Optimal. Leaf size=19 \[ \frac{\left (b x+d x^3\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.0089349, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}
Mathematica [C] time = 0.0645414, 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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Maple [A] time = 0.003, size = 26, normalized size = 1.4 \begin{align*}{\frac{x \left ( d{x}^{2}+b \right ) \left ( d{x}^{3}+bx \right ) ^{n}}{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64072, size = 53, normalized size = 2.79 \begin{align*} \frac{{\left (d x^{3} + b x\right )}{\left (d x^{3} + b x\right )}^{n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.4504, size = 73, normalized size = 3.84 \begin{align*} \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{\left (x \right )} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20395, size = 50, normalized size = 2.63 \begin{align*} \frac{{\left (d x^{3} + b x\right )}^{n} d x^{3} +{\left (d x^{3} + b x\right )}^{n} b x}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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