Optimal. Leaf size=91 \[ \frac{A (e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a^2 e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a^2 e^2 (m+2)} \]
[Out]
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Rubi [A] time = 0.113586, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{A (e x)^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{a^2 e (m+1)}+\frac{B (e x)^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{a^2 e^2 (m+2)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x))/(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 13.2885, size = 71, normalized size = 0.78 \[ \frac{A \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{a^{2} e \left (m + 1\right )} + \frac{B \left (e x\right )^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{a^{2} e^{2} \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(B*x+A)/(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0900858, size = 82, normalized size = 0.9 \[ \frac{x (e x)^m \left (A (m+2) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )+B (m+1) x \, _2F_1\left (2,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{a^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x))/(a + c*x^2)^2,x]
[Out]
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Maple [F] time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( Bx+A \right ) }{ \left ( c{x}^{2}+a \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(B*x+A)/(c*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^m/(c*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \left (e x\right )^{m}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^m/(c*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 133.248, size = 770, normalized size = 8.46 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(B*x+A)/(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x)^m/(c*x^2 + a)^2,x, algorithm="giac")
[Out]