Optimal. Leaf size=66 \[ \frac {2 a^3}{b^4 \sqrt {a+b x}}+\frac {6 a^2 \sqrt {a+b x}}{b^4}-\frac {2 a (a+b x)^{3/2}}{b^4}+\frac {2 (a+b x)^{5/2}}{5 b^4} \]
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Rubi [A] time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} \frac {2 a^3}{b^4 \sqrt {a+b x}}+\frac {6 a^2 \sqrt {a+b x}}{b^4}-\frac {2 a (a+b x)^{3/2}}{b^4}+\frac {2 (a+b x)^{5/2}}{5 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^{3/2}} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^{3/2}}+\frac {3 a^2}{b^3 \sqrt {a+b x}}-\frac {3 a \sqrt {a+b x}}{b^3}+\frac {(a+b x)^{3/2}}{b^3}\right ) \, dx\\ &=\frac {2 a^3}{b^4 \sqrt {a+b x}}+\frac {6 a^2 \sqrt {a+b x}}{b^4}-\frac {2 a (a+b x)^{3/2}}{b^4}+\frac {2 (a+b x)^{5/2}}{5 b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.68 \begin {gather*} \frac {2 \left (16 a^3+8 a^2 b x-2 a b^2 x^2+b^3 x^3\right )}{5 b^4 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 49, normalized size = 0.74 \begin {gather*} \frac {2 \left (5 a^3+15 a^2 (a+b x)-5 a (a+b x)^2+(a+b x)^3\right )}{5 b^4 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 51, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b^{3} x^{3} - 2 \, a b^{2} x^{2} + 8 \, a^{2} b x + 16 \, a^{3}\right )} \sqrt {b x + a}}{5 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 61, normalized size = 0.92 \begin {gather*} \frac {2 \, a^{3}}{\sqrt {b x + a} b^{4}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {5}{2}} b^{16} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{16} + 15 \, \sqrt {b x + a} a^{2} b^{16}\right )}}{5 \, b^{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 0.64 \begin {gather*} \frac {\frac {2}{5} b^{3} x^{3}-\frac {4}{5} a \,b^{2} x^{2}+\frac {16}{5} a^{2} b x +\frac {32}{5} a^{3}}{b^{4} \sqrt {b x +a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 56, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{b^{4}} + \frac {6 \, \sqrt {b x + a} a^{2}}{b^{4}} + \frac {2 \, a^{3}}{\sqrt {b x + a} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 56, normalized size = 0.85 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}+\frac {6\,a^2\,\sqrt {a+b\,x}}{b^4}+\frac {2\,a^3}{b^4\,\sqrt {a+b\,x}}-\frac {2\,a\,{\left (a+b\,x\right )}^{3/2}}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.94, size = 1538, normalized size = 23.30
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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