Optimal. Leaf size=28 \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]
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Rubi [A] time = 0.0032819, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {23, 30} \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{x \sqrt{a+b x}}{\sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int x \, dx}{\sqrt{-a-b x}}\\ &=\frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}}\\ \end{align*}
Mathematica [A] time = 0.004716, size = 28, normalized size = 1. \[ \frac{x^2 \sqrt{a+b x}}{2 \sqrt{-a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 23, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12023, size = 4, normalized size = 0.14 \begin{align*} 0 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.14335, size = 27, normalized size = 0.96 \begin{align*} \frac{i a^{2}}{b^{2}} + \frac{i a x}{b} - \frac{i \left (a + b x\right )^{2}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.55571, size = 28, normalized size = 1. \begin{align*} -\frac{i \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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