Optimal. Leaf size=100 \[ -\frac{a d \left (a+c x^2\right )^{p+1}}{2 c^2 (p+1)}+\frac{d \left (a+c x^2\right )^{p+2}}{2 c^2 (p+2)}+\frac{1}{5} e x^5 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right ) \]
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Rubi [A] time = 0.0609412, antiderivative size = 100, 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, 266, 43, 365, 364} \[ -\frac{a d \left (a+c x^2\right )^{p+1}}{2 c^2 (p+1)}+\frac{d \left (a+c x^2\right )^{p+2}}{2 c^2 (p+2)}+\frac{1}{5} e x^5 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 764
Rule 266
Rule 43
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^3 (d+e x) \left (a+c x^2\right )^p \, dx &=d \int x^3 \left (a+c x^2\right )^p \, dx+e \int x^4 \left (a+c x^2\right )^p \, dx\\ &=\frac{1}{2} d \operatorname{Subst}\left (\int x (a+c x)^p \, dx,x,x^2\right )+\left (e \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int x^4 \left (1+\frac{c x^2}{a}\right )^p \, dx\\ &=\frac{1}{5} e x^5 \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right )+\frac{1}{2} d \operatorname{Subst}\left (\int \left (-\frac{a (a+c x)^p}{c}+\frac{(a+c x)^{1+p}}{c}\right ) \, dx,x,x^2\right )\\ &=-\frac{a d \left (a+c x^2\right )^{1+p}}{2 c^2 (1+p)}+\frac{d \left (a+c x^2\right )^{2+p}}{2 c^2 (2+p)}+\frac{1}{5} e x^5 \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0790483, size = 87, normalized size = 0.87 \[ \frac{1}{10} \left (a+c x^2\right )^p \left (2 e x^5 \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{c x^2}{a}\right )-\frac{5 d \left (a+c x^2\right ) \left (a-c (p+1) x^2\right )}{c^2 (p+1) (p+2)}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( ex+d \right ) \left ( c{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*} e \int{\left (c x^{2} + a\right )}^{p} x^{4}\,{d x} + \frac{{\left (c^{2}{\left (p + 1\right )} x^{4} + a c p x^{2} - a^{2}\right )}{\left (c x^{2} + a\right )}^{p} d}{2 \,{\left (p^{2} + 3 \, p + 2\right )} c^{2}} \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^{4} + d x^{3}\right )}{\left (c 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 = 19.8383, size = 394, normalized size = 3.94 \begin{align*} \frac{a^{p} e x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, - p \\ \frac{7}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{5} + d \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: c = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} + \frac{x^{2}}{2 c} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{a c p x^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} p x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} & \text{otherwise} \end{cases}\right ) \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 (c x^{2} + a\right )}^{p} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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