Optimal. Leaf size=108 \[ -\frac {32 a^3 \left (a x^2+b x^3\right )^{5/2}}{1155 b^4 x^5}+\frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{231 b^3 x^4}-\frac {4 a \left (a x^2+b x^3\right )^{5/2}}{33 b^2 x^3}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2} \]
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Rubi [A] time = 0.14, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2002, 2016, 2014} \[ -\frac {32 a^3 \left (a x^2+b x^3\right )^{5/2}}{1155 b^4 x^5}+\frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{231 b^3 x^4}-\frac {4 a \left (a x^2+b x^3\right )^{5/2}}{33 b^2 x^3}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \left (a x^2+b x^3\right )^{3/2} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}-\frac {(6 a) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x} \, dx}{11 b}\\ &=-\frac {4 a \left (a x^2+b x^3\right )^{5/2}}{33 b^2 x^3}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}+\frac {\left (8 a^2\right ) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^2} \, dx}{33 b^2}\\ &=\frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{231 b^3 x^4}-\frac {4 a \left (a x^2+b x^3\right )^{5/2}}{33 b^2 x^3}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}-\frac {\left (16 a^3\right ) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^3} \, dx}{231 b^3}\\ &=-\frac {32 a^3 \left (a x^2+b x^3\right )^{5/2}}{1155 b^4 x^5}+\frac {16 a^2 \left (a x^2+b x^3\right )^{5/2}}{231 b^3 x^4}-\frac {4 a \left (a x^2+b x^3\right )^{5/2}}{33 b^2 x^3}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{11 b x^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.54 \[ \frac {2 x (a+b x)^3 \left (-16 a^3+40 a^2 b x-70 a b^2 x^2+105 b^3 x^3\right )}{1155 b^4 \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 73, normalized size = 0.68 \[ \frac {2 \, {\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{1155 \, b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 210, normalized size = 1.94 \[ \frac {32 \, a^{\frac {11}{2}} \mathrm {sgn}\relax (x)}{1155 \, b^{4}} + \frac {2 \, {\left (\frac {99 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{2} \mathrm {sgn}\relax (x)}{b^{3}} + \frac {22 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a \mathrm {sgn}\relax (x)}{b^{3}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} \mathrm {sgn}\relax (x)}{b^{3}}\right )}}{3465 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.53 \[ -\frac {2 \left (b x +a \right ) \left (-105 b^{3} x^{3}+70 a \,b^{2} x^{2}-40 a^{2} b x +16 a^{3}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{1155 b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 64, normalized size = 0.59 \[ \frac {2 \, {\left (105 \, b^{5} x^{5} + 140 \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{3} - 6 \, a^{3} b^{2} x^{2} + 8 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt {b x + a}}{1155 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 58, normalized size = 0.54 \[ -\frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^2\,\left (16\,a^3-40\,a^2\,b\,x+70\,a\,b^2\,x^2-105\,b^3\,x^3\right )}{1155\,b^4\,x} \]
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 \[ \int \left (a x^{2} + b x^{3}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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