3.2288 \(\int \frac{(d+e x)^m (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=205 \[ \frac{(d+e x)^m (-b e+c d-c e x) \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (m+1) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.801897, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{(d+e x)^m (-b e+c d-c e x) \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (m+1) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{g (d+e x)^m \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 133.862, size = 197, normalized size = 0.96 \[ - \frac{g \left (d + e x\right )^{m} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{c e^{2} \left (m + 1\right )} - \frac{\left (\frac{c \left (- d - e x\right )}{b e - 2 c d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )} \left (2 b e g m + b e g - 2 c d g m - 2 c e f m - 2 c e f\right ){{}_{2}F_{1}\left (\begin{matrix} - m + \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b e - c d + c e x}{b e - 2 c d}} \right )}}{c e^{2} \sqrt{d + e x} \left (m + 1\right ) \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.501583, size = 183, normalized size = 0.89 \[ \frac{2 (d+e x)^m (b e-c d+c e x) \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} \left (3 (-b e g+c d g+c e f) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )+g (b e-c d+c e x) \, _2F_1\left (\frac{3}{2},\frac{1}{2}-m;\frac{5}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )\right )}{3 c^2 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^m*((c*(d + e*x))/(2*c*d - b*e))^(1/2 - m)*(-(c*d) + b*e + c*e*x)*(3
*(c*e*f + c*d*g - b*e*g)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (-(c*d) + b*e + c*
e*x)/(-2*c*d + b*e)] + g*(-(c*d) + b*e + c*e*x)*Hypergeometric2F1[3/2, 1/2 - m,
5/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]))/(3*c^2*e^2*Sqrt[(d + e*x)*(-(b*e)
+ c*(d - e*x))])

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Maple [F]  time = 0.293, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( gx+f \right ){\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((d + e*x)**m*(f + g*x)/sqrt(-(d + e*x)*(b*e - c*d + c*e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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