3.7.89 \(\int \frac {(f+g x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [689]

Optimal. Leaf size=336 \[ -\frac {128 (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}} \]

[Out]

-128/15015*(-a*e*g+c*d*f)^3*(2*a*e^2*g-c*d*(-5*d*g+7*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^5/d^5/e/(
e*x+d)^(5/2)+128/3003*g*(-a*e*g+c*d*f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4/e/(e*x+d)^(3/2)+32/42
9*(-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)^(5/2)/c^3/d^3/(e*x+d)^(5/2)+16/143*(-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)^(5/2)/c^2/d^2/(e*x+d)^(5/2)+2/13*(g*x+f)^4*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(5/2)/c/d/(e*x+d)^(5/2)

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Rubi [A]
time = 0.39, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {884, 808, 662} \begin {gather*} -\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3}{3003 c^4 d^4 e (d+e x)^{3/2}}+\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 )^{5/2} (c d f-a e g)^2}{429 c^3 d^3 (d+e x)^{5/2}}+\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 )^{5/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{13 c d (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1501
5*c^5*d^5*e*(d + e*x)^(5/2)) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c
^4*d^4*e*(d + e*x)^(3/2)) + (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)^(5/2))/(
429*c^3*d^3*(d + e*x)^(5/2)) + (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)^(5/2))/
(143*c^2*d^2*(d + e*x)^(5/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d*(d + e*x
)^(5/2))

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*} \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 )^{3/2}}{(d+e x)^{3/2}} \, dx &=\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 )^{5/2}}{13 c d (d+e x)^{5/2}}+\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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{13 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 )^{5/2}}{143 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {\left (48 (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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{143 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 )^{5/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\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 )^{5/2}}{143 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{13 c d (d+e x)^{5/2}}+\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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{429 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 )^{5/2}}{3003 c^4 d^4 e (d+e x)^{3/2}}+\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 )^{5/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\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 )^{5/2}}{143 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {\left (64 (c d f-a e g)^3 \left (7 f-\frac {5 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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 c^3 d^3}\\ &=\frac {128 (c d f-a e g)^3 \left (7 f-\frac {5 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 )^{5/2}}{15015 c^4 d^4 (d+e x)^{5/2}}+\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 )^{5/2}}{3003 c^4 d^4 e (d+e x)^{3/2}}+\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 )^{5/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\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 )^{5/2}}{143 c^2 d^2 (d+e x)^{5/2}}+\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 )^{5/2}}{13 c d (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 195, normalized size = 0.58 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (13 f+5 g x)+16 a^2 c^2 d^2 e^2 g^2 \left (143 f^2+130 f g x+35 g^2 x^2\right )-8 a c^3 d^3 e g \left (429 f^3+715 f^2 g x+455 f g^2 x^2+105 g^3 x^3\right )+c^4 d^4 \left (3003 f^4+8580 f^3 g x+10010 f^2 g^2 x^2+5460 f g^3 x^3+1155 g^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(13*f + 5*g*x) + 16*a^2*c^2*d^2*e^2*g
^2*(143*f^2 + 130*f*g*x + 35*g^2*x^2) - 8*a*c^3*d^3*e*g*(429*f^3 + 715*f^2*g*x + 455*f*g^2*x^2 + 105*g^3*x^3)
+ c^4*d^4*(3003*f^4 + 8580*f^3*g*x + 10010*f^2*g^2*x^2 + 5460*f*g^3*x^3 + 1155*g^4*x^4)))/(15015*c^5*d^5*(d +
e*x)^(5/2))

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Maple [A]
time = 0.14, size = 275, normalized size = 0.82

