3.2289 \(\int \frac{(d+e x)^m (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{g (d+e x)^m}{c e^2 (1-m) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (1-2 m)-2 c (e f (1-m)-d g m)) \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c e^2 (1-m) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.817255, antiderivative size = 210, 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 \[ \frac{g (d+e x)^m}{c e^2 (1-m) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(d+e x)^m \left (\frac{c (d+e x)}{2 c d-b e}\right )^{\frac{1}{2}-m} (b e g (1-2 m)-2 c (e f (1-m)-d g m)) \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{c e^2 (1-m) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(g*(d + e*x)^m)/(c*e^2*(1 - m)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - ((b*
e*g*(1 - 2*m) - 2*c*(e*f*(1 - m) - d*g*m))*(d + e*x)^m*((c*(d + e*x))/(2*c*d - b
*e))^(1/2 - m)*Hypergeometric2F1[-1/2, 3/2 - m, 1/2, (c*d - b*e - c*e*x)/(2*c*d
- b*e)])/(c*e^2*(2*c*d - b*e)*(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 = 132.702, size = 209, normalized size = 1. \[ \frac{\left (\frac{c \left (- d - e x\right )}{b e - 2 c d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )} \left (- 2 b e g m + b e g + 2 c d g m + 2 c e f m - 2 c e f\right ){{}_{2}F_{1}\left (\begin{matrix} - m + \frac{3}{2}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b e - c d + c e x}{b e - 2 c d}} \right )}}{e^{2} \sqrt{d + e x} \left (- m + 1\right ) \left (b e - 2 c d\right )^{2} \left (b e - c d + c e x\right )} + \frac{g \left (d + e x\right )^{m}}{c e^{2} \left (- m + 1\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

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)**(3/2),x)

[Out]

(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 + 3/2, -1/2), (1/2,), (b*e - c*d + c*e*x)/(b*e - 2*c*d))/(e**2*sqr
t(d + e*x)*(-m + 1)*(b*e - 2*c*d)**2*(b*e - c*d + c*e*x)) + g*(d + e*x)**m/(c*e*
*2*(-m + 1)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)))

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Mathematica [A]  time = 0.620969, size = 178, normalized size = 0.85 \[ \frac{2 (d+e x)^{m+1} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} \left ((-b e g+c d g+c e f) \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )+g (c (d-e x)-b e) \, _2F_1\left (\frac{1}{2},\frac{3}{2}-m;\frac{3}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )\right )}{e^2 (b e-2 c d)^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]

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)^(3/2),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)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x + f\right )}{\left (e x + d\right )}^{m}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(e*x + d)^m/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)*(e*x + d)^m/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(e*x + d)^m/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="fricas")

[Out]

integral(-(g*x + f)*(e*x + d)^m/((c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)*sqrt(-c*e
^2*x^2 - b*e^2*x + c*d^2 - b*d*e)), 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)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x + f\right )}{\left (e x + d\right )}^{m}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(e*x + d)^m/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="giac")

[Out]

integrate((g*x + f)*(e*x + d)^m/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2), x)