3.2287 \(\int (d+e x)^m (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

Optimal. Leaf size=208 \[ \frac{(d+e x)^m (-b e+c d-c e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+3)-2 c (d g m+e f (m+3))) \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{3 c^2 e^2 (m+3)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.794938, antiderivative size = 208, 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 \[ \frac{(d+e x)^m (-b e+c d-c e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+3)-2 c (d g m+e f (m+3))) \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{3 c^2 e^2 (m+3)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (m+3)} \]

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

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Rubi in Sympy [A]  time = 123.622, size = 207, normalized size = 1. \[ - \frac{g \left (d + e x\right )^{m} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{c e^{2} \left (m + 3\right )} - \frac{\left (\frac{c \left (- d - e x\right )}{b e - 2 c d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \left (b e - c d + c e x\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )} \left (2 b e g m + 3 b e g - 2 c d g m - 2 c e f m - 6 c e f\right ){{}_{2}F_{1}\left (\begin{matrix} - m - \frac{1}{2}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b e - c d + c e x}{b e - 2 c d}} \right )}}{3 c^{2} e^{2} \sqrt{d + e x} \left (m + 3\right )} \]

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*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(c*e**2*(m + 3
)) - (c*(-d - e*x)/(b*e - 2*c*d))**(-m - 1/2)*(d + e*x)**(m + 1/2)*(b*e - c*d +
c*e*x)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))*(2*b*e*g*m + 3*b*e*g - 2*c
*d*g*m - 2*c*e*f*m - 6*c*e*f)*hyper((-m - 1/2, 3/2), (5/2,), (b*e - c*d + c*e*x)
/(b*e - 2*c*d))/(3*c**2*e**2*sqrt(d + e*x)*(m + 3))

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Mathematica [A]  time = 0.528236, size = 184, normalized size = 0.88 \[ \frac{2 (d+e x)^m (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} \left (5 (-b e g+c d g+c e f) \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )+3 g (b e-c d+c e x) \, _2F_1\left (\frac{5}{2},-m-\frac{1}{2};\frac{7}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )\right )}{15 c^2 e^2} \]

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)*S
qrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(5*(c*e*f + c*d*g - b*e*g)*Hypergeometric2
F1[3/2, -1/2 - m, 5/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + 3*g*(-(c*d) + b*
e + c*e*x)*Hypergeometric2F1[5/2, -1/2 - m, 7/2, (-(c*d) + b*e + c*e*x)/(-2*c*d
+ b*e)]))/(15*c^2*e^2)

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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