Optimal. Leaf size=102 \[ \frac{c x \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{4}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a}+\frac{d x^2 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a}+\frac{e \left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]
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Rubi [A] time = 0.0762832, antiderivative size = 120, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1886, 261, 1893, 246, 245, 365, 364} \[ c x \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};-\frac{b x^3}{a}\right )+\frac{1}{2} d x^2 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )+\frac{e \left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 1886
Rule 261
Rule 1893
Rule 246
Rule 245
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^p \, dx &=e \int x^2 \left (a+b x^3\right )^p \, dx+\int (c+d x) \left (a+b x^3\right )^p \, dx\\ &=\frac{e \left (a+b x^3\right )^{1+p}}{3 b (1+p)}+\int \left (c \left (a+b x^3\right )^p+d x \left (a+b x^3\right )^p\right ) \, dx\\ &=\frac{e \left (a+b x^3\right )^{1+p}}{3 b (1+p)}+c \int \left (a+b x^3\right )^p \, dx+d \int x \left (a+b x^3\right )^p \, dx\\ &=\frac{e \left (a+b x^3\right )^{1+p}}{3 b (1+p)}+\left (c \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p}\right ) \int \left (1+\frac{b x^3}{a}\right )^p \, dx+\left (d \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p}\right ) \int x \left (1+\frac{b x^3}{a}\right )^p \, dx\\ &=\frac{e \left (a+b x^3\right )^{1+p}}{3 b (1+p)}+c x \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p} \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};-\frac{b x^3}{a}\right )+\frac{1}{2} d x^2 \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p} \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0616251, size = 114, normalized size = 1.12 \[ \frac{\left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \left (6 b c (p+1) x \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};-\frac{b x^3}{a}\right )+3 b d (p+1) x^2 \, _2F_1\left (\frac{2}{3},-p;\frac{5}{3};-\frac{b x^3}{a}\right )+2 e \left (a+b x^3\right ) \left (\frac{b x^3}{a}+1\right )^p\right )}{6 b (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.264, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+dx+c \right ) \left ( b{x}^{3}+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^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p}\,{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^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 91.6781, size = 112, normalized size = 1.1 \begin{align*} \frac{a^{p} c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - p \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{a^{p} d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, - p \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} + e \left (\begin{cases} \frac{a^{p} x^{3}}{3} & \text{for}\: b = 0 \\\frac{\begin{cases} \frac{\left (a + b x^{3}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b x^{3} \right )} & \text{otherwise} \end{cases}}{3 b} & \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^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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