Optimal. Leaf size=72 \[ -\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {16 a \sqrt {a x^2+b x^3}}{3 b^3 x} \]
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Rubi [A]
time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2040, 2041,
1602} \begin {gather*} -\frac {16 a \sqrt {a x^2+b x^3}}{3 b^3 x}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {2 x^3}{b \sqrt {a x^2+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rule 2040
Rule 2041
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=-\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {4 \int \frac {x^2}{\sqrt {a x^2+b x^3}} \, dx}{b}\\ &=-\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {(8 a) \int \frac {x}{\sqrt {a x^2+b x^3}} \, dx}{3 b^2}\\ &=-\frac {2 x^3}{b \sqrt {a x^2+b x^3}}+\frac {8 \sqrt {a x^2+b x^3}}{3 b^2}-\frac {16 a \sqrt {a x^2+b x^3}}{3 b^3 x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 0.54 \begin {gather*} \frac {2 x \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 46, normalized size = 0.64
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right ) \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right ) x^{3}}{3 b^{3} \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}\) | \(46\) |
default | \(-\frac {2 \left (b x +a \right ) \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right ) x^{3}}{3 b^{3} \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}\) | \(46\) |
trager | \(-\frac {2 \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{3 \left (b x +a \right ) b^{3} x}\) | \(48\) |
risch | \(-\frac {2 \left (-b x +5 a \right ) \left (b x +a \right ) x}{3 b^{3} \sqrt {x^{2} \left (b x +a \right )}}-\frac {2 a^{2} x}{b^{3} \sqrt {x^{2} \left (b x +a \right )}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 30, normalized size = 0.42 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )}}{3 \, \sqrt {b x + a} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.86, size = 49, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )} \sqrt {b x^{3} + a x^{2}}}{3 \, {\left (b^{4} x^{2} + a b^{3} x\right )}} \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 {x^{5}}{\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 64, normalized size = 0.89 \begin {gather*} \frac {16 \, a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, b^{3}} - \frac {2 \, a^{2}}{\sqrt {b x + a} b^{3} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x + a} a b^{6}\right )}}{3 \, b^{9} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.22, size = 47, normalized size = 0.65 \begin {gather*} -\frac {2\,\sqrt {b\,x^3+a\,x^2}\,\left (8\,a^2+4\,a\,b\,x-b^2\,x^2\right )}{3\,b^3\,x\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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