method	result	size
default	\(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2} \left (1155 g^{4} x^{4} c^{4} d^{4}-840 a \,c^{3} d^{3} e \,g^{4} x^{3}+5460 c^{4} d^{4} f \,g^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-3640 a \,c^{3} d^{3} e f \,g^{3} x^{2}+10010 c^{4} d^{4} f^{2} g^{2} x^{2}-320 a^{3} c d \,e^{3} g^{4} x +2080 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -5720 a \,c^{3} d^{3} e \,f^{2} g^{2} x +8580 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-832 a^{3} c d \,e^{3} f \,g^{3}+2288 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-3432 a \,c^{3} d^{3} e \,f^{3} g +3003 f^{4} c^{4} d^{4}\right )}{15015 \sqrt {e x +d}\, c^{5} d^{5}}\)	\(275\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (1155 g^{4} x^{4} c^{4} d^{4}-840 a \,c^{3} d^{3} e \,g^{4} x^{3}+5460 c^{4} d^{4} f \,g^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-3640 a \,c^{3} d^{3} e f \,g^{3} x^{2}+10010 c^{4} d^{4} f^{2} g^{2} x^{2}-320 a^{3} c d \,e^{3} g^{4} x +2080 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -5720 a \,c^{3} d^{3} e \,f^{2} g^{2} x +8580 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-832 a^{3} c d \,e^{3} f \,g^{3}+2288 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-3432 a \,c^{3} d^{3} e \,f^{3} g +3003 f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{15015 c^{5} d^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/15015*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^2*(1155*c^4*d^4*g^4*x^4-840*a*c^3*d^3*e*g^4*x^3+
5460*c^4*d^4*f*g^3*x^3+560*a^2*c^2*d^2*e^2*g^4*x^2-3640*a*c^3*d^3*e*f*g^3*x^2+10010*c^4*d^4*f^2*g^2*x^2-320*a^
3*c*d*e^3*g^4*x+2080*a^2*c^2*d^2*e^2*f*g^3*x-5720*a*c^3*d^3*e*f^2*g^2*x+8580*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-8
32*a^3*c*d*e^3*f*g^3+2288*a^2*c^2*d^2*e^2*f^2*g^2-3432*a*c^3*d^3*e*f^3*g+3003*c^4*d^4*f^4)/c^5/d^5

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Maxima [A]
time = 0.36, size = 408, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d x e + a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{4}}{5 \, c d} + \frac {8 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} x^{2} e + a^{2} c d x e^{2} - 2 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{3} g}{35 \, c^{2} d^{2}} + \frac {4 \, {\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} x^{3} e + 3 \, a^{2} c^{2} d^{2} x^{2} e^{2} - 4 \, a^{3} c d x e^{3} + 8 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{105 \, c^{3} d^{3}} + \frac {8 \, {\left (105 \, c^{5} d^{5} x^{5} + 140 \, a c^{4} d^{4} x^{4} e + 5 \, a^{2} c^{3} d^{3} x^{3} e^{2} - 6 \, a^{3} c^{2} d^{2} x^{2} e^{3} + 8 \, a^{4} c d x e^{4} - 16 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f g^{3}}{1155 \, c^{4} d^{4}} + \frac {2 \, {\left (1155 \, c^{6} d^{6} x^{6} + 1470 \, a c^{5} d^{5} x^{5} e + 35 \, a^{2} c^{4} d^{4} x^{4} e^{2} - 40 \, a^{3} c^{3} d^{3} x^{3} e^{3} + 48 \, a^{4} c^{2} d^{2} x^{2} e^{4} - 64 \, a^{5} c d x e^{5} + 128 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} g^{4}}{15015 \, c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*x*e + a^2*e^2)*sqrt(c*d*x + a*e)*f^4/(c*d) + 8/35*(5*c^3*d^3*x^3 + 8*a*c^2*d^2*x^2*
e + a^2*c*d*x*e^2 - 2*a^3*e^3)*sqrt(c*d*x + a*e)*f^3*g/(c^2*d^2) + 4/105*(35*c^4*d^4*x^4 + 50*a*c^3*d^3*x^3*e
+ 3*a^2*c^2*d^2*x^2*e^2 - 4*a^3*c*d*x*e^3 + 8*a^4*e^4)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*d^3) + 8/1155*(105*c^5*d
^5*x^5 + 140*a*c^4*d^4*x^4*e + 5*a^2*c^3*d^3*x^3*e^2 - 6*a^3*c^2*d^2*x^2*e^3 + 8*a^4*c*d*x*e^4 - 16*a^5*e^5)*s
qrt(c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/15015*(1155*c^6*d^6*x^6 + 1470*a*c^5*d^5*x^5*e + 35*a^2*c^4*d^4*x^4*e^2 -
40*a^3*c^3*d^3*x^3*e^3 + 48*a^4*c^2*d^2*x^2*e^4 - 64*a^5*c*d*x*e^5 + 128*a^6*e^6)*sqrt(c*d*x + a*e)*g^4/(c^5*
d^5)

