\(\int \frac {(f+g x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [700]

Optimal result

Integrand size = 46, antiderivative size = 336 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {128 (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]

[Out]

-128/45045*(-a*e*g+c*d*f)^3*(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^5/d^5/e/(
e*x+d)^(7/2)+128/6435*g*(-a*e*g+c*d*f)^3*(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)^(5/2)+32/71
5*(-a*e*g+c*d*f)^2*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/(e*x+d)^(7/2)+16/195*(-a*e*g+c*d*
f)*(g*x+f)^3*(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)+2/15*(g*x+f)^4*(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 = 0.36 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {884, 808, 662} \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=-\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]

[In]

Int[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-128*(c*d*f - a*e*g)^3*(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))/(4504
5*c^5*d^5*e*(d + e*x)^(7/2)) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c
^4*d^4*e*(d + e*x)^(5/2)) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(
715*c^3*d^3*(d + e*x)^(7/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/
(195*c^2*d^2*(d + e*x)^(7/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*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)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {(8 (c d f-a e g)) \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}{15 c d} \\ & = \frac {16 (c d f-a e g) (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}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {\left (16 (c d f-a e g)^2\right ) \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}{65 c^2 d^2} \\ & = \frac {32 (c d f-a e g)^2 (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}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (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}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {\left (64 (c d f-a e g)^3\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}{715 c^3 d^3} \\ & = \frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (c d f-a e g)^2 (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}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (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}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {\left (64 (c d f-a e g)^3 \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}{6435 c^3 d^3} \\ & = \frac {128 (c d f-a e g)^3 \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}}{45045 c^4 d^4 (d+e x)^{7/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (c d f-a e g)^2 (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}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (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}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.61 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (15 f+7 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (65 f^2+70 f g x+21 g^2 x^2\right )-8 a c^3 d^3 e g \left (715 f^3+1365 f^2 g x+945 f g^2 x^2+231 g^3 x^3\right )+c^4 d^4 \left (6435 f^4+20020 f^3 g x+24570 f^2 g^2 x^2+13860 f g^3 x^3+3003 g^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt {d+e x}} \]

[In]

Integrate[((f + g*x)^4*(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)]*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(15*f + 7*g*x) + 48*a^2
*c^2*d^2*e^2*g^2*(65*f^2 + 70*f*g*x + 21*g^2*x^2) - 8*a*c^3*d^3*e*g*(715*f^3 + 1365*f^2*g*x + 945*f*g^2*x^2 +
231*g^3*x^3) + c^4*d^4*(6435*f^4 + 20020*f^3*g*x + 24570*f^2*g^2*x^2 + 13860*f*g^3*x^3 + 3003*g^4*x^4)))/(4504
5*c^5*d^5*Sqrt[d + e*x])

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (3003 g^{4} x^{4} c^{4} d^{4}-1848 a \,c^{3} d^{3} e \,g^{4} x^{3}+13860 c^{4} d^{4} f \,g^{3} x^{3}+1008 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-7560 a \,c^{3} d^{3} e f \,g^{3} x^{2}+24570 c^{4} d^{4} f^{2} g^{2} x^{2}-448 a^{3} c d \,e^{3} g^{4} x +3360 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -10920 a \,c^{3} d^{3} e \,f^{2} g^{2} x +20020 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-960 a^{3} c d \,e^{3} f \,g^{3}+3120 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-5720 a \,c^{3} d^{3} e \,f^{3} g +6435 f^{4} c^{4} d^{4}\right )}{45045 \sqrt {e x +d}\, c^{5} d^{5}}\)	\(275\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (3003 g^{4} x^{4} c^{4} d^{4}-1848 a \,c^{3} d^{3} e \,g^{4} x^{3}+13860 c^{4} d^{4} f \,g^{3} x^{3}+1008 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-7560 a \,c^{3} d^{3} e f \,g^{3} x^{2}+24570 c^{4} d^{4} f^{2} g^{2} x^{2}-448 a^{3} c d \,e^{3} g^{4} x +3360 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -10920 a \,c^{3} d^{3} e \,f^{2} g^{2} x +20020 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-960 a^{3} c d \,e^{3} f \,g^{3}+3120 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-5720 a \,c^{3} d^{3} e \,f^{3} g +6435 f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{45045 c^{5} d^{5} \left (e x +d \right )^{\frac {5}{2}}}\)	\(283\)

[In]

int((g*x+f)^4*(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]

