Optimal. Leaf size=221 \[ \frac{(2 c d-b e)^2 (d+e x)^m (-b e+c d-c e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+7)-2 c (d g m+e f (m+7))) \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (m+7)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (m+7)} \]
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Rubi [A] time = 0.877531, antiderivative size = 221, 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 \[ \frac{(2 c d-b e)^2 (d+e x)^m (-b e+c d-c e x)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} (b e g (2 m+7)-2 c (d g m+e f (m+7))) \, _2F_1\left (\frac{7}{2},-m-\frac{5}{2};\frac{9}{2};\frac{c d-b e-c e x}{2 c d-b e}\right )}{7 c^4 e^2 (m+7)}-\frac{g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{c e^2 (m+7)} \]
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)^(5/2),x]
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Rubi in Sympy [A] time = 139.247, size = 219, normalized size = 0.99 \[ - \frac{g \left (d + e x\right )^{m} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{c e^{2} \left (m + 7\right )} - \frac{\left (\frac{c \left (- d - e x\right )}{b e - 2 c d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \left (b e - 2 c d\right )^{2} \left (b e - c d + c e x\right )^{3} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )} \left (2 b e g m + 7 b e g - 2 c d g m - 2 c e f m - 14 c e f\right ){{}_{2}F_{1}\left (\begin{matrix} - m - \frac{5}{2}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b e - c d + c e x}{b e - 2 c d}} \right )}}{7 c^{4} e^{2} \sqrt{d + e x} \left (m + 7\right )} \]
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)**(5/2),x)
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Mathematica [A] time = 2.71858, size = 346, normalized size = 1.57 \[ \frac{2 (d+e x)^m (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{c (d+e x)}{2 c d-b e}\right )^{-m-\frac{1}{2}} \left (1287 (b e-2 c d)^2 (-b e g+c d g+c e f) \, _2F_1\left (\frac{7}{2},-m-\frac{1}{2};\frac{9}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )-7 (c (d-e x)-b e) \left (143 (2 c d-b e) (-3 b e g+4 c d g+2 c e f) \, _2F_1\left (\frac{9}{2},-m-\frac{1}{2};\frac{11}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )-9 (c (d-e x)-b e) \left (13 (-3 b e g+5 c d g+c e f) \, _2F_1\left (\frac{11}{2},-m-\frac{1}{2};\frac{13}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )+11 g (b e-c d+c e x) \, _2F_1\left (\frac{13}{2},-m-\frac{1}{2};\frac{15}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )\right )\right )\right )}{9009 c^4 e^2} \]
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)^(5/2),x]
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Maple [F] time = 1.108, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}}\, dx \]
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)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}{\left (g x + f\right )}{\left (e x + d\right )}^{m}\,{d x} \]
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)^(5/2)*(g*x + f)*(e*x + d)^m,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} e^{4} g x^{5} +{\left (c^{2} e^{4} f + 2 \, b c e^{4} g\right )} x^{4} +{\left (2 \, b c e^{4} f -{\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} - b^{2} e^{4}\right )} g\right )} x^{3} -{\left ({\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} - b^{2} e^{4}\right )} f + 2 \,{\left (b c d^{2} e^{2} - b^{2} d e^{3}\right )} g\right )} x^{2} +{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} f -{\left (2 \,{\left (b c d^{2} e^{2} - b^{2} d e^{3}\right )} f -{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e x + d\right )}^{m}, x\right ) \]
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)^(5/2)*(g*x + f)*(e*x + d)^m,x, algorithm="fricas")
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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)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
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)^(5/2)*(g*x + f)*(e*x + d)^m,x, algorithm="giac")
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