Optimal. Leaf size=75 \[ \frac {d \left (a+b x^2\right )^{p+1}}{2 b (p+1)}+\frac {1}{3} e x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {764, 261, 365, 364} \[ \frac {d \left (a+b x^2\right )^{p+1}}{2 b (p+1)}+\frac {1}{3} e x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 261
Rule 364
Rule 365
Rule 764
Rubi steps
\begin {align*} \int x (d+e x) \left (a+b x^2\right )^p \, dx &=d \int x \left (a+b x^2\right )^p \, dx+e \int x^2 \left (a+b x^2\right )^p \, dx\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\left (e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {1}{3} e x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 0.95 \[ \frac {1}{6} \left (a+b x^2\right )^p \left (\frac {3 d \left (a+b x^2\right )}{b (p+1)}+2 e x^3 \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x^{2} + d x\right )} {\left (b x^{2} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) x \left (b \,x^{2}+a \right )^{p}\, dx \]
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 \[ e \int {\left (b x^{2} + a\right )}^{p} x^{2}\,{d x} + \frac {{\left (b x^{2} + a\right )}^{p + 1} d}{2 \, b {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (b\,x^2+a\right )}^p\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.01, size = 65, normalized size = 0.87 \[ \frac {a^{p} e x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + d \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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