3.784 \(\int \frac{(d+e x)^{3/2} (f+g x)^3}{\sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=412 \[ -\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt{d+e x}}-\frac{4 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}+\frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}} \]

[Out]

(16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2])/(315*c^5*d^5*e*g*Sqrt[d + e*x]) - (16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*S
qrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(315*c^4*d^4*e) - (4*(c*d*f - a*e*g)*(8*a*e^2*g + c*
d*(e*f - 9*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^3*d^3*g*Sqrt[d + e*x]) - (2*(
8*a*e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(63*c^2*d^2*g*Sqrt[d +
e*x]) + (2*e*(f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(9*c*d*g*Sqrt[d + e*x])

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Rubi [A]  time = 0.627098, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {880, 870, 794, 648} \[ -\frac{2 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (8 a e^2 g+c d (e f-9 d g)\right )}{63 c^2 d^2 g \sqrt{d+e x}}-\frac{4 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{16 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )}{315 c^4 d^4 e}+\frac{16 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{315 c^5 d^5 e g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{9 c d g \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2])/(315*c^5*d^5*e*g*Sqrt[d + e*x]) - (16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*S
qrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(315*c^4*d^4*e) - (4*(c*d*f - a*e*g)*(8*a*e^2*g + c*
d*(e*f - 9*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^3*d^3*g*Sqrt[d + e*x]) - (2*(
8*a*e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(63*c^2*d^2*g*Sqrt[d +
e*x]) + (2*e*(f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(9*c*d*g*Sqrt[d + e*x])

Rule 880

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 - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(c*g*(n + p + 2)), x] - Dist[(b*e*g*(
n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(c*g*(n + p + 2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x +
c*x^2)^p, x], 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] && Eq
Q[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p - 1, 0] &&  !LtQ[n, -1] && IntegerQ[2*p]

Rule 870

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] &&  !Integ
erQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rule 794

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 648

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]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{3/2} (f+g x)^3}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}-\frac{1}{9} \left (-9 d+\frac{8 a e^2}{c d}+\frac{e f}{g}\right ) \int \frac{\sqrt{d+e x} (f+g x)^3}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}-\frac{\left (2 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right )\right ) \int \frac{\sqrt{d+e x} (f+g x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{21 c^2 d^2 g}\\ &=-\frac{4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}-\frac{\left (8 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right )\right ) \int \frac{\sqrt{d+e x} (f+g x)}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{105 c^3 d^3 g}\\ &=-\frac{16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^4 d^4 e}-\frac{4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}+\frac{\left (8 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{315 c^4 d^4 e g}\\ &=\frac{16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^5 d^5 e g \sqrt{d+e x}}-\frac{16 (c d f-a e g)^2 \left (8 a e^2 g+c d (e f-9 d g)\right ) \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{315 c^4 d^4 e}-\frac{4 (c d f-a e g) \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{105 c^3 d^3 g \sqrt{d+e x}}-\frac{2 \left (8 a e^2 g+c d (e f-9 d g)\right ) (f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{63 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{9 c d g \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.278514, size = 264, normalized size = 0.64 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (24 a^2 c^2 d^2 e^2 g \left (3 d g (7 f+g x)+e \left (21 f^2+9 f g x+2 g^2 x^2\right )\right )-16 a^3 c d e^3 g^2 (9 d g+27 e f+4 e g x)+128 a^4 e^5 g^3-2 a c^3 d^3 e \left (9 d g \left (35 f^2+14 f g x+3 g^2 x^2\right )+e \left (126 f^2 g x+105 f^3+81 f g^2 x^2+20 g^3 x^3\right )\right )+c^4 d^4 \left (9 d \left (35 f^2 g x+35 f^3+21 f g^2 x^2+5 g^3 x^3\right )+e x \left (189 f^2 g x+105 f^3+135 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{315 c^5 d^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^5*g^3 - 16*a^3*c*d*e^3*g^2*(27*e*f + 9*d*g + 4*e*g*x) + 24*a^2*c^2
*d^2*e^2*g*(3*d*g*(7*f + g*x) + e*(21*f^2 + 9*f*g*x + 2*g^2*x^2)) - 2*a*c^3*d^3*e*(9*d*g*(35*f^2 + 14*f*g*x +
3*g^2*x^2) + e*(105*f^3 + 126*f^2*g*x + 81*f*g^2*x^2 + 20*g^3*x^3)) + c^4*d^4*(9*d*(35*f^3 + 35*f^2*g*x + 21*f
*g^2*x^2 + 5*g^3*x^3) + e*x*(105*f^3 + 189*f^2*g*x + 135*f*g^2*x^2 + 35*g^3*x^3))))/(315*c^5*d^5*Sqrt[d + e*x]
)

