Optimal. Leaf size=125 \[ \frac{a^2 e \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{e \left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)}+\frac{1}{5} d x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right ) \]
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Rubi [A] time = 0.0848504, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {764, 365, 364, 266, 43} \[ \frac{a^2 e \left (a+b x^2\right )^{p+1}}{2 b^3 (p+1)}-\frac{a e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac{e \left (a+b x^2\right )^{p+3}}{2 b^3 (p+3)}+\frac{1}{5} d x^5 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 764
Rule 365
Rule 364
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^4 (d+e x) \left (a+b x^2\right )^p \, dx &=d \int x^4 \left (a+b x^2\right )^p \, dx+e \int x^5 \left (a+b x^2\right )^p \, dx\\ &=\frac{1}{2} e \operatorname{Subst}\left (\int x^2 (a+b x)^p \, dx,x,x^2\right )+\left (d \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int x^4 \left (1+\frac{b x^2}{a}\right )^p \, dx\\ &=\frac{1}{5} d x^5 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )+\frac{1}{2} e \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^p}{b^2}-\frac{2 a (a+b x)^{1+p}}{b^2}+\frac{(a+b x)^{2+p}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 e \left (a+b x^2\right )^{1+p}}{2 b^3 (1+p)}-\frac{a e \left (a+b x^2\right )^{2+p}}{b^3 (2+p)}+\frac{e \left (a+b x^2\right )^{3+p}}{2 b^3 (3+p)}+\frac{1}{5} d x^5 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0714254, size = 112, normalized size = 0.9 \[ \frac{1}{10} \left (a+b x^2\right )^p \left (\frac{5 e \left (a+b x^2\right ) \left (2 a^2-2 a b (p+1) x^2+b^2 \left (p^2+3 p+2\right ) x^4\right )}{b^3 (p+1) (p+2) (p+3)}+2 d x^5 \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.418, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( ex+d \right ) \left ( b{x}^{2}+a \right ) ^{p}\, 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{\left (e x + d\right )}{\left (b x^{2} + a\right )}^{p} x^{4}\,{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 ({\left (e x^{5} + d x^{4}\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 37.9123, size = 1010, normalized size = 8.08 \begin{align*} \text{result too large to display} \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{\left (e x + d\right )}{\left (b x^{2} + a\right )}^{p} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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