Optimal. Leaf size=137 \[ -\frac {3 a^2}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{7 b^3 \left (a+b \sqrt [3]{x}\right )^6 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {1}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1341, 646, 43} \[ -\frac {3 a^2}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{7 b^3 \left (a+b \sqrt [3]{x}\right )^6 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {1}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1341
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{9/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{9/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b^9 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac {\left (3 b^9 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname {Subst}\left (\int \left (\frac {a^2}{b^{11} (a+b x)^9}-\frac {2 a}{b^{11} (a+b x)^8}+\frac {1}{b^{11} (a+b x)^7}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac {3 a^2}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{7 b^3 \left (a+b \sqrt [3]{x}\right )^6 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {1}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.42 \[ \frac {-a^2-8 a b \sqrt [3]{x}-28 b^2 x^{2/3}}{56 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 275, normalized size = 2.01 \[ -\frac {28 \, b^{18} x^{6} - 2856 \, a^{3} b^{15} x^{5} + 18186 \, a^{6} b^{12} x^{4} - 20608 \, a^{9} b^{9} x^{3} + 4200 \, a^{12} b^{6} x^{2} - 48 \, a^{15} b^{3} x + a^{18} - 27 \, {\left (8 \, a b^{17} x^{5} - 244 \, a^{4} b^{14} x^{4} + 840 \, a^{7} b^{11} x^{3} - 553 \, a^{10} b^{8} x^{2} + 56 \, a^{13} b^{5} x\right )} x^{\frac {2}{3}} + 27 \, {\left (35 \, a^{2} b^{16} x^{5} - 448 \, a^{5} b^{13} x^{4} + 876 \, a^{8} b^{10} x^{3} - 328 \, a^{11} b^{7} x^{2} + 14 \, a^{14} b^{4} x\right )} x^{\frac {1}{3}}}{56 \, {\left (b^{27} x^{8} + 8 \, a^{3} b^{24} x^{7} + 28 \, a^{6} b^{21} x^{6} + 56 \, a^{9} b^{18} x^{5} + 70 \, a^{12} b^{15} x^{4} + 56 \, a^{15} b^{12} x^{3} + 28 \, a^{18} b^{9} x^{2} + 8 \, a^{21} b^{6} x + a^{24} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 43, normalized size = 0.31 \[ -\frac {28 \, b^{2} x^{\frac {2}{3}} + 8 \, a b x^{\frac {1}{3}} + a^{2}}{56 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.39 \[ -\frac {\sqrt {b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}}\, \left (28 b^{2} x^{\frac {2}{3}}+8 a b \,x^{\frac {1}{3}}+a^{2}\right )}{56 \left (b \,x^{\frac {1}{3}}+a \right )^{9} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 53, normalized size = 0.39 \[ -\frac {1}{2 \, b^{9} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{6}} + \frac {6 \, a}{7 \, b^{10} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{7}} - \frac {3 \, a^{2}}{8 \, b^{11} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 53, normalized size = 0.39 \[ -\frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^2+28\,b^2\,x^{2/3}+8\,a\,b\,x^{1/3}\right )}{56\,b^3\,{\left (a+b\,x^{1/3}\right )}^9} \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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