Optimal. Leaf size=234 \[ \frac {b x^{m+1} (b c (3-m)-a d (7-m))}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {b x^{m+1} \left (a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (m^2-6 m+5\right )+b^2 c^2 \left (m^2-4 m+3\right )\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{8 a^3 (m+1) (b c-a d)^3}-\frac {d^3 x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c (m+1) (b c-a d)^3}+\frac {b x^{m+1}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.37, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {472, 579, 584, 364} \[ \frac {b x^{m+1} \left (a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (m^2-6 m+5\right )+b^2 c^2 \left (m^2-4 m+3\right )\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{8 a^3 (m+1) (b c-a d)^3}+\frac {b x^{m+1} (b c (3-m)-a d (7-m))}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {d^3 x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{c (m+1) (b c-a d)^3}+\frac {b x^{m+1}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 472
Rule 579
Rule 584
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx &=\frac {b x^{1+m}}{4 a (b c-a d) \left (a+b x^2\right )^2}-\frac {\int \frac {x^m \left (4 a d-b c (3-m)-b d (3-m) x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx}{4 a (b c-a d)}\\ &=\frac {b x^{1+m}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (b c (3-m)-a d (7-m)) x^{1+m}}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\int \frac {x^m \left (8 a^2 d^2-a b c d \left (7-8 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )+b d (b c (3-m)-a d (7-m)) (1-m) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac {b x^{1+m}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (b c (3-m)-a d (7-m)) x^{1+m}}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\int \left (\frac {b \left (a^2 d^2 \left (15-8 m+m^2\right )-2 a b c d \left (5-6 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right ) x^m}{(b c-a d) \left (a+b x^2\right )}+\frac {8 a^2 d^3 x^m}{(-b c+a d) \left (c+d x^2\right )}\right ) \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac {b x^{1+m}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (b c (3-m)-a d (7-m)) x^{1+m}}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3 \int \frac {x^m}{c+d x^2} \, dx}{(b c-a d)^3}+\frac {\left (b \left (a^2 d^2 \left (15-8 m+m^2\right )-2 a b c d \left (5-6 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right ) \int \frac {x^m}{a+b x^2} \, dx}{8 a^2 (b c-a d)^3}\\ &=\frac {b x^{1+m}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (b c (3-m)-a d (7-m)) x^{1+m}}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {b \left (a^2 d^2 \left (15-8 m+m^2\right )-2 a b c d \left (5-6 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right ) x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{8 a^3 (b c-a d)^3 (1+m)}-\frac {d^3 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)^3 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 170, normalized size = 0.73 \[ \frac {x^{m+1} \left (a^3 d^3 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )-a^2 b c d^2 \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )-b c (a d-b c) \left ((a d-b c) \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )+a d \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )\right )\right )}{a^3 c (m+1) (a d-b c)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{b^{3} d x^{8} + {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{6} + 3 \, {\left (a b^{2} c + a^{2} b d\right )} x^{4} + a^{3} c + {\left (3 \, a^{2} b c + a^{3} d\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {x^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \,x^{2}+a \right )^{3} \left (d \,x^{2}+c \right )}\, dx \]
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 \[ \int \frac {x^{m}}{{\left (b x^{2} + a\right )}^{3} {\left (d x^{2} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^m}{{\left (b\,x^2+a\right )}^3\,\left (d\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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