Optimal. Leaf size=51 \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
[Out]
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Rubi [A] time = 0.0369977, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2),x]
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Rubi in Sympy [A] time = 21.1856, size = 37, normalized size = 0.73 \[ \frac{\left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{\left (m + 1\right ) \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0344961, size = 51, normalized size = 1. \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^m)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [F] time = 0.276, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) \left ( ex+d \right ) ^{m}}{{b}^{2}{x}^{2}+2\,abx+{a}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^m/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**m/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]