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

Optimal. Leaf size=104 \[ -\frac{\sqrt{d+e x} (f+g x)^{n+1} (a e+c d x) \, _2F_1\left (1,n+\frac{3}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right )}{(n+1) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[Out]

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

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Rubi [A]  time = 0.34536, antiderivative size = 118, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 \sqrt{d+e x} (f+g x)^n (a e+c d x) \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*e + c*d*x)*Sqrt[d + e*x]*(f + g*x)^n*Hypergeometric2F1[1/2, -n, 3/2, -((g*
(a*e + c*d*x))/(c*d*f - a*e*g))])/(c*d*((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 = 61.8758, size = 97, normalized size = 0.93 \[ \frac{2 \left (\frac{c d \left (- f - g x\right )}{a e g - c d f}\right )^{- n} \left (f + g x\right )^{n} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{g \left (a e + c d x\right )}{a e g - c d f}} \right )}}{c d \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.0949758, size = 98, normalized size = 0.94 \[ \frac{2 (f+g x)^n \sqrt{(d+e x) (a e+c d x)} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int{ \left ( gx+f \right ) ^{n}\sqrt{ex+d}{\frac{1}{\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)*(g*x+f)^n/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(e*x + d)*(g*x + f)^n/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)**(1/2)*(g*x+f)**n/(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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}{\left (g x + f\right )}^{n}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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