Optimal. Leaf size=19 \[ \frac {\left (b x+d x^3\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1588} \[ \frac {\left (b x+d x^3\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \left (b+3 d x^2\right ) \left (b x+d x^3\right )^n \, dx &=\frac {\left (b x+d x^3\right )^{1+n}}{1+n}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 106, normalized size = 5.58 \[ \frac {x \left (x \left (b+d x^2\right )\right )^n \left (\frac {d x^2}{b}+1\right )^{-n} \left (3 d (n+1) x^2 \, _2F_1\left (-n,\frac {n+3}{2};\frac {n+5}{2};-\frac {d x^2}{b}\right )+b (n+3) \, _2F_1\left (-n,\frac {n+1}{2};\frac {n+3}{2};-\frac {d x^2}{b}\right )\right )}{(n+1) (n+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 26, normalized size = 1.37 \[ \frac {{\left (d x^{3} + b x\right )} {\left (d x^{3} + b x\right )}^{n}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 19, normalized size = 1.00 \[ \frac {{\left (d x^{3} + b x\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.37 \[ \frac {\left (d \,x^{2}+b \right ) x \left (d \,x^{3}+b x \right )^{n}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 19, normalized size = 1.00 \[ \frac {{\left (d x^{3} + b x\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 25, normalized size = 1.32 \[ \frac {x\,{\left (d\,x^3+b\,x\right )}^n\,\left (d\,x^2+b\right )}{n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 11.34, size = 73, normalized size = 3.84 \[ \begin {cases} \frac {b x \left (b x + d x^{3}\right )^{n}}{n + 1} + \frac {d x^{3} \left (b x + d x^{3}\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\\log {\relax (x )} + \log {\left (- i \sqrt {b} \sqrt {\frac {1}{d}} + x \right )} + \log {\left (i \sqrt {b} \sqrt {\frac {1}{d}} + x \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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