Optimal. Leaf size=234 \[ \frac{b x^{m+1} \left (a^2 d^2 \left (m^2-8 m+15\right )-2 a b c d \left (m^2-6 m+5\right )+b^2 c^2 \left (m^2-4 m+3\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{8 a^3 (m+1) (b c-a d)^3}+\frac{b x^{m+1} (b c (3-m)-a d (7-m))}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d^3 x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c (m+1) (b c-a d)^3}+\frac{b x^{m+1}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.373813, antiderivative size = 234, normalized size of antiderivative = 1., 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 472
Rule 579
Rule 584
Rule 364
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*}
Mathematica [C] time = 0.0509, size = 54, normalized size = 0.23 \[ \frac{x^{m+1} F_1\left (\frac{m+1}{2};3,1;\frac{m+1}{2}+1;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{a^3 c (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) ^{3} \left ( d{x}^{2}+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x^{2} + a\right )}^{3}{\left (d x^{2} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{{\left (b x^{2} + a\right )}^{3}{\left (d x^{2} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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