Optimal. Leaf size=21 \[ -\frac{\left (b x^2\right )^p}{2 (1-p) x^2} \]
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Rubi [A] time = 0.0051819, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ -\frac{\left (b x^2\right )^p}{2 (1-p) x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (b x^2\right )^p}{x^3} \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{-3+2 p} \, dx\\ &=-\frac{\left (b x^2\right )^p}{2 (1-p) x^2}\\ \end{align*}
Mathematica [A] time = 0.0020254, size = 18, normalized size = 0.86 \[ \frac{\left (b x^2\right )^p}{(2 p-2) x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 18, normalized size = 0.9 \begin{align*}{\frac{ \left ( b{x}^{2} \right ) ^{p}}{2\,{x}^{2} \left ( p-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981526, size = 24, normalized size = 1.14 \begin{align*} \frac{b^{p}{\left (x^{2}\right )}^{p}}{2 \,{\left (p - 1\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87311, size = 39, normalized size = 1.86 \begin{align*} \frac{\left (b x^{2}\right )^{p}}{2 \,{\left (p - 1\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.484649, size = 24, normalized size = 1.14 \begin{align*} \begin{cases} \frac{b^{p} \left (x^{2}\right )^{p}}{2 p x^{2} - 2 x^{2}} & \text{for}\: p \neq 1 \\b \log{\left (x \right )} & \text{otherwise} \end{cases} \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 \frac{\left (b x^{2}\right )^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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