Optimal. Leaf size=158 \[ \frac {729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt {x}}-\frac {243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8} \]
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Rubi [A] time = 0.06, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt {x}}-\frac {243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx &=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}-\frac {(15 b) \int \frac {1}{x^{15/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{16 a}\\ &=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}+\frac {\left (45 b^2\right ) \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx}{52 a^2}\\ &=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}-\frac {\left (81 b^3\right ) \int \frac {1}{x^{9/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{104 a^3}\\ &=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}+\frac {\left (243 b^4\right ) \int \frac {1}{x^3 \left (a+b x^{3/2}\right )^{2/3}} \, dx}{364 a^4}\\ &=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac {243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}-\frac {\left (729 b^5\right ) \int \frac {1}{x^{3/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{1456 a^5}\\ &=-\frac {\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac {15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac {9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac {243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac {729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 0.53 \[ \frac {\sqrt [3]{a+b x^{3/2}} \left (-91 a^5+105 a^4 b x^{3/2}-126 a^3 b^2 x^3+162 a^2 b^3 x^{9/2}-243 a b^4 x^6+729 b^5 x^{15/2}\right )}{728 a^6 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 75, normalized size = 0.47 \[ -\frac {{\left (243 \, a b^{4} x^{6} + 126 \, a^{3} b^{2} x^{3} + 91 \, a^{5} - 3 \, {\left (243 \, b^{5} x^{7} + 54 \, a^{2} b^{3} x^{4} + 35 \, a^{4} b x\right )} \sqrt {x}\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{728 \, a^{6} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{\frac {3}{2}}+a \right )^{\frac {2}{3}} x^{9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 103, normalized size = 0.65 \[ \frac {\frac {1456 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{5}}{\sqrt {x}} - \frac {1820 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} b^{4}}{x^{2}} + \frac {2080 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} b^{3}}{x^{\frac {7}{2}}} - \frac {1456 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}} b^{2}}{x^{5}} + \frac {560 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {13}{3}} b}{x^{\frac {13}{2}}} - \frac {91 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {16}{3}}}{x^{8}}}{728 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^9\,{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 55.10, size = 1554, normalized size = 9.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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