2/45045*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(3003*c^4*d^4*g^4*x^4-1848*a*c^3*d^3*e*g^4*x^3
+13860*c^4*d^4*f*g^3*x^3+1008*a^2*c^2*d^2*e^2*g^4*x^2-7560*a*c^3*d^3*e*f*g^3*x^2+24570*c^4*d^4*f^2*g^2*x^2-448
*a^3*c*d*e^3*g^4*x+3360*a^2*c^2*d^2*e^2*f*g^3*x-10920*a*c^3*d^3*e*f^2*g^2*x+20020*c^4*d^4*f^3*g*x+128*a^4*e^4*
g^4-960*a^3*c*d*e^3*f*g^3+3120*a^2*c^2*d^2*e^2*f^2*g^2-5720*a*c^3*d^3*e*f^3*g+6435*c^4*d^4*f^4)/c^5/d^5

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.69 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3003 \, c^{7} d^{7} g^{4} x^{7} + 6435 \, a^{3} c^{4} d^{4} e^{3} f^{4} - 5720 \, a^{4} c^{3} d^{3} e^{4} f^{3} g + 3120 \, a^{5} c^{2} d^{2} e^{5} f^{2} g^{2} - 960 \, a^{6} c d e^{6} f g^{3} + 128 \, a^{7} e^{7} g^{4} + 231 \, {\left (60 \, c^{7} d^{7} f g^{3} + 31 \, a c^{6} d^{6} e g^{4}\right )} x^{6} + 63 \, {\left (390 \, c^{7} d^{7} f^{2} g^{2} + 540 \, a c^{6} d^{6} e f g^{3} + 71 \, a^{2} c^{5} d^{5} e^{2} g^{4}\right )} x^{5} + 35 \, {\left (572 \, c^{7} d^{7} f^{3} g + 1794 \, a c^{6} d^{6} e f^{2} g^{2} + 636 \, a^{2} c^{5} d^{5} e^{2} f g^{3} + a^{3} c^{4} d^{4} e^{3} g^{4}\right )} x^{4} + 5 \, {\left (1287 \, c^{7} d^{7} f^{4} + 10868 \, a c^{6} d^{6} e f^{3} g + 8814 \, a^{2} c^{5} d^{5} e^{2} f^{2} g^{2} + 60 \, a^{3} c^{4} d^{4} e^{3} f g^{3} - 8 \, a^{4} c^{3} d^{3} e^{4} g^{4}\right )} x^{3} + 3 \, {\left (6435 \, a c^{6} d^{6} e f^{4} + 14300 \, a^{2} c^{5} d^{5} e^{2} f^{3} g + 390 \, a^{3} c^{4} d^{4} e^{3} f^{2} g^{2} - 120 \, a^{4} c^{3} d^{3} e^{4} f g^{3} + 16 \, a^{5} c^{2} d^{2} e^{5} g^{4}\right )} x^{2} + {\left (19305 \, a^{2} c^{5} d^{5} e^{2} f^{4} + 2860 \, a^{3} c^{4} d^{4} e^{3} f^{3} g - 1560 \, a^{4} c^{3} d^{3} e^{4} f^{2} g^{2} + 480 \, a^{5} c^{2} d^{2} e^{5} f g^{3} - 64 \, a^{6} c d e^{6} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{45045 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

[In]

integrate((g*x+f)^4*(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]