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Fricas [A]
time = 1.82, size = 469, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (1155 \, c^{6} d^{6} g^{4} x^{6} + 5460 \, c^{6} d^{6} f g^{3} x^{5} + 10010 \, c^{6} d^{6} f^{2} g^{2} x^{4} + 8580 \, c^{6} d^{6} f^{3} g x^{3} + 3003 \, c^{6} d^{6} f^{4} x^{2} + 128 \, a^{6} g^{4} e^{6} - 64 \, {\left (a^{5} c d g^{4} x + 13 \, a^{5} c d f g^{3}\right )} e^{5} + 16 \, {\left (3 \, a^{4} c^{2} d^{2} g^{4} x^{2} + 26 \, a^{4} c^{2} d^{2} f g^{3} x + 143 \, a^{4} c^{2} d^{2} f^{2} g^{2}\right )} e^{4} - 8 \, {\left (5 \, a^{3} c^{3} d^{3} g^{4} x^{3} + 39 \, a^{3} c^{3} d^{3} f g^{3} x^{2} + 143 \, a^{3} c^{3} d^{3} f^{2} g^{2} x + 429 \, a^{3} c^{3} d^{3} f^{3} g\right )} e^{3} + {\left (35 \, a^{2} c^{4} d^{4} g^{4} x^{4} + 260 \, a^{2} c^{4} d^{4} f g^{3} x^{3} + 858 \, a^{2} c^{4} d^{4} f^{2} g^{2} x^{2} + 1716 \, a^{2} c^{4} d^{4} f^{3} g x + 3003 \, a^{2} c^{4} d^{4} f^{4}\right )} e^{2} + 2 \, {\left (735 \, a c^{5} d^{5} g^{4} x^{5} + 3640 \, a c^{5} d^{5} f g^{3} x^{4} + 7150 \, a c^{5} d^{5} f^{2} g^{2} x^{3} + 6864 \, a c^{5} d^{5} f^{3} g x^{2} + 3003 \, a c^{5} d^{5} f^{4} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{15015 \, {\left (c^{5} d^{5} x e + c^{5} d^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*c^6*d^6*g^4*x^6 + 5460*c^6*d^6*f*g^3*x^5 + 10010*c^6*d^6*f^2*g^2*x^4 + 8580*c^6*d^6*f^3*g*x^3 +
3003*c^6*d^6*f^4*x^2 + 128*a^6*g^4*e^6 - 64*(a^5*c*d*g^4*x + 13*a^5*c*d*f*g^3)*e^5 + 16*(3*a^4*c^2*d^2*g^4*x^2
+ 26*a^4*c^2*d^2*f*g^3*x + 143*a^4*c^2*d^2*f^2*g^2)*e^4 - 8*(5*a^3*c^3*d^3*g^4*x^3 + 39*a^3*c^3*d^3*f*g^3*x^2
+ 143*a^3*c^3*d^3*f^2*g^2*x + 429*a^3*c^3*d^3*f^3*g)*e^3 + (35*a^2*c^4*d^4*g^4*x^4 + 260*a^2*c^4*d^4*f*g^3*x^
3 + 858*a^2*c^4*d^4*f^2*g^2*x^2 + 1716*a^2*c^4*d^4*f^3*g*x + 3003*a^2*c^4*d^4*f^4)*e^2 + 2*(735*a*c^5*d^5*g^4*
x^5 + 3640*a*c^5*d^5*f*g^3*x^4 + 7150*a*c^5*d^5*f^2*g^2*x^3 + 6864*a*c^5*d^5*f^3*g*x^2 + 3003*a*c^5*d^5*f^4*x)
*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^5*d^5*x*e + c^5*d^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )^{4}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)*(f + g*x)**4/(d + e*x)**(3/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2489 vs. \(2 (317) = 634\).
time = 3.68, size = 2489, normalized size = 7.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/45045*(1716*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)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e -
c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*
d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^(-1) - 858*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)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a
^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)
*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4))*e^(-1) + 52*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)*e^(-4)/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)*c*
d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e + d
)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5))*e^(
-1) - 5*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)*e^(-5)/(c^6*d^6) +
(3003*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^5*e^15 - 9009*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^4
*e^12 + 12870*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^3*e^9 - 10010*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9
/2)*a^2*e^6 + 4095*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2)*a*e^3 - 693*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^
(13/2))*e^(-11)/(c^6*d^6))*e^(-1) - 3003*c*d*f^4*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (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))*e^(-2) - 12012*a*f^3*g*((5*((x*e + d)*c*d*e -
c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (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))*e^(-1)
+ 15015*a*f^4*(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d) + (sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(
-c*d^2*e + a*e^3)*a*e^2)/(c*d)) + 2574*a*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*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)*e^(-2)/(c^3*d^3)
+ (35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 +
15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3)) - 572*a*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)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e - c*d^2*e +
a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d*e - c*d^2*e
+ a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4)) + 13*a*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)*e^(-4)/(c^5*d^5) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)
*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5)))
*e^(-1)

