Optimal. Leaf size=22 \[ \frac{x^{n+1} \left (b+d x^2\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.0080456, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {449} \[ \frac{x^{n+1} \left (b+d x^2\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin{align*} \int x^n \left (b+d x^2\right )^n \left (b+3 d x^2\right ) \, dx &=\frac{x^{1+n} \left (b+d x^2\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [C] time = 0.0354555, size = 108, normalized size = 4.91 \[ \frac{x^{n+1} \left (b+d x^2\right )^n \left (\frac{d x^2}{b}+1\right )^{-n} \left (3 d (n+1) x^2 \, _2F_1\left (-n,\frac{n+3}{2};\frac{n+5}{2};-\frac{d x^2}{b}\right )+b (n+3) \, _2F_1\left (-n,\frac{n+1}{2};\frac{n+3}{2};-\frac{d x^2}{b}\right )\right )}{(n+1) (n+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+n} \left ( d{x}^{2}+b \right ) ^{1+n}}{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19996, size = 42, normalized size = 1.91 \begin{align*} \frac{{\left (d x^{3} + b x\right )} e^{\left (n \log \left (d x^{2} + b\right ) + n \log \left (x\right )\right )}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6317, size = 55, normalized size = 2.5 \begin{align*} \frac{{\left (d x^{3} + b x\right )}{\left (d x^{2} + b\right )}^{n} x^{n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 58.6106, size = 76, normalized size = 3.45 \begin{align*} \begin{cases} \frac{b x x^{n} \left (b + d x^{2}\right )^{n}}{n + 1} + \frac{d x^{3} x^{n} \left (b + d x^{2}\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 = 1.15311, size = 53, normalized size = 2.41 \begin{align*} \frac{{\left (d x^{2} + b\right )}^{n} d x^{3} x^{n} +{\left (d x^{2} + b\right )}^{n} b x x^{n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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