Optimal. Leaf size=102 \[ \frac{b x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)} \]
[Out]
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Rubi [A] time = 0.130967, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^m/((a + b*x^2)*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 21.2751, size = 76, normalized size = 0.75 \[ \frac{d x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{d x^{2}}{c}} \right )}}{c \left (m + 1\right ) \left (a d - b c\right )} - \frac{b x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a \left (m + 1\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(b*x**2+a)/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0761294, size = 85, normalized size = 0.83 \[ \frac{x^{m+1} \left (a d \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )-b c \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )\right )}{a c (m+1) (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/((a + b*x^2)*(c + d*x^2)),x]
[Out]
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Maple [F] time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(b*x^2+a)/(d*x^2+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b d x^{4} +{\left (b c + a d\right )} x^{2} + a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 93.4697, size = 354, normalized size = 3.47 \[ \frac{a m x^{m} \Phi \left (\frac{a e^{i \pi }}{b x^{2}}, 1, - \frac{m}{2} + \frac{3}{2}\right ) \Gamma ^{2}\left (- \frac{m}{2} + \frac{3}{2}\right )}{x^{3} \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} - \frac{3 a x^{m} \Phi \left (\frac{a e^{i \pi }}{b x^{2}}, 1, - \frac{m}{2} + \frac{3}{2}\right ) \Gamma ^{2}\left (- \frac{m}{2} + \frac{3}{2}\right )}{x^{3} \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} + \frac{b m x^{m} \Phi \left (\frac{c e^{i \pi }}{d x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )}{x \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} - \frac{b x^{m} \Phi \left (\frac{c e^{i \pi }}{d x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )}{x \left (4 a b d \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right ) - 4 b^{2} c \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{5}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(b*x**2+a)/(d*x**2+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="giac")
[Out]