e^6*f^8*g^2 + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^2*e^8*f^8*g^2 - 20*sqrt(c^2*d^2*e^2*f - c^2*d^3*e
*g)*c^4*d^7*e^3*f^7*g^3 - 60*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^5*e^5*f^7*g^3 - 36*sqrt(c^2*d^2*e^2*f -
c^2*d^3*e*g)*a^2*c^2*d^3*e^7*f^7*g^3 - 4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d*e^9*f^7*g^3 + 15*sqrt(c^2*
d^2*e^2*f - c^2*d^3*e*g)*c^4*d^8*e^2*f^6*g^4 + 80*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^6*e^4*f^6*g^4 + 90
*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^4*e^6*f^6*g^4 + 24*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^2*e^
8*f^6*g^4 + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*e^10*f^6*g^4 - 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^9*e
*f^5*g^5 - 60*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^7*e^3*f^5*g^5 - 120*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*
a^2*c^2*d^5*e^5*f^5*g^5 - 60*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^3*e^7*f^5*g^5 - 6*sqrt(c^2*d^2*e^2*f -
c^2*d^3*e*g)*a^4*d*e^9*f^5*g^5 + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^4*d^10*f^4*g^6 + 24*sqrt(c^2*d^2*e^2*f -
c^2*d^3*e*g)*a*c^3*d^8*e^2*f^4*g^6 + 90*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^6*e^4*f^4*g^6 + 80*sqrt(c^
2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^4*e^6*f^4*g^6 + 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d^2*e^8*f^4*g^6 -
4*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c^3*d^9*e*f^3*g^7 - 36*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^7*e^3
*f^3*g^7 - 60*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^3*c*d^5*e^5*f^3*g^7 - 20*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a
^4*d^3*e^7*f^3*g^7 + 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^2*c^2*d^8*e^2*f^2*g^8 + 24*sqrt(c^2*d^2*e^2*f - c^2
*d^3*e*g)*a^3*c*d^6*e^4*f^2*g^8 + 15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d^4*e^6*f^2*g^8 - 4*sqrt(c^2*d^2*e^
2*f - c^2*d^3*e*g)*a^3*c*d^7*e^3*f*g^9 - 6*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a^4*d^5*e^5*f*g^9 + sqrt(c^2*d^2*
e^2*f - c^2*d^3*e*g)*a^4*d^6*e^4*g^10) + 2/3003*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*(2*((e*x + d)*c*d*e
- c*d^2*e + a*e^3)*(4*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*(2*(c^14*d^14*e^12*f^2*g^9*abs(c)*abs(d) - 2*a*c^13*
d^13*e^13*f*g^10*abs(c)*abs(d) + a^2*c^12*d^12*e^14*g^11*abs(c)*abs(d))*((e*x + d)*c*d*e - c*d^2*e + a*e^3)/(c
^6*d^6*e^12*f^6*g^6 - 6*a*c^5*d^5*e^13*f^5*g^7 + 15*a^2*c^4*d^4*e^14*f^4*g^8 - 20*a^3*c^3*d^3*e^15*f^3*g^9 + 1
5*a^4*c^2*d^2*e^16*f^2*g^10 - 6*a^5*c*d*e^17*f*g^11 + a^6*e^18*g^12) + 13*(c^15*d^15*e^14*f^3*g^8*abs(c)*abs(d
) - 3*a*c^14*d^14*e^15*f^2*g^9*abs(c)*abs(d) + 3*a^2*c^13*d^13*e^16*f*g^10*abs(c)*abs(d) - a^3*c^12*d^12*e^17*
g^11*abs(c)*abs(d))/(c^6*d^6*e^12*f^6*g^6 - 6*a*c^5*d^5*e^13*f^5*g^7 + 15*a^2*c^4*d^4*e^14*f^4*g^8 - 20*a^3*c^
3*d^3*e^15*f^3*g^9 + 15*a^4*c^2*d^2*e^16*f^2*g^10 - 6*a^5*c*d*e^17*f*g^11 + a^6*e^18*g^12)) + 143*(c^16*d^16*e
^16*f^4*g^7*abs(c)*abs(d) - 4*a*c^15*d^15*e^17*f^3*g^8*abs(c)*abs(d) + 6*a^2*c^14*d^14*e^18*f^2*g^9*abs(c)*abs
(d) - 4*a^3*c^13*d^13*e^19*f*g^10*abs(c)*abs(d) + a^4*c^12*d^12*e^20*g^11*abs(c)*abs(d))/(c^6*d^6*e^12*f^6*g^6
- 6*a*c^5*d^5*e^13*f^5*g^7 + 15*a^2*c^4*d^4*e^14*f^4*g^8 - 20*a^3*c^3*d^3*e^15*f^3*g^9 + 15*a^4*c^2*d^2*e^16*
f^2*g^10 - 6*a^5*c*d*e^17*f*g^11 + a^6*e^18*g^12)) + 429*(c^17*d^17*e^18*f^5*g^6*abs(c)*abs(d) - 5*a*c^16*d^16
*e^19*f^4*g^7*abs(c)*abs(d) + 10*a^2*c^15*d^15*e^20*f^3*g^8*abs(c)*abs(d) - 10*a^3*c^14*d^14*e^21*f^2*g^9*abs(
c)*abs(d) + 5*a^4*c^13*d^13*e^22*f*g^10*abs(c)*abs(d) - a^5*c^12*d^12*e^23*g^11*abs(c)*abs(d))/(c^6*d^6*e^12*f
^6*g^6 - 6*a*c^5*d^5*e^13*f^5*g^7 + 15*a^2*c^4*d^4*e^14*f^4*g^8 - 20*a^3*c^3*d^3*e^15*f^3*g^9 + 15*a^4*c^2*d^2
*e^16*f^2*g^10 - 6*a^5*c*d*e^17*f*g^11 + a^6*e^18*g^12))/(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)^(13/2)

Mupad [B] (verification not implemented)

Time = 13.62 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {462\,a^6\,e^6\,g^3-1638\,a^5\,c\,d\,e^5\,f\,g^2+2002\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^3\,\left (-10\,a^3\,c^3\,d^3\,e^3\,g^3+78\,a^2\,c^4\,d^4\,e^2\,f\,g^2-286\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^6\,d^6\,x^6}{3003\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^4\,d^4\,x^4\,\left (3\,a^2\,e^2\,g^2-26\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{3003\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^5\,d^5\,x^5\,\left (a\,e\,g-13\,c\,d\,f\right )}{3003\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (567\,a^3\,e^3\,g^3-2093\,a^2\,c\,d\,e^2\,f\,g^2+2717\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (371\,a^3\,e^3\,g^3-1469\,a^2\,c\,d\,e^2\,f\,g^2+2145\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^6}+\frac {6\,f\,x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {6\,f^5\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {15\,f^2\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {20\,f^3\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {15\,f^4\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}} \]

[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)