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Maple [A]  time = 0.051, size = 425, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 35\,e{g}^{3}{x}^{4}{c}^{4}{d}^{4}-40\,a{c}^{3}{d}^{3}{e}^{2}{g}^{3}{x}^{3}+45\,{c}^{4}{d}^{5}{g}^{3}{x}^{3}+135\,{c}^{4}{d}^{4}ef{g}^{2}{x}^{3}+48\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}{g}^{3}{x}^{2}-54\,a{c}^{3}{d}^{4}e{g}^{3}{x}^{2}-162\,a{c}^{3}{d}^{3}{e}^{2}f{g}^{2}{x}^{2}+189\,{c}^{4}{d}^{5}f{g}^{2}{x}^{2}+189\,{c}^{4}{d}^{4}e{f}^{2}g{x}^{2}-64\,{a}^{3}cd{e}^{4}{g}^{3}x+72\,{a}^{2}{c}^{2}{d}^{3}{e}^{2}{g}^{3}x+216\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}f{g}^{2}x-252\,a{c}^{3}{d}^{4}ef{g}^{2}x-252\,a{c}^{3}{d}^{3}{e}^{2}{f}^{2}gx+315\,{c}^{4}{d}^{5}{f}^{2}gx+105\,{c}^{4}{d}^{4}e{f}^{3}x+128\,{a}^{4}{e}^{5}{g}^{3}-144\,{a}^{3}c{d}^{2}{e}^{3}{g}^{3}-432\,{a}^{3}cd{e}^{4}f{g}^{2}+504\,{a}^{2}{c}^{2}{d}^{3}{e}^{2}f{g}^{2}+504\,{a}^{2}{c}^{2}{d}^{2}{e}^{3}{f}^{2}g-630\,a{c}^{3}{d}^{4}e{f}^{2}g-210\,a{c}^{3}{d}^{3}{e}^{2}{f}^{3}+315\,{d}^{5}{f}^{3}{c}^{4} \right ) }{315\,{c}^{5}{d}^{5}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

