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

Optimal. Leaf size=104 \[ -\frac{(d+e x)^{5/2} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left (1,n-\frac{1}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right )}{(n+1) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)} \]

[Out]

-(((a*e + c*d*x)*(d + e*x)^(5/2)*(f + g*x)^(1 + n)*Hypergeometric2F1[1, -1/2 + n
, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)])/((c*d*f - a*e*g)*(1 + n)*(a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)))

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Rubi [A]  time = 0.369578, antiderivative size = 122, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 \sqrt{d+e x} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{3 c d (a e+c d x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(f + g*x)^n*Hypergeometric2F1[-3/2, -n, -1/2, -((g*(a*e + c*d*
x))/(c*d*f - a*e*g))])/(3*c*d*(a*e + c*d*x)*((c*d*(f + g*x))/(c*d*f - a*e*g))^n*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi in Sympy [A]  time = 61.6644, size = 114, normalized size = 1.1 \[ - \frac{2 \left (\frac{c d \left (- f - g x\right )}{a e g - c d f}\right )^{- n} \left (f + g x\right )^{n} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}{{}_{2}F_{1}\left (\begin{matrix} - n, - \frac{3}{2} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{g \left (a e + c d x\right )}{a e g - c d f}} \right )}}{3 c d \sqrt{d + e x} \left (a e + c d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*d*(-f - g*x)/(a*e*g - c*d*f))**(-n)*(f + g*x)**n*sqrt(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))*hyper((-n, -3/2), (-1/2,), g*(a*e + c*d*x)/(a*e*g - c*d*f)
)/(3*c*d*sqrt(d + e*x)*(a*e + c*d*x)**2)

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Mathematica [A]  time = 0.208506, size = 100, normalized size = 0.96 \[ -\frac{2 (d+e x)^{3/2} (f+g x)^n \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (-\frac{3}{2},-n;-\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{3 c d ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^(3/2)*(f + g*x)^n*Hypergeometric2F1[-3/2, -n, -1/2, (g*(a*e + c*d*
x))/(-(c*d*f) + a*e*g)])/(3*c*d*((a*e + c*d*x)*(d + e*x))^(3/2)*((c*d*(f + g*x))
/(c*d*f - a*e*g))^n)

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Maple [F]  time = 0.157, size = 0, normalized size = 0. \[ \int{ \left ( gx+f \right ) ^{n} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x + d}{\left (g x + f\right )}^{n}}{{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(e*x + d)*(g*x + f)^n/((c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c
*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)), 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)**(5/2)*(g*x+f)**n/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.805773, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x