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

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

[Out]

-(((a*e + c*d*x)*(f + g*x)^(1 + n)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]*H
ypergeometric2F1[1, 5/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)])/((c*d*f -
a*e*g)*(1 + n)*Sqrt[d + e*x]))

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Rubi [A]  time = 0.339298, antiderivative size = 120, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 (f+g x)^n (a e+c d x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{3 c d \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*e + c*d*x)*(f + g*x)^n*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]*Hyperge
ometric2F1[3/2, -n, 5/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))])/(3*c*d*Sqrt[d +
e*x]*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)

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Rubi in Sympy [A]  time = 62.0315, size = 107, normalized size = 1.03 \[ \frac{2 \left (\frac{c d \left (- f - g x\right )}{a e g - c d f}\right )^{- n} \left (f + g x\right )^{n} \left (a e + c d x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{g \left (a e + c d x\right )}{a e g - c d f}} \right )}}{3 c d \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d*(-f - g*x)/(a*e*g - c*d*f))**(-n)*(f + g*x)**n*(a*e + c*d*x)*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))*hyper((-n, 3/2), (5/2,), g*(a*e + c*d*x)/(a*e
*g - c*d*f))/(3*c*d*sqrt(d + e*x))

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Mathematica [A]  time = 0.124929, size = 100, normalized size = 0.96 \[ \frac{2 (f+g x)^n ((d+e x) (a e+c d x))^{3/2} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{3 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(f + g*x)^n*Hypergeometric2F1[3/2, -n, 5/2, (
g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(3*c*d*(d + e*x)^(3/2)*((c*d*(f + g*x))/(c
*d*f - a*e*g))^n)

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{ \left ( gx+f \right ) ^{n}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}{\frac{1}{\sqrt{ex+d}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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