2/45045*(3003*c^7*d^7*g^4*x^7 + 6435*a^3*c^4*d^4*e^3*f^4 - 5720*a^4*c^3*d^3*e^4*f^3*g + 3120*a^5*c^2*d^2*e^5*f
^2*g^2 - 960*a^6*c*d*e^6*f*g^3 + 128*a^7*e^7*g^4 + 231*(60*c^7*d^7*f*g^3 + 31*a*c^6*d^6*e*g^4)*x^6 + 63*(390*c
^7*d^7*f^2*g^2 + 540*a*c^6*d^6*e*f*g^3 + 71*a^2*c^5*d^5*e^2*g^4)*x^5 + 35*(572*c^7*d^7*f^3*g + 1794*a*c^6*d^6*
e*f^2*g^2 + 636*a^2*c^5*d^5*e^2*f*g^3 + a^3*c^4*d^4*e^3*g^4)*x^4 + 5*(1287*c^7*d^7*f^4 + 10868*a*c^6*d^6*e*f^3
*g + 8814*a^2*c^5*d^5*e^2*f^2*g^2 + 60*a^3*c^4*d^4*e^3*f*g^3 - 8*a^4*c^3*d^3*e^4*g^4)*x^3 + 3*(6435*a*c^6*d^6*
e*f^4 + 14300*a^2*c^5*d^5*e^2*f^3*g + 390*a^3*c^4*d^4*e^3*f^2*g^2 - 120*a^4*c^3*d^3*e^4*f*g^3 + 16*a^5*c^2*d^2
*e^5*g^4)*x^2 + (19305*a^2*c^5*d^5*e^2*f^4 + 2860*a^3*c^4*d^4*e^3*f^3*g - 1560*a^4*c^3*d^3*e^4*f^2*g^2 + 480*a
^5*c^2*d^2*e^5*f*g^3 - 64*a^6*c*d*e^6*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d
^5*e*x + c^5*d^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**4*(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 = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.48 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{4}}{7 \, c d} + \frac {8 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{3} g}{63 \, c^{2} d^{2}} + \frac {4 \, {\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{231 \, c^{3} d^{3}} + \frac {8 \, {\left (231 \, c^{6} d^{6} x^{6} + 567 \, a c^{5} d^{5} e x^{5} + 371 \, a^{2} c^{4} d^{4} e^{2} x^{4} + 5 \, a^{3} c^{3} d^{3} e^{3} x^{3} - 6 \, a^{4} c^{2} d^{2} e^{4} x^{2} + 8 \, a^{5} c d e^{5} x - 16 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} f g^{3}}{3003 \, c^{4} d^{4}} + \frac {2 \, {\left (3003 \, c^{7} d^{7} x^{7} + 7161 \, a c^{6} d^{6} e x^{6} + 4473 \, a^{2} c^{5} d^{5} e^{2} x^{5} + 35 \, a^{3} c^{4} d^{4} e^{3} x^{4} - 40 \, a^{4} c^{3} d^{3} e^{4} x^{3} + 48 \, a^{5} c^{2} d^{2} e^{5} x^{2} - 64 \, a^{6} c d e^{6} x + 128 \, a^{7} e^{7}\right )} \sqrt {c d x + a e} g^{4}}{45045 \, c^{5} d^{5}} \]

[In]

integrate((g*x+f)^4*(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^4/(c*d) + 8/63*(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^3*g/(c^2*
d^2) + 4/231*(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^2*g^2/(c^3*d^3) + 8/3003*(231*c^6*d^6*x^6 + 567*a*c^5*d^5*e*x^5 + 37
1*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)*f*g^3/(c^4*d^4) + 2/45045*(3003*c^7*d^7*x^7 + 7161*a*c^6*d^6*e*x^6 + 4473*a^2*c^5*d^5*e^2*x^5 + 35*a
^3*c^4*d^4*e^3*x^4 - 40*a^4*c^3*d^3*e^4*x^3 + 48*a^5*c^2*d^2*e^5*x^2 - 64*a^6*c*d*e^6*x + 128*a^7*e^7)*sqrt(c*
d*x + a*e)*g^4/(c^5*d^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4287 vs. \(2 (306) = 612\).

