Optimal. Leaf size=57 \[ \frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c} \]
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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {430, 429} \[ \frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^p}{c+d x^2} \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p}{c+d x^2} \, dx\\ &=\frac {x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}\\ \end {align*}
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Mathematica [B] time = 0.18, size = 162, normalized size = 2.84 \[ -\frac {3 a c x \left (a+b x^2\right )^p F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{\left (c+d x^2\right ) \left (2 x^2 \left (a d F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-b c p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )-3 a c F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p}}{d x^{2} + c}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{p}}{d \,x^{2}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x^2+a\right )}^p}{d\,x^2+c} \,d x \]
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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