Optimal. Leaf size=378 \[ \frac{(d+e x)^{m+1} (b e (1-m)+4 c d)}{2 b^2 d^2 x (b+c x)^2}-\frac{(d+e x)^{m+1} \left (-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{2 b^5 d^3 (m+1)}+\frac{c^3 (d+e x)^{m+1} \left (b^2 e^2 \left (m^2-7 m+12\right )-6 b c d e (4-m)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (m+1) (c d-b e)^3}+\frac{c (2 c d-b e) (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right )}{2 b^4 d^2 (b+c x) (c d-b e)^2}+\frac{c (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right )}{2 b^3 d^2 (b+c x)^2 (c d-b e)}-\frac{(d+e x)^{m+1}}{2 b d x^2 (b+c x)^2} \]
[Out]
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Rubi [A] time = 1.51904, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{(d+e x)^{m+1} (b e (1-m)+4 c d)}{2 b^2 d^2 x (b+c x)^2}-\frac{(d+e x)^{m+1} \left (-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{2 b^5 d^3 (m+1)}+\frac{c^3 (d+e x)^{m+1} \left (b^2 e^2 \left (m^2-7 m+12\right )-6 b c d e (4-m)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (m+1) (c d-b e)^3}+\frac{c (2 c d-b e) (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right )}{2 b^4 d^2 (b+c x) (c d-b e)^2}+\frac{c (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right )}{2 b^3 d^2 (b+c x)^2 (c d-b e)}-\frac{(d+e x)^{m+1}}{2 b d x^2 (b+c x)^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/(b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m/(c*x**2+b*x)**3,x)
[Out]
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Mathematica [A] time = 0.691492, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (b x+c x^2\right )^3} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(d + e*x)^m/(b*x + c*x^2)^3,x]
[Out]
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Maple [F] time = 0.183, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m/(c*x^2+b*x)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*x^2 + b*x)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*x^2 + b*x)^3,x, algorithm="fricas")
[Out]
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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/(c*x**2+b*x)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*x^2 + b*x)^3,x, algorithm="giac")
[Out]