Optimal. Leaf size=56 \[ -\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}}+\frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 x^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2041, 2039}
\begin {gather*} \frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 x^{3/2}}-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^3}} \, dx &=-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}}-\frac {(2 b) \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{3 a x^{5/2}}+\frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 31, normalized size = 0.55 \begin {gather*} -\frac {2 (a-2 b x) \sqrt {x^2 (a+b x)}}{3 a^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 33, normalized size = 0.59
method | result | size |
risch | \(-\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 \sqrt {x^{2} \left (b x +a \right )}\, \sqrt {x}\, a^{2}}\) | \(31\) |
gosper | \(-\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 \sqrt {x}\, a^{2} \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(33\) |
default | \(-\frac {2 \left (b x +a \right ) \left (-2 b x +a \right )}{3 \sqrt {x}\, a^{2} \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.73, size = 29, normalized size = 0.52 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a x^{2}} {\left (2 \, b x - a\right )}}{3 \, a^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {3}{2}} \sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 54, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{3}}{a^{2}} - \frac {3 \, b^{3}}{a}\right )} \sqrt {b x + a} b}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} {\left | b \right |} \mathrm {sgn}\left (x\right )} \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 = -0.02 \begin {gather*} \int \frac {1}{x^{3/2}\,\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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