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

Optimal. Leaf size=103 \[ \frac{2 \sqrt{f+g x} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (\frac{1}{2},m;\frac{3}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{g} \]

[Out]

(2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*Sqrt[f + g*x]*Hypergeome
tric2F1[1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^m)

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Rubi [A]  time = 0.204133, antiderivative size = 103, 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 \[ \frac{2 \sqrt{f+g x} (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac{g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (\frac{1}{2},m;\frac{3}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{g} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*Sqrt[f + g*x]*Hypergeome
tric2F1[1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^m)

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Rubi in Sympy [A]  time = 47.0762, size = 92, normalized size = 0.89 \[ \frac{2 \left (\frac{g \left (a e + c d x\right )}{a e g - c d f}\right )^{m} \left (d + e x\right )^{m} \sqrt{f + g x} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{c d \left (- f - g x\right )}{a e g - c d f}} \right )}}{g} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0923525, size = 91, normalized size = 0.88 \[ \frac{2 \sqrt{f+g x} (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left (\frac{g (a e+c d x)}{a e g-c d f}\right )^m \, _2F_1\left (\frac{1}{2},m;\frac{3}{2};\frac{c d (f+g x)}{c d f-a e g}\right )}{g} \]

Antiderivative was successfully verified.

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

[Out]

(2*((g*(a*e + c*d*x))/(-(c*d*f) + a*e*g))^m*(d + e*x)^m*Sqrt[f + g*x]*Hypergeome
tric2F1[1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)])/(g*((a*e + c*d*x)*(d + e*
x))^m)

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}}{\frac{1}{\sqrt{gx+f}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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