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

Optimal. Leaf size=181 \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.606305, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (3 e f-d g)\right )}{3 c^3 d^3 e \sqrt{d+e x}}+\frac{8 g^2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e}-\frac{2 \sqrt{d+e x} (f+g x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(3/2)*(f + g*x)^2)/(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)^2)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
- (8*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])/(3*c^3*d^3*e*Sqrt[d + e*x]) + (8*g^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2])/(3*c^2*d^2*e)

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Rubi in Sympy [A]  time = 61.106, size = 175, normalized size = 0.97 \[ - \frac{2 \sqrt{d + e x} \left (f + g x\right )^{2}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{8 g^{2} \sqrt{d + e x} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 c^{2} d^{2} e} - \frac{8 g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g + c d^{2} g - 3 c d e f\right )}{3 c^{3} d^{3} e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(g*x+f)**2/(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)**2/(c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)
)) + 8*g**2*sqrt(d + e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(3*c**2
*d**2*e) - 8*g*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)/(3*c**3*d**3*e*sqrt(d + e*x))

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Mathematica [A]  time = 0.118233, size = 88, normalized size = 0.49 \[ \frac{2 \sqrt{d+e x} \left (-8 a^2 e^2 g^2-4 a c d e g (g x-3 f)+c^2 d^2 \left (-3 f^2+6 f g x+g^2 x^2\right )\right )}{3 c^3 d^3 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 116, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -{g}^{2}{x}^{2}{c}^{2}{d}^{2}+4\,acde{g}^{2}x-6\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-12\,acdefg+3\,{c}^{2}{d}^{2}{f}^{2} \right ) }{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*d*x+a*e)*(-c^2*d^2*g^2*x^2+4*a*c*d*e*g^2*x-6*c^2*d^2*f*g*x+8*a^2*e^2*g^2
-12*a*c*d*e*f*g+3*c^2*d^2*f^2)*(e*x+d)^(3/2)/c^3/d^3/(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 = 0.775449, size = 132, normalized size = 0.73 \[ -\frac{2 \, f^{2}}{\sqrt{c d x + a e} c d} + \frac{4 \,{\left (c d x + 2 \, a e\right )} f g}{\sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (c^{2} d^{2} x^{2} - 4 \, a c d e x - 8 \, a^{2} e^{2}\right )} g^{2}}{3 \, \sqrt{c d x + a e} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*f^2/(sqrt(c*d*x + a*e)*c*d) + 4*(c*d*x + 2*a*e)*f*g/(sqrt(c*d*x + a*e)*c^2*d^
2) + 2/3*(c^2*d^2*x^2 - 4*a*c*d*e*x - 8*a^2*e^2)*g^2/(sqrt(c*d*x + a*e)*c^3*d^3)

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Fricas [A]  time = 0.269696, size = 247, normalized size = 1.36 \[ \frac{2 \,{\left (c^{2} d^{2} e g^{2} x^{3} - 3 \, c^{2} d^{3} f^{2} + 12 \, a c d^{2} e f g - 8 \, a^{2} d e^{2} g^{2} +{\left (6 \, c^{2} d^{2} e f g +{\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} g^{2}\right )} x^{2} -{\left (3 \, c^{2} d^{2} e f^{2} - 6 \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} f g + 4 \,{\left (a c d^{2} e + 2 \, a^{2} e^{3}\right )} g^{2}\right )} x\right )}}{3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c^2*d^2*e*g^2*x^3 - 3*c^2*d^3*f^2 + 12*a*c*d^2*e*f*g - 8*a^2*d*e^2*g^2 + (6
*c^2*d^2*e*f*g + (c^2*d^3 - 4*a*c*d*e^2)*g^2)*x^2 - (3*c^2*d^2*e*f^2 - 6*(c^2*d^
3 + 2*a*c*d*e^2)*f*g + 4*(a*c*d^2*e + 2*a^2*e^3)*g^2)*x)/(sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^3*d^3)

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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)**2/(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 = 0.715027, 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)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x