\(\int \frac {(f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [701]

Optimal result

Integrand size = 46, antiderivative size = 269 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {16 (c d f-a e g)^2 \left (2 a e^2 g-c d (9 e f-7 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 c^4 d^4 e (d+e x)^{7/2}}+\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}} \]

[Out]

-16/3003*(-a*e*g+c*d*f)^2*(2*a*e^2*g-c*d*(-7*d*g+9*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^4/d^4/e/(e*
x+d)^(7/2)+16/429*g*(-a*e*g+c*d*f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/e/(e*x+d)^(5/2)+12/143*(-
a*e*g+c*d*f)*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/(e*x+d)^(7/2)+2/13*(g*x+f)^3*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*x+d)^(7/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {884, 808, 662} \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{3003 c^4 d^4 e (d+e x)^{7/2}}+\frac {16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}} \]

[In]

Int[((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*
c^4*d^4*e*(d + e*x)^(7/2)) + (16*g*(c*d*f - a*e*g)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(429*c^3*d
^3*e*(d + e*x)^(5/2)) + (12*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*c^
2*d^2*(d + e*x)^(7/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*c*d*(d + e*x)^(7/2)
)

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 808

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 884

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n*((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Dist[n*((c*e*f + c*d
*g - b*e*g)/(c*e*(m - n - 1))), 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] &&  !Int
egerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = \frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}+\frac {(6 (c d f-a e g)) \int \frac {(f+g x)^2 \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}} \, dx}{13 c d} \\ & = \frac {12 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}+\frac {\left (24 (c d f-a e g)^2\right ) \int \frac {(f+g x) \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}} \, dx}{143 c^2 d^2} \\ & = \frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}}+\frac {\left (8 (c d f-a e g)^2 \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right )\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}} \, dx}{429 c^2 d^2} \\ & = \frac {16 (c d f-a e g)^2 \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 c^3 d^3 (d+e x)^{7/2}}+\frac {16 g (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 e (d+e x)^{5/2}}+\frac {12 (c d f-a e g) (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d (d+e x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.55 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (13 f+7 g x)-2 a c^2 d^2 e g \left (143 f^2+182 f g x+63 g^2 x^2\right )+c^3 d^3 \left (429 f^3+1001 f^2 g x+819 f g^2 x^2+231 g^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt {d+e x}} \]

[In]

Integrate[((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(13*f + 7*g*x) - 2*a*c^2
*d^2*e*g*(143*f^2 + 182*f*g*x + 63*g^2*x^2) + c^3*d^3*(429*f^3 + 1001*f^2*g*x + 819*f*g^2*x^2 + 231*g^3*x^3)))
/(3003*c^4*d^4*Sqrt[d + e*x])

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (-231 g^{3} x^{3} c^{3} d^{3}+126 a \,c^{2} d^{2} e \,g^{3} x^{2}-819 c^{3} d^{3} f \,g^{2} x^{2}-56 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -1001 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-104 a^{2} c d \,e^{2} f \,g^{2}+286 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} c^{3} d^{3}\right )}{3003 \sqrt {e x +d}\, c^{4} d^{4}}\)	\(180\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-231 g^{3} x^{3} c^{3} d^{3}+126 a \,c^{2} d^{2} e \,g^{3} x^{2}-819 c^{3} d^{3} f \,g^{2} x^{2}-56 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -1001 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-104 a^{2} c d \,e^{2} f \,g^{2}+286 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 c^{4} d^{4} \left (e x +d \right )^{\frac {5}{2}}}\)	\(188\)

[In]

int((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3003*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(-231*c^3*d^3*g^3*x^3+126*a*c^2*d^2*e*g^3*x^2-
819*c^3*d^3*f*g^2*x^2-56*a^2*c*d*e^2*g^3*x+364*a*c^2*d^2*e*f*g^2*x-1001*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-104*a^2
*c*d*e^2*f*g^2+286*a*c^2*d^2*e*f^2*g-429*c^3*d^3*f^3)/c^4/d^4

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.55 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (231 \, c^{6} d^{6} g^{3} x^{6} + 429 \, a^{3} c^{3} d^{3} e^{3} f^{3} - 286 \, a^{4} c^{2} d^{2} e^{4} f^{2} g + 104 \, a^{5} c d e^{5} f g^{2} - 16 \, a^{6} e^{6} g^{3} + 63 \, {\left (13 \, c^{6} d^{6} f g^{2} + 9 \, a c^{5} d^{5} e g^{3}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{6} f^{2} g + 299 \, a c^{5} d^{5} e f g^{2} + 53 \, a^{2} c^{4} d^{4} e^{2} g^{3}\right )} x^{4} + {\left (429 \, c^{6} d^{6} f^{3} + 2717 \, a c^{5} d^{5} e f^{2} g + 1469 \, a^{2} c^{4} d^{4} e^{2} f g^{2} + 5 \, a^{3} c^{3} d^{3} e^{3} g^{3}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{5} e f^{3} + 715 \, a^{2} c^{4} d^{4} e^{2} f^{2} g + 13 \, a^{3} c^{3} d^{3} e^{3} f g^{2} - 2 \, a^{4} c^{2} d^{2} e^{4} g^{3}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{4} e^{2} f^{3} + 143 \, a^{3} c^{3} d^{3} e^{3} f^{2} g - 52 \, a^{4} c^{2} d^{2} e^{4} f g^{2} + 8 \, a^{5} c d e^{5} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3003 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]

