Optimal. Leaf size=136 \[ \frac {256 a^4 \left (a x^2+b x^3\right )^{5/2}}{15015 b^5 x^5}-\frac {128 a^3 \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac {32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac {16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x} \]
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Rubi [A] time = 0.17, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2016, 2002, 2014} \begin {gather*} \frac {256 a^4 \left (a x^2+b x^3\right )^{5/2}}{15015 b^5 x^5}-\frac {128 a^3 \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac {32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac {16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x \left (a x^2+b x^3\right )^{3/2} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {(8 a) \int \left (a x^2+b x^3\right )^{3/2} \, dx}{13 b}\\ &=-\frac {16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}+\frac {\left (48 a^2\right ) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x} \, dx}{143 b^2}\\ &=\frac {32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac {16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}-\frac {\left (64 a^3\right ) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^2} \, dx}{429 b^3}\\ &=-\frac {128 a^3 \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac {32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac {16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}+\frac {\left (128 a^4\right ) \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^3} \, dx}{3003 b^4}\\ &=\frac {256 a^4 \left (a x^2+b x^3\right )^{5/2}}{15015 b^5 x^5}-\frac {128 a^3 \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac {32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac {16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{13 b x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 69, normalized size = 0.51 \begin {gather*} \frac {2 x (a+b x)^3 \left (128 a^4-320 a^3 b x+560 a^2 b^2 x^2-840 a b^3 x^3+1155 b^4 x^4\right )}{15015 b^5 \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.56, size = 75, normalized size = 0.55 \begin {gather*} \frac {2 (a+b x) \left (x^2 (a+b x)\right )^{3/2} \left (3003 a^4-8580 a^3 (a+b x)+10010 a^2 (a+b x)^2-5460 a (a+b x)^3+1155 (a+b x)^4\right )}{15015 b^5 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 84, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (1155 \, b^{6} x^{6} + 1470 \, a b^{5} x^{5} + 35 \, a^{2} b^{4} x^{4} - 40 \, a^{3} b^{3} x^{3} + 48 \, a^{4} b^{2} x^{2} - 64 \, a^{5} b x + 128 \, a^{6}\right )} \sqrt {b x^{3} + a x^{2}}}{15015 \, b^{5} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 246, normalized size = 1.81 \begin {gather*} -\frac {256 \, a^{\frac {13}{2}} \mathrm {sgn}\relax (x)}{15015 \, b^{5}} + \frac {2 \, {\left (\frac {143 \, {\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^{2} \mathrm {sgn}\relax (x)}{b^{4}} + \frac {130 \, {\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 )} a \mathrm {sgn}\relax (x)}{b^{4}} + \frac {15 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} \mathrm {sgn}\relax (x)}{b^{4}}\right )}}{45045 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 68, normalized size = 0.50 \begin {gather*} \frac {2 \left (b x +a \right ) \left (1155 x^{4} b^{4}-840 a \,b^{3} x^{3}+560 a^{2} x^{2} b^{2}-320 a^{3} x b +128 a^{4}\right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{15015 b^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 75, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left (1155 \, b^{6} x^{6} + 1470 \, a b^{5} x^{5} + 35 \, a^{2} b^{4} x^{4} - 40 \, a^{3} b^{3} x^{3} + 48 \, a^{4} b^{2} x^{2} - 64 \, a^{5} b x + 128 \, a^{6}\right )} \sqrt {b x + a}}{15015 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 69, normalized size = 0.51 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a\,x^2}\,{\left (a+b\,x\right )}^2\,\left (128\,a^4-320\,a^3\,b\,x+560\,a^2\,b^2\,x^2-840\,a\,b^3\,x^3+1155\,b^4\,x^4\right )}{15015\,b^5\,x} \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 x \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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