3.725 \(\int \frac{(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=192 \[ -\frac{16 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}-\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[Out]

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

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Rubi [A]  time = 0.793229, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{16 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}-\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 69.3041, size = 185, normalized size = 0.96 \[ \frac{16 c d g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \sqrt{f + g x} \left (a e g - c d f\right )^{3}} - \frac{8 g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{2}} + \frac{2 \sqrt{d + e x}}{\left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.231669, size = 105, normalized size = 0.55 \[ -\frac{2 \sqrt{d+e x} \left (-a^2 e^2 g^2+2 a c d e g (3 f+2 g x)+c^2 d^2 \left (3 f^2+12 f g x+8 g^2 x^2\right )\right )}{3 (f+g x)^{3/2} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 168, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}-4\,acde{g}^{2}x-12\,{c}^{2}{d}^{2}fgx+{a}^{2}{e}^{2}{g}^{2}-6\,acdefg-3\,{c}^{2}{d}^{2}{f}^{2} \right ) }{3\,{a}^{3}{e}^{3}{g}^{3}-9\,{a}^{2}cd{e}^{2}f{g}^{2}+9\,a{c}^{2}{d}^{2}e{f}^{2}g-3\,{c}^{3}{d}^{3}{f}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(g*x+f)^(5/2)/(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)*(-8*c^2*d^2*g^2*x^2-4*a*c*d*e*g^2*x-12*c^2*d^2*f*g*x+a^2*e^2*g^
2-6*a*c*d*e*f*g-3*c^2*d^2*f^2)*(e*x+d)^(3/2)/(g*x+f)^(3/2)/(a^3*e^3*g^3-3*a^2*c*
d*e^2*f*g^2+3*a*c^2*d^2*e*f^2*g-c^3*d^3*f^3)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(
3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(5/2)),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x +
f)^(5/2)), x)

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Fricas [A]  time = 0.298727, size = 876, normalized size = 4.56 \[ -\frac{2 \,{\left (8 \, c^{2} d^{2} g^{2} x^{2} + 3 \, c^{2} d^{2} f^{2} + 6 \, a c d e f g - a^{2} e^{2} g^{2} + 4 \,{\left (3 \, c^{2} d^{2} f g + a c d e g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{3 \,{\left (a c^{3} d^{4} e f^{5} - 3 \, a^{2} c^{2} d^{3} e^{2} f^{4} g + 3 \, a^{3} c d^{2} e^{3} f^{3} g^{2} - a^{4} d e^{4} f^{2} g^{3} +{\left (c^{4} d^{4} e f^{3} g^{2} - 3 \, a c^{3} d^{3} e^{2} f^{2} g^{3} + 3 \, a^{2} c^{2} d^{2} e^{3} f g^{4} - a^{3} c d e^{4} g^{5}\right )} x^{4} +{\left (2 \, c^{4} d^{4} e f^{4} g +{\left (c^{4} d^{5} - 5 \, a c^{3} d^{3} e^{2}\right )} f^{3} g^{2} - 3 \,{\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g^{3} +{\left (3 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g^{4} -{\left (a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{5}\right )} x^{3} +{\left (c^{4} d^{4} e f^{5} - a^{4} d e^{4} g^{5} +{\left (2 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} f^{4} g -{\left (5 \, a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{3} g^{2} +{\left (3 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} f^{2} g^{3} +{\left (a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5}\right )} f g^{4}\right )} x^{2} -{\left (2 \, a^{4} d e^{4} f g^{4} -{\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} f^{5} +{\left (a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{4} g + 3 \,{\left (a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{3} g^{2} -{\left (5 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{2} g^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(5/2)),x, algorithm="fricas")

[Out]

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

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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)**(5/2)/(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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError