Optimal. Leaf size=140 \[ \frac {B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)}-\frac {x^{m+1} \left (\frac {c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p (b B (m+2 p+1)-A c (m+4 p+3)) \, _2F_1\left (-p,\frac {1}{2} (m+2 p+1);\frac {1}{2} (m+2 p+3);-\frac {c x^2}{b}\right )}{c (m+2 p+1) (m+4 p+3)} \]
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Rubi [A] time = 0.14, antiderivative size = 126, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2039, 2032, 365, 364} \[ x^{m+1} \left (\frac {c x^2}{b}+1\right )^{-p} \left (b x^2+c x^4\right )^p \left (\frac {A}{m+2 p+1}-\frac {b B}{c (m+4 p+3)}\right ) \, _2F_1\left (-p,\frac {1}{2} (m+2 p+1);\frac {1}{2} (m+2 p+3);-\frac {c x^2}{b}\right )+\frac {B x^{m-1} \left (b x^2+c x^4\right )^{p+1}}{c (m+4 p+3)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int x^m \left (A+B x^2\right ) \left (b x^2+c x^4\right )^p \, dx &=\frac {B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\left (-A+\frac {b B (1+m+2 p)}{c (3+m+4 p)}\right ) \int x^m \left (b x^2+c x^4\right )^p \, dx\\ &=\frac {B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\left (\left (-A+\frac {b B (1+m+2 p)}{c (3+m+4 p)}\right ) x^{-2 p} \left (b+c x^2\right )^{-p} \left (b x^2+c x^4\right )^p\right ) \int x^{m+2 p} \left (b+c x^2\right )^p \, dx\\ &=\frac {B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}-\left (\left (-A+\frac {b B (1+m+2 p)}{c (3+m+4 p)}\right ) x^{-2 p} \left (1+\frac {c x^2}{b}\right )^{-p} \left (b x^2+c x^4\right )^p\right ) \int x^{m+2 p} \left (1+\frac {c x^2}{b}\right )^p \, dx\\ &=\frac {B x^{-1+m} \left (b x^2+c x^4\right )^{1+p}}{c (3+m+4 p)}+\left (\frac {A}{1+m+2 p}-\frac {b B}{c (3+m+4 p)}\right ) x^{1+m} \left (1+\frac {c x^2}{b}\right )^{-p} \left (b x^2+c x^4\right )^p \, _2F_1\left (-p,\frac {1}{2} (1+m+2 p);\frac {1}{2} (3+m+2 p);-\frac {c x^2}{b}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 135, normalized size = 0.96 \[ \frac {x^{m+1} \left (x^2 \left (b+c x^2\right )\right )^p \left (\frac {c x^2}{b}+1\right )^{-p} \left (A (m+2 p+3) \, _2F_1\left (-p,\frac {1}{2} (m+2 p+1);\frac {1}{2} (m+2 p+3);-\frac {c x^2}{b}\right )+B x^2 (m+2 p+1) \, _2F_1\left (-p,\frac {1}{2} (m+2 p+3);\frac {1}{2} (m+2 p+5);-\frac {c x^2}{b}\right )\right )}{(m+2 p+1) (m+2 p+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x^{2} + A\right )} {\left (c x^{4} + b x^{2}\right )}^{p} x^{m}, x\right ) \]
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 \[ \int {\left (B x^{2} + A\right )} {\left (c x^{4} + b x^{2}\right )}^{p} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \left (B \,x^{2}+A \right ) x^{m} \left (c \,x^{4}+b \,x^{2}\right )^{p}\, dx \]
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 \[ \int {\left (B x^{2} + A\right )} {\left (c x^{4} + b x^{2}\right )}^{p} x^{m}\,{d x} \]
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 \[ \int x^m\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (x^{2} \left (b + c x^{2}\right )\right )^{p} \left (A + B x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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