[In]

integrate((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/3003*(231*c^6*d^6*g^3*x^6 + 429*a^3*c^3*d^3*e^3*f^3 - 286*a^4*c^2*d^2*e^4*f^2*g + 104*a^5*c*d*e^5*f*g^2 - 16
*a^6*e^6*g^3 + 63*(13*c^6*d^6*f*g^2 + 9*a*c^5*d^5*e*g^3)*x^5 + 7*(143*c^6*d^6*f^2*g + 299*a*c^5*d^5*e*f*g^2 +
53*a^2*c^4*d^4*e^2*g^3)*x^4 + (429*c^6*d^6*f^3 + 2717*a*c^5*d^5*e*f^2*g + 1469*a^2*c^4*d^4*e^2*f*g^2 + 5*a^3*c
^3*d^3*e^3*g^3)*x^3 + 3*(429*a*c^5*d^5*e*f^3 + 715*a^2*c^4*d^4*e^2*f^2*g + 13*a^3*c^3*d^3*e^3*f*g^2 - 2*a^4*c^
2*d^2*e^4*g^3)*x^2 + (1287*a^2*c^4*d^4*e^2*f^3 + 143*a^3*c^3*d^3*e^3*f^2*g - 52*a^4*c^2*d^2*e^4*f*g^2 + 8*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)/(c^4*d^4*e*x + c^4*d^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.35 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{3}}{7 \, c d} + \frac {2 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{2} g}{21 \, c^{2} d^{2}} + \frac {2 \, {\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f g^{2}}{231 \, c^{3} d^{3}} + \frac {2 \, {\left (231 \, c^{6} d^{6} x^{6} + 567 \, a c^{5} d^{5} e x^{5} + 371 \, a^{2} c^{4} d^{4} e^{2} x^{4} + 5 \, a^{3} c^{3} d^{3} e^{3} x^{3} - 6 \, a^{4} c^{2} d^{2} e^{4} x^{2} + 8 \, a^{5} c d e^{5} x - 16 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} g^{3}}{3003 \, c^{4} d^{4}} \]

[In]

integrate((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*x + a*e)*f^3/(c*d) + 2/21*(7*c^4*d^
4*x^4 + 19*a*c^3*d^3*e*x^3 + 15*a^2*c^2*d^2*e^2*x^2 + a^3*c*d*e^3*x - 2*a^4*e^4)*sqrt(c*d*x + a*e)*f^2*g/(c^2*
d^2) + 2/231*(63*c^5*d^5*x^5 + 161*a*c^4*d^4*e*x^4 + 113*a^2*c^3*d^3*e^2*x^3 + 3*a^3*c^2*d^2*e^3*x^2 - 4*a^4*c
*d*e^4*x + 8*a^5*e^5)*sqrt(c*d*x + a*e)*f*g^2/(c^3*d^3) + 2/3003*(231*c^6*d^6*x^6 + 567*a*c^5*d^5*e*x^5 + 371*
a^2*c^4*d^4*e^2*x^4 + 5*a^3*c^3*d^3*e^3*x^3 - 6*a^4*c^2*d^2*e^4*x^2 + 8*a^5*c*d*e^5*x - 16*a^6*e^6)*sqrt(c*d*x
+ a*e)*g^3/(c^4*d^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3058 vs. \(2 (245) = 490\).

