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

Optimal. Leaf size=257 \[ -\frac{16 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{5 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*Sqrt[d + e*x]*(f + g*x)^3)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
- (16*g*(c*d*f - a*e*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])/(5*c^4*d^4*e*Sqrt[d + e*x]) + (16*g^2*(c*d*f - a*e*g)*Sqrt
[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^3*d^3*e) + (12*g*(f
+ g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^2*d^2*Sqrt[d + e*x])

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Rubi [A]  time = 1.01343, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{16 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (2 a e^2 g-c d (3 e f-d g)\right )}{5 c^4 d^4 e \sqrt{d+e x}}+\frac{16 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{5 c^3 d^3 e}+\frac{12 g (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c^2 d^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^3}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(f + g*x)^3)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
- (16*g*(c*d*f - a*e*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])/(5*c^4*d^4*e*Sqrt[d + e*x]) + (16*g^2*(c*d*f - a*e*g)*Sqrt
[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^3*d^3*e) + (12*g*(f
+ g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c^2*d^2*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 90.5588, size = 252, normalized size = 0.98 \[ - \frac{2 \sqrt{d + e x} \left (f + g x\right )^{3}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{12 g \left (f + g x\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 c^{2} d^{2} \sqrt{d + e x}} - \frac{16 g^{2} \sqrt{d + e x} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 c^{3} d^{3} e} + \frac{16 g \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g + c d^{2} g - 3 c d e f\right )}{5 c^{4} d^{4} e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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)**(3/2),x)

[Out]

-2*sqrt(d + e*x)*(f + g*x)**3/(c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)
)) + 12*g*(f + g*x)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(5*c**2*d*
*2*sqrt(d + e*x)) - 16*g**2*sqrt(d + e*x)*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**
2 + x*(a*e**2 + c*d**2))/(5*c**3*d**3*e) + 16*g*(a*e*g - c*d*f)*sqrt(a*d*e + c*d
*e*x**2 + x*(a*e**2 + c*d**2))*(2*a*e**2*g + c*d**2*g - 3*c*d*e*f)/(5*c**4*d**4*
e*sqrt(d + e*x))

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Mathematica [A]  time = 0.186859, size = 134, normalized size = 0.52 \[ \frac{2 \sqrt{d+e x} \left (16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (g x-5 f)-2 a c^2 d^2 e g \left (-15 f^2+10 f g x+g^2 x^2\right )+c^3 d^3 \left (-5 f^3+15 f^2 g x+5 f g^2 x^2+g^3 x^3\right )\right )}{5 c^4 d^4 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(-5*f + g*x) - 2*a*c^2*d^2*
e*g*(-15*f^2 + 10*f*g*x + g^2*x^2) + c^3*d^3*(-5*f^3 + 15*f^2*g*x + 5*f*g^2*x^2
+ g^3*x^3)))/(5*c^4*d^4*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.012, size = 187, normalized size = 0.7 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ({g}^{3}{x}^{3}{c}^{3}{d}^{3}-2\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}+5\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}+8\,{a}^{2}cd{e}^{2}{g}^{3}x-20\,a{c}^{2}{d}^{2}ef{g}^{2}x+15\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-40\,{a}^{2}cd{e}^{2}f{g}^{2}+30\,a{c}^{2}{d}^{2}e{f}^{2}g-5\,{f}^{3}{c}^{3}{d}^{3} \right ) }{5\,{c}^{4}{d}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

2/5*(c*d*x+a*e)*(c^3*d^3*g^3*x^3-2*a*c^2*d^2*e*g^3*x^2+5*c^3*d^3*f*g^2*x^2+8*a^2
*c*d*e^2*g^3*x-20*a*c^2*d^2*e*f*g^2*x+15*c^3*d^3*f^2*g*x+16*a^3*e^3*g^3-40*a^2*c
*d*e^2*f*g^2+30*a*c^2*d^2*e*f^2*g-5*c^3*d^3*f^3)*(e*x+d)^(3/2)/c^4/d^4/(c*d*e*x^
2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)

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Maxima [A]  time = 0.786263, size = 223, normalized size = 0.87 \[ -\frac{2 \, f^{3}}{\sqrt{c d x + a e} c d} + \frac{6 \,{\left (c d x + 2 \, a e\right )} f^{2} g}{\sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - 4 \, a c d e x - 8 \, a^{2} e^{2}\right )} f g^{2}}{\sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (c^{3} d^{3} x^{3} - 2 \, a c^{2} d^{2} e x^{2} + 8 \, a^{2} c d e^{2} x + 16 \, a^{3} e^{3}\right )} g^{3}}{5 \, \sqrt{c d x + a e} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*f^3/(sqrt(c*d*x + a*e)*c*d) + 6*(c*d*x + 2*a*e)*f^2*g/(sqrt(c*d*x + a*e)*c^2*
d^2) + 2*(c^2*d^2*x^2 - 4*a*c*d*e*x - 8*a^2*e^2)*f*g^2/(sqrt(c*d*x + a*e)*c^3*d^
3) + 2/5*(c^3*d^3*x^3 - 2*a*c^2*d^2*e*x^2 + 8*a^2*c*d*e^2*x + 16*a^3*e^3)*g^3/(s
qrt(c*d*x + a*e)*c^4*d^4)

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Fricas [A]  time = 0.28209, size = 424, normalized size = 1.65 \[ \frac{2 \,{\left (c^{3} d^{3} e g^{3} x^{4} - 5 \, c^{3} d^{4} f^{3} + 30 \, a c^{2} d^{3} e f^{2} g - 40 \, a^{2} c d^{2} e^{2} f g^{2} + 16 \, a^{3} d e^{3} g^{3} +{\left (5 \, c^{3} d^{3} e f g^{2} +{\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} g^{3}\right )} x^{3} +{\left (15 \, c^{3} d^{3} e f^{2} g + 5 \,{\left (c^{3} d^{4} - 4 \, a c^{2} d^{2} e^{2}\right )} f g^{2} - 2 \,{\left (a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} g^{3}\right )} x^{2} -{\left (5 \, c^{3} d^{3} e f^{3} - 15 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} f^{2} g + 20 \,{\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f g^{2} - 8 \,{\left (a^{2} c d^{2} e^{2} + 2 \, a^{3} e^{4}\right )} g^{3}\right )} x\right )}}{5 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(c^3*d^3*e*g^3*x^4 - 5*c^3*d^4*f^3 + 30*a*c^2*d^3*e*f^2*g - 40*a^2*c*d^2*e^2
*f*g^2 + 16*a^3*d*e^3*g^3 + (5*c^3*d^3*e*f*g^2 + (c^3*d^4 - 2*a*c^2*d^2*e^2)*g^3
)*x^3 + (15*c^3*d^3*e*f^2*g + 5*(c^3*d^4 - 4*a*c^2*d^2*e^2)*f*g^2 - 2*(a*c^2*d^3
*e - 4*a^2*c*d*e^3)*g^3)*x^2 - (5*c^3*d^3*e*f^3 - 15*(c^3*d^4 + 2*a*c^2*d^2*e^2)
*f^2*g + 20*(a*c^2*d^3*e + 2*a^2*c*d*e^3)*f*g^2 - 8*(a^2*c*d^2*e^2 + 2*a^3*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^4*d^4)

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

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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.73925, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x