3.2286 \(\int (d+e x)^m (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=219 \[ \frac{(2 c d-b e) (d+e x)^m (-b e+c d-c e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+5)-2 c (d g m+e f (m+5))) \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{5 c^3 e^2 (m+5)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{c e^2 (m+5)} \]

[Out]

-((g*(d + e*x)^m*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(c*e^2*(5 + m))) +
((2*c*d - b*e)*(b*e*g*(5 + 2*m) - 2*c*(d*g*m + e*f*(5 + m)))*(d + e*x)^m*((c*(d
+ e*x))/(2*c*d - b*e))^(-1/2 - m)*(c*d - b*e - c*e*x)^2*Sqrt[d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2]*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (c*d - b*e - c*e*x)/(2*
c*d - b*e)])/(5*c^3*e^2*(5 + m))

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Rubi [A]  time = 0.859606, antiderivative size = 219, 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 \[ \frac{(2 c d-b e) (d+e x)^m (-b e+c d-c e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+5)-2 c (d g m+e f (m+5))) \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{5 c^3 e^2 (m+5)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{c e^2 (m+5)} \]

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*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(c*e^2*(5 + m))) +
((2*c*d - b*e)*(b*e*g*(5 + 2*m) - 2*c*(d*g*m + e*f*(5 + m)))*(d + e*x)^m*((c*(d
+ e*x))/(2*c*d - b*e))^(-1/2 - m)*(c*d - b*e - c*e*x)^2*Sqrt[d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2]*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (c*d - b*e - c*e*x)/(2*
c*d - b*e)])/(5*c^3*e^2*(5 + m))

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Rubi in Sympy [A]  time = 125.491, size = 218, normalized size = 1. \[ - \frac{g \left (d + e x\right )^{m} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{c e^{2} \left (m + 5\right )} - \frac{\left (\frac{c \left (- d - e x\right )}{b e - 2 c d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \left (b e - 2 c d\right ) \left (b e - c d + c e x\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )} \left (2 b e g m + 5 b e g - 2 c d g m - 2 c e f m - 10 c e f\right ){{}_{2}F_{1}\left (\begin{matrix} - m - \frac{3}{2}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b e - c d + c e x}{b e - 2 c d}} \right )}}{5 c^{3} e^{2} \sqrt{d + e x} \left (m + 5\right )} \]

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]

-g*(d + e*x)**m*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(c*e**2*(m + 5
)) - (c*(-d - e*x)/(b*e - 2*c*d))**(-m - 1/2)*(d + e*x)**(m + 1/2)*(b*e - 2*c*d)
*(b*e - c*d + c*e*x)**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))*(2*b*e*g*
m + 5*b*e*g - 2*c*d*g*m - 2*c*e*f*m - 10*c*e*f)*hyper((-m - 3/2, 5/2), (7/2,), (
b*e - c*d + c*e*x)/(b*e - 2*c*d))/(5*c**3*e**2*sqrt(d + e*x)*(m + 5))

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Mathematica [A]  time = 1.20166, size = 265, normalized size = 1.21 \[ -\frac{2 (d+e x)^m (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} \left (63 (2 c d-b e) (-b e g+c d g+c e f) \, _2F_1\left (\frac{5}{2},-m-\frac{1}{2};\frac{7}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )-5 (c (d-e x)-b e) \left (9 (-2 b e g+3 c d g+c e f) \, _2F_1\left (\frac{7}{2},-m-\frac{1}{2};\frac{9}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )+7 g (b e-c d+c e x) \, _2F_1\left (\frac{9}{2},-m-\frac{1}{2};\frac{11}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )\right )\right )}{315 c^3 e^2} \]

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)^m*((c*(d + e*x))/(2*c*d - b*e))^(-1/2 - m)*(-(c*d) + b*e + c*e*x)^
2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(63*(2*c*d - b*e)*(c*e*f + c*d*g - b*e*
g)*Hypergeometric2F1[5/2, -1/2 - m, 7/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]
- 5*(-(b*e) + c*(d - e*x))*(9*(c*e*f + 3*c*d*g - 2*b*e*g)*Hypergeometric2F1[7/2,
-1/2 - m, 9/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 7*g*(-(c*d) + b*e + c*e
*x)*Hypergeometric2F1[9/2, -1/2 - m, 11/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)
])))/(315*c^3*e^2)

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Maple [F]  time = 0.106, 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. \[ \int{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{3}{2}}{\left (g x + f\right )}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(c*e^2*g*x^3 + (c*e^2*f + b*e^2*g)*x^2 - (c*d^2 - b*d*e)*f + (b*e^2*f
- (c*d^2 - b*d*e)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^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)*(-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. \[ \int{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{3}{2}}{\left (g x + f\right )}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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