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Mupad [B]
time = 3.80, size = 445, normalized size = 1.32 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g^3\,x^5\,\left (7\,a\,e\,g+26\,c\,d\,f\right )}{143}+\frac {256\,a^6\,e^6\,g^4-1664\,a^5\,c\,d\,e^5\,f\,g^3+4576\,a^4\,c^2\,d^2\,e^4\,f^2\,g^2-6864\,a^3\,c^3\,d^3\,e^3\,f^3\,g+6006\,a^2\,c^4\,d^4\,e^2\,f^4}{15015\,c^5\,d^5}+\frac {x^2\,\left (96\,a^4\,c^2\,d^2\,e^4\,g^4-624\,a^3\,c^3\,d^3\,e^3\,f\,g^3+1716\,a^2\,c^4\,d^4\,e^2\,f^2\,g^2+27456\,a\,c^5\,d^5\,e\,f^3\,g+6006\,c^6\,d^6\,f^4\right )}{15015\,c^5\,d^5}+\frac {x\,\left (-128\,a^5\,c\,d\,e^5\,g^4+832\,a^4\,c^2\,d^2\,e^4\,f\,g^3-2288\,a^3\,c^3\,d^3\,e^3\,f^2\,g^2+3432\,a^2\,c^4\,d^4\,e^2\,f^3\,g+12012\,a\,c^5\,d^5\,e\,f^4\right )}{15015\,c^5\,d^5}+\frac {2\,c\,d\,g^4\,x^6}{13}+\frac {8\,g\,x^3\,\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 )}{3003\,c^2\,d^2}+\frac {2\,g^2\,x^4\,\left (a^2\,e^2\,g^2+208\,a\,c\,d\,e\,f\,g+286\,c^2\,d^2\,f^2\right )}{429\,c\,d}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*g^3*x^5*(7*a*e*g + 26*c*d*f))/143 + (256*a^6*e^6*g^4 + 6006
*a^2*c^4*d^4*e^2*f^4 - 6864*a^3*c^3*d^3*e^3*f^3*g - 1664*a^5*c*d*e^5*f*g^3 + 4576*a^4*c^2*d^2*e^4*f^2*g^2)/(15
015*c^5*d^5) + (x^2*(6006*c^6*d^6*f^4 + 96*a^4*c^2*d^2*e^4*g^4 - 624*a^3*c^3*d^3*e^3*f*g^3 + 27456*a*c^5*d^5*e
*f^3*g + 1716*a^2*c^4*d^4*e^2*f^2*g^2))/(15015*c^5*d^5) + (x*(12012*a*c^5*d^5*e*f^4 - 128*a^5*c*d*e^5*g^4 + 34
32*a^2*c^4*d^4*e^2*f^3*g + 832*a^4*c^2*d^2*e^4*f*g^3 - 2288*a^3*c^3*d^3*e^3*f^2*g^2))/(15015*c^5*d^5) + (2*c*d
*g^4*x^6)/13 + (8*g*x^3*(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))/(300
3*c^2*d^2) + (2*g^2*x^4*(a^2*e^2*g^2 + 286*c^2*d^2*f^2 + 208*a*c*d*e*f*g))/(429*c*d)))/(d + e*x)^(1/2)

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