Optimal. Leaf size=28 \[ \frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30} \begin {gather*} \frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx &=\frac {\sqrt {a+b x} \int x^2 \, dx}{\sqrt {-a-b x}}\\ &=\frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {x^3 \sqrt {a+b x}}{3 \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.07, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.56, size = 33, normalized size = 1.18 \begin {gather*} -\frac {i \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2}\right )}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.82 \begin {gather*} \frac {\sqrt {b x +a}\, x^{3}}{3 \sqrt {-b x -a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2\,\sqrt {a+b\,x}}{\sqrt {-a-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.68, size = 44, normalized size = 1.57 \begin {gather*} - \frac {i a^{3}}{b^{3}} - \frac {i a^{2} x}{b^{2}} + \frac {i a \left (a + b x\right )^{2}}{b^{3}} - \frac {i \left (a + b x\right )^{3}}{3 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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