2/315*(c*d*x+a*e)*(35*c^4*d^4*e*g^3*x^4-40*a*c^3*d^3*e^2*g^3*x^3+45*c^4*d^5*g^3*x^3+135*c^4*d^4*e*f*g^2*x^3+48
*a^2*c^2*d^2*e^3*g^3*x^2-54*a*c^3*d^4*e*g^3*x^2-162*a*c^3*d^3*e^2*f*g^2*x^2+189*c^4*d^5*f*g^2*x^2+189*c^4*d^4*
e*f^2*g*x^2-64*a^3*c*d*e^4*g^3*x+72*a^2*c^2*d^3*e^2*g^3*x+216*a^2*c^2*d^2*e^3*f*g^2*x-252*a*c^3*d^4*e*f*g^2*x-
252*a*c^3*d^3*e^2*f^2*g*x+315*c^4*d^5*f^2*g*x+105*c^4*d^4*e*f^3*x+128*a^4*e^5*g^3-144*a^3*c*d^2*e^3*g^3-432*a^
3*c*d*e^4*f*g^2+504*a^2*c^2*d^3*e^2*f*g^2+504*a^2*c^2*d^2*e^3*f^2*g-630*a*c^3*d^4*e*f^2*g-210*a*c^3*d^3*e^2*f^
3+315*c^4*d^5*f^3)*(e*x+d)^(1/2)/c^5/d^5/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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Maxima [A]  time = 1.67286, size = 653, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{3}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{2} g}{5 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \,{\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} -{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f g^{2}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} + \frac{2 \,{\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \,{\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} -{\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \,{\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \,{\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} g^{3}}{315 \, \sqrt{c d x + a e} c^{5} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/3*(c^2*d^2*e*x^2 + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)*f^3/(sqrt(c*d*x + a*e)*c^2*d^2) + 2/
5*(3*c^3*d^3*e*x^3 - 10*a^2*c*d^2*e^2 + 8*a^3*e^4 + (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*c
*d*e^3)*x)*f^2*g/(sqrt(c*d*x + a*e)*c^3*d^3) + 2/35*(15*c^4*d^4*e*x^4 + 56*a^3*c*d^2*e^3 - 48*a^4*e^5 + 3*(7*c
^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*x^2 + 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)*
x)*f*g^2/(sqrt(c*d*x + a*e)*c^4*d^4) + 2/315*(35*c^5*d^5*e*x^5 - 144*a^4*c*d^2*e^4 + 128*a^5*e^6 + 5*(9*c^5*d^
6 - a*c^4*d^4*e^2)*x^4 - (9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*x^3 + 2*(9*a^2*c^3*d^4*e^2 - 8*a^3*c^2*d^2*e^4)*x
^2 - 8*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*x)*g^3/(sqrt(c*d*x + a*e)*c^5*d^5)

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Fricas [A]  time = 1.53569, size = 837, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (35 \, c^{4} d^{4} e g^{3} x^{4} + 105 \,{\left (3 \, c^{4} d^{5} - 2 \, a c^{3} d^{3} e^{2}\right )} f^{3} - 126 \,{\left (5 \, a c^{3} d^{4} e - 4 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g + 72 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} f g^{2} - 16 \,{\left (9 \, a^{3} c d^{2} e^{3} - 8 \, a^{4} e^{5}\right )} g^{3} + 5 \,{\left (27 \, c^{4} d^{4} e f g^{2} +{\left (9 \, c^{4} d^{5} - 8 \, a c^{3} d^{3} e^{2}\right )} g^{3}\right )} x^{3} + 3 \,{\left (63 \, c^{4} d^{4} e f^{2} g + 9 \,{\left (7 \, c^{4} d^{5} - 6 \, a c^{3} d^{3} e^{2}\right )} f g^{2} - 2 \,{\left (9 \, a c^{3} d^{4} e - 8 \, a^{2} c^{2} d^{2} e^{3}\right )} g^{3}\right )} x^{2} +{\left (105 \, c^{4} d^{4} e f^{3} + 63 \,{\left (5 \, c^{4} d^{5} - 4 \, a c^{3} d^{3} e^{2}\right )} f^{2} g - 36 \,{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} f g^{2} + 8 \,{\left (9 \, a^{2} c^{2} d^{3} e^{2} - 8 \, a^{3} c d e^{4}\right )} 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}}{315 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*c^4*d^4*e*g^3*x^4 + 105*(3*c^4*d^5 - 2*a*c^3*d^3*e^2)*f^3 - 126*(5*a*c^3*d^4*e - 4*a^2*c^2*d^2*e^3)*
f^2*g + 72*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)*f*g^2 - 16*(9*a^3*c*d^2*e^3 - 8*a^4*e^5)*g^3 + 5*(27*c^4*d^4*e*
f*g^2 + (9*c^4*d^5 - 8*a*c^3*d^3*e^2)*g^3)*x^3 + 3*(63*c^4*d^4*e*f^2*g + 9*(7*c^4*d^5 - 6*a*c^3*d^3*e^2)*f*g^2
- 2*(9*a*c^3*d^4*e - 8*a^2*c^2*d^2*e^3)*g^3)*x^2 + (105*c^4*d^4*e*f^3 + 63*(5*c^4*d^5 - 4*a*c^3*d^3*e^2)*f^2*
g - 36*(7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*f*g^2 + 8*(9*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*g^3)*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{3}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

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