Optimal. Leaf size=123 \[ \frac{x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{1}{4};-p,1;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{c}-\frac{e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{3}{4};-p,1;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{3 c^2} \]
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Rubi [A] time = 0.126569, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1240, 430, 429, 511, 510} \[ \frac{x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{1}{4};-p,1;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{c}-\frac{e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{3}{4};-p,1;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 1240
Rule 430
Rule 429
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^p}{c+e x^2} \, dx &=\int \left (\frac{c \left (a+b x^4\right )^p}{c^2-e^2 x^4}+\frac{e x^2 \left (a+b x^4\right )^p}{-c^2+e^2 x^4}\right ) \, dx\\ &=c \int \frac{\left (a+b x^4\right )^p}{c^2-e^2 x^4} \, dx+e \int \frac{x^2 \left (a+b x^4\right )^p}{-c^2+e^2 x^4} \, dx\\ &=\left (c \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b x^4}{a}\right )^p}{c^2-e^2 x^4} \, dx+\left (e \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p}\right ) \int \frac{x^2 \left (1+\frac{b x^4}{a}\right )^p}{-c^2+e^2 x^4} \, dx\\ &=\frac{x \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p} F_1\left (\frac{1}{4};-p,1;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{c}-\frac{e x^3 \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p} F_1\left (\frac{3}{4};-p,1;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{3 c^2}\\ \end{align*}
Mathematica [F] time = 0.127536, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x^4\right )^p}{c+e x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{4}+a \right ) ^{p}}{e{x}^{2}+c}}\, 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{{\left (b x^{4} + a\right )}^{p}}{e x^{2} + c}\,{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{{\left (b x^{4} + a\right )}^{p}}{e x^{2} + c}, 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{{\left (b x^{4} + a\right )}^{p}}{e x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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