Time = 0.48 (sec) , antiderivative size = 4287, normalized size of antiderivative = 12.76 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^4*(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^4*((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^4*((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 + 3432*a*c*d*f^3*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 + 2574*a^2*f^2*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*s
qrt(-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) - 572*c^2*d^2*f^3*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 - 1716*a*c*d*f^2*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*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
- 572*a^2*f*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) + 78*c^2*d^2*f^2*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*s
qrt(-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 + 104*a*c*d*f*g^3*((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 + 13*a^2*g^4*(
(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 - 154
0*((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) - 20*c^2*d^2*f*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 - 10*a*c*d*g^4*((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 - 1
0010*((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 + c^2*d^2*g^4*((3003*sqrt(-c*d^2*e
+ a*e^3)*c^7*d^14 - 231*sqrt(-c*d^2*e + a*e^3)*a*c^6*d^12*e^2 - 252*sqrt(-c*d^2*e + a*e^3)*a^2*c^5*d^10*e^4 -
280*sqrt(-c*d^2*e + a*e^3)*a^3*c^4*d^8*e^6 - 320*sqrt(-c*d^2*e + a*e^3)*a^4*c^3*d^6*e^8 - 384*sqrt(-c*d^2*e +
a*e^3)*a^5*c^2*d^4*e^10 - 512*sqrt(-c*d^2*e + a*e^3)*a^6*c*d^2*e^12 - 1024*sqrt(-c*d^2*e + a*e^3)*a^7*e^14)/(
c^7*d^7*e^6) + (15015*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^6*e^18 - 54054*((e*x + d)*c*d*e - c*d^2*e +
a*e^3)^(5/2)*a^5*e^15 + 96525*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^4*e^12 - 100100*((e*x + d)*c*d*e - c
*d^2*e + a*e^3)^(9/2)*a^3*e^9 + 61425*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(11/2)*a^2*e^6 - 20790*((e*x + d)*c*
d*e - c*d^2*e + a*e^3)^(13/2)*a*e^3 + 3003*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(15/2))/(c^7*d^7*e^13))*abs(e)/
e^2 - 6006*a*c*d*f^4*((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 - 12012*a^2*f^3*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 = 12.83 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.56 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^2\,x^5\,\left (71\,a^2\,e^2\,g^2+540\,a\,c\,d\,e\,f\,g+390\,c^2\,d^2\,f^2\right )}{715}+\frac {256\,a^7\,e^7\,g^4-1920\,a^6\,c\,d\,e^6\,f\,g^3+6240\,a^5\,c^2\,d^2\,e^5\,f^2\,g^2-11440\,a^4\,c^3\,d^3\,e^4\,f^3\,g+12870\,a^3\,c^4\,d^4\,e^3\,f^4}{45045\,c^5\,d^5}+\frac {x^3\,\left (-80\,a^4\,c^3\,d^3\,e^4\,g^4+600\,a^3\,c^4\,d^4\,e^3\,f\,g^3+88140\,a^2\,c^5\,d^5\,e^2\,f^2\,g^2+108680\,a\,c^6\,d^6\,e\,f^3\,g+12870\,c^7\,d^7\,f^4\right )}{45045\,c^5\,d^5}+\frac {2\,c^2\,d^2\,g^4\,x^7}{15}+\frac {2\,c\,d\,g^3\,x^6\,\left (31\,a\,e\,g+60\,c\,d\,f\right )}{195}+\frac {2\,g\,x^4\,\left (a^3\,e^3\,g^3+636\,a^2\,c\,d\,e^2\,f\,g^2+1794\,a\,c^2\,d^2\,e\,f^2\,g+572\,c^3\,d^3\,f^3\right )}{1287\,c\,d}+\frac {2\,a^2\,e^2\,x\,\left (-64\,a^4\,e^4\,g^4+480\,a^3\,c\,d\,e^3\,f\,g^3-1560\,a^2\,c^2\,d^2\,e^2\,f^2\,g^2+2860\,a\,c^3\,d^3\,e\,f^3\,g+19305\,c^4\,d^4\,f^4\right )}{45045\,c^4\,d^4}+\frac {2\,a\,e\,x^2\,\left (16\,a^4\,e^4\,g^4-120\,a^3\,c\,d\,e^3\,f\,g^3+390\,a^2\,c^2\,d^2\,e^2\,f^2\,g^2+14300\,a\,c^3\,d^3\,e\,f^3\,g+6435\,c^4\,d^4\,f^4\right )}{15015\,c^3\,d^3}\right )}{\sqrt {d+e\,x}} \]

[In]

int(((f + g*x)^4*(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^2*x^5*(71*a^2*e^2*g^2 + 390*c^2*d^2*f^2 + 540*a*c*d*e*f*g
))/715 + (256*a^7*e^7*g^4 + 12870*a^3*c^4*d^4*e^3*f^4 - 11440*a^4*c^3*d^3*e^4*f^3*g - 1920*a^6*c*d*e^6*f*g^3 +
6240*a^5*c^2*d^2*e^5*f^2*g^2)/(45045*c^5*d^5) + (x^3*(12870*c^7*d^7*f^4 - 80*a^4*c^3*d^3*e^4*g^4 + 600*a^3*c^
4*d^4*e^3*f*g^3 + 108680*a*c^6*d^6*e*f^3*g + 88140*a^2*c^5*d^5*e^2*f^2*g^2))/(45045*c^5*d^5) + (2*c^2*d^2*g^4*
x^7)/15 + (2*c*d*g^3*x^6*(31*a*e*g + 60*c*d*f))/195 + (2*g*x^4*(a^3*e^3*g^3 + 572*c^3*d^3*f^3 + 1794*a*c^2*d^2
*e*f^2*g + 636*a^2*c*d*e^2*f*g^2))/(1287*c*d) + (2*a^2*e^2*x*(19305*c^4*d^4*f^4 - 64*a^4*e^4*g^4 + 2860*a*c^3*
d^3*e*f^3*g + 480*a^3*c*d*e^3*f*g^3 - 1560*a^2*c^2*d^2*e^2*f^2*g^2))/(45045*c^4*d^4) + (2*a*e*x^2*(16*a^4*e^4*
g^4 + 6435*c^4*d^4*f^4 + 14300*a*c^3*d^3*e*f^3*g - 120*a^3*c*d*e^3*f*g^3 + 390*a^2*c^2*d^2*e^2*f^2*g^2))/(1501
5*c^3*d^3)))/(d + e*x)^(1/2)