Optimal. Leaf size=102 \[ \frac {b x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a (m+1) (b c-a d)}-\frac {d x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c (m+1) (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {482, 364} \[ \frac {b x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a (m+1) (b c-a d)}-\frac {d x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 482
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {b \int \frac {x^m}{a+b x^2} \, dx}{b c-a d}-\frac {d \int \frac {x^m}{c+d x^2} \, dx}{b c-a d}\\ &=\frac {b x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a (b c-a d) (1+m)}-\frac {d x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{c (b c-a d) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, 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.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{\left (b\,x^2+a\right )\,\left (d\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.02, size = 354, normalized size = 3.47 \[ \frac {a m x^{m} \Phi \left (\frac {a e^{i \pi }}{b x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma ^{2}\left (\frac {3}{2} - \frac {m}{2}\right )}{x^{3} \left (4 a b d \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) - 4 b^{2} c \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )\right )} - \frac {3 a x^{m} \Phi \left (\frac {a e^{i \pi }}{b x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma ^{2}\left (\frac {3}{2} - \frac {m}{2}\right )}{x^{3} \left (4 a b d \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) - 4 b^{2} c \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )\right )} + \frac {b m x^{m} \Phi \left (\frac {c e^{i \pi }}{d x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )}{x \left (4 a b d \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) - 4 b^{2} c \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )\right )} - \frac {b x^{m} \Phi \left (\frac {c e^{i \pi }}{d x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )}{x \left (4 a b d \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) - 4 b^{2} c \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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