Time = 0.41 (sec) , antiderivative size = 3058, normalized size of antiderivative = 11.37 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/45045*(15015*a^2*f^3*((sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^3)*a*e^2)/(c*d) + ((e*x + d)*c*d*e
- c*d^2*e + a*e^3)^(3/2)/(c*d*e))*abs(e) + 429*c^2*d^2*f^3*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^
2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d
^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)
*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e)/e^2 + 2574*a*c*d*f^2*g*((15*sqrt(
-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 -
8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*(
(e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*
abs(e)/e + 1287*a^2*f*g^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqr
t(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c
*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d
^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e) - 429*c^2*d^2*f^2*g*((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c
*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*
e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4*e^3) + (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^
9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^
3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))/(c^4*d^4*e^7))*abs(e)/e^2 - 858*a*c*d*f*g^2*((35*sqrt(-c*d^2
*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sq
rt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4*e^3) + (105*((e*x + d)*c*d*e
- c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*
e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))/(c^4*d^4*e^7))*abs(e)/e - 143
*a^2*g^3*((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^
3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4*e^3)
+ (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*
e^6 + 135*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))/(c^4
*d^4*e^7))*abs(e) + 39*c^2*d^2*f*g^2*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^10 - 35*sqrt(-c*d^2*e + a*e^3)*a*c^4*d
^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e + a*e^3)*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^
2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)/(c^5*d^5*e^4) + (1155*((e*x + d)*c*d*e - c*d
^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((e*x + d)*c*d*e
- c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((e*x + d)*c*d*e
- c*d^2*e + a*e^3)^(11/2))/(c^5*d^5*e^9))*abs(e)/e^2 + 26*a*c*d*g^3*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^10 - 3
5*sqrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e + a*e^3)
*a^3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)/(c^5*d^5*e^4
) + (1155*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*
a^3*e^9 + 2970*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9
/2)*a*e^3 + 315*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))/(c^5*d^5*e^9))*abs(e)/e - 5*c^2*d^2*g^3*((693*sqrt
(-c*d^2*e + a*e^3)*c^6*d^12 - 63*sqrt(-c*d^2*e + a*e^3)*a*c^5*d^10*e^2 - 70*sqrt(-c*d^2*e + a*e^3)*a^2*c^4*d^8
*e^4 - 80*sqrt(-c*d^2*e + a*e^3)*a^3*c^3*d^6*e^6 - 96*sqrt(-c*d^2*e + a*e^3)*a^4*c^2*d^4*e^8 - 128*sqrt(-c*d^2
*e + a*e^3)*a^5*c*d^2*e^10 - 256*sqrt(-c*d^2*e + a*e^3)*a^6*e^12)/(c^6*d^6*e^5) + (3003*((e*x + d)*c*d*e - c*d
^2*e + a*e^3)^(3/2)*a^5*e^15 - 9009*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^4*e^12 + 12870*((e*x + d)*c*d*
e - c*d^2*e + a*e^3)^(7/2)*a^3*e^9 - 10010*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a^2*e^6 + 4095*((e*x + d)
*c*d*e - c*d^2*e + a*e^3)^(11/2)*a*e^3 - 693*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(13/2))/(c^6*d^6*e^11))*abs(e
)/e^2 - 6006*a*c*d*f^3*((3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2
*e + a*e^3)*a^2*e^4)/(c^2*d^2) + (5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*d
^2*e + a*e^3)^(5/2))/(c^2*d^2*e^2))*abs(e)/e^2 - 9009*a^2*f^2*g*((3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d
^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2) + (5*((e*x + d)*c*d*e - c*d^2*e + a*e^
3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))/(c^2*d^2*e^2))*abs(e)/e)/e

Mupad [B] (verification not implemented)

Time = 12.93 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.41 \[ \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g\,x^4\,\left (53\,a^2\,e^2\,g^2+299\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{429}-\frac {32\,a^6\,e^6\,g^3-208\,a^5\,c\,d\,e^5\,f\,g^2+572\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,c^4\,d^4}+\frac {x^3\,\left (10\,a^3\,c^3\,d^3\,e^3\,g^3+2938\,a^2\,c^4\,d^4\,e^2\,f\,g^2+5434\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,c^4\,d^4}+\frac {2\,c^2\,d^2\,g^3\,x^6}{13}+\frac {6\,c\,d\,g^2\,x^5\,\left (9\,a\,e\,g+13\,c\,d\,f\right )}{143}+\frac {2\,a^2\,e^2\,x\,\left (8\,a^3\,e^3\,g^3-52\,a^2\,c\,d\,e^2\,f\,g^2+143\,a\,c^2\,d^2\,e\,f^2\,g+1287\,c^3\,d^3\,f^3\right )}{3003\,c^3\,d^3}+\frac {2\,a\,e\,x^2\,\left (-2\,a^3\,e^3\,g^3+13\,a^2\,c\,d\,e^2\,f\,g^2+715\,a\,c^2\,d^2\,e\,f^2\,g+429\,c^3\,d^3\,f^3\right )}{1001\,c^2\,d^2}\right )}{\sqrt {d+e\,x}} \]

[In]

int(((f + g*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*g*x^4*(53*a^2*e^2*g^2 + 143*c^2*d^2*f^2 + 299*a*c*d*e*f*g))
/429 - (32*a^6*e^6*g^3 - 858*a^3*c^3*d^3*e^3*f^3 + 572*a^4*c^2*d^2*e^4*f^2*g - 208*a^5*c*d*e^5*f*g^2)/(3003*c^
4*d^4) + (x^3*(858*c^6*d^6*f^3 + 10*a^3*c^3*d^3*e^3*g^3 + 2938*a^2*c^4*d^4*e^2*f*g^2 + 5434*a*c^5*d^5*e*f^2*g)
)/(3003*c^4*d^4) + (2*c^2*d^2*g^3*x^6)/13 + (6*c*d*g^2*x^5*(9*a*e*g + 13*c*d*f))/143 + (2*a^2*e^2*x*(8*a^3*e^3
*g^3 + 1287*c^3*d^3*f^3 + 143*a*c^2*d^2*e*f^2*g - 52*a^2*c*d*e^2*f*g^2))/(3003*c^3*d^3) + (2*a*e*x^2*(429*c^3*
d^3*f^3 - 2*a^3*e^3*g^3 + 715*a*c^2*d^2*e*f^2*g + 13*a^2*c*d*e^2*f*g^2))/(1001*c^2*d^2)))/(d + e*x)^(1/2)