Optimal. Leaf size=410 \[ \frac {3 b^7 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac {7 b^6 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac {63 b^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac {105 b^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac {105 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {45 b^2 \sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^7 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}-\frac {15 b x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {x \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}} \]
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Rubi [A] time = 0.27, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1341, 1355, 263, 43} \[ \frac {3 b^7 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac {7 b^6 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac {63 b^5 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac {105 b^4 \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}+\frac {45 b^2 \sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^7 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}-\frac {15 b x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}+\frac {x \left (a+\frac {b}{\sqrt [3]{x}}\right )}{a^5 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}-\frac {105 b^3 \left (a+\frac {b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 1341
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a b+\frac {b^2}{x}\right )^5} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ &=\frac {\left (3 b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (b^2+a b x\right )^5} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ &=\frac {\left (3 b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )\right ) \operatorname {Subst}\left (\int \left (\frac {15}{a^7 b^3}-\frac {5 x}{a^6 b^4}+\frac {x^2}{a^5 b^5}-\frac {b^2}{a^7 (b+a x)^5}+\frac {7 b}{a^7 (b+a x)^4}-\frac {21}{a^7 (b+a x)^3}+\frac {35}{a^7 b (b+a x)^2}-\frac {35}{a^7 b^2 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ &=\frac {3 \left (a b^7+\frac {b^8}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^4}-\frac {7 \left (a b^6+\frac {b^7}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^3}+\frac {63 \left (a b^5+\frac {b^6}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^2}-\frac {105 \left (a b^4+\frac {b^5}{\sqrt [3]{x}}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )}+\frac {45 \left (a b^2+\frac {b^3}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^7 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {15 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^6 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}+\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}{a^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}-\frac {105 \left (a b^3+\frac {b^4}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 152, normalized size = 0.37 \[ \frac {\left (a \sqrt [3]{x}+b\right ) \left (4 a^7 x^{7/3}-14 a^6 b x^2+84 a^5 b^2 x^{5/3}+556 a^4 b^3 x^{4/3}+544 a^3 b^4 x-444 a^2 b^5 x^{2/3}-856 a b^6 \sqrt [3]{x}-420 b^3 \left (a \sqrt [3]{x}+b\right )^4 \log \left (a \sqrt [3]{x}+b\right )-319 b^7\right )}{4 a^8 x^{5/3} \left (\frac {\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 141, normalized size = 0.34 \[ -\frac {105 \, b^{3} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{8} \mathrm {sgn}\left (a x^{\frac {2}{3}} + b x^{\frac {1}{3}}\right )} - \frac {420 \, a^{3} b^{4} x + 1134 \, a^{2} b^{5} x^{\frac {2}{3}} + 1036 \, a b^{6} x^{\frac {1}{3}} + 319 \, b^{7}}{4 \, {\left (a x^{\frac {1}{3}} + b\right )}^{4} a^{8} \mathrm {sgn}\left (a x^{\frac {2}{3}} + b x^{\frac {1}{3}}\right )} + \frac {2 \, a^{10} x - 15 \, a^{9} b x^{\frac {2}{3}} + 90 \, a^{8} b^{2} x^{\frac {1}{3}}}{2 \, a^{15} \mathrm {sgn}\left (a x^{\frac {2}{3}} + b x^{\frac {1}{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 199, normalized size = 0.49 \[ \frac {\left (4 a^{7} x^{\frac {7}{3}}-420 a^{4} b^{3} x^{\frac {4}{3}} \ln \left (a \,x^{\frac {1}{3}}+b \right )-14 a^{6} b \,x^{2}-1680 a^{3} b^{4} x \ln \left (a \,x^{\frac {1}{3}}+b \right )+84 a^{5} b^{2} x^{\frac {5}{3}}-2520 a^{2} b^{5} x^{\frac {2}{3}} \ln \left (a \,x^{\frac {1}{3}}+b \right )+556 a^{4} b^{3} x^{\frac {4}{3}}-1680 a \,b^{6} x^{\frac {1}{3}} \ln \left (a \,x^{\frac {1}{3}}+b \right )+544 a^{3} b^{4} x -420 b^{7} \ln \left (a \,x^{\frac {1}{3}}+b \right )-444 a^{2} b^{5} x^{\frac {2}{3}}-856 a \,b^{6} x^{\frac {1}{3}}-319 b^{7}\right ) \left (a \,x^{\frac {1}{3}}+b \right )}{4 \left (\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}} a^{8} x^{\frac {5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 139, normalized size = 0.34 \[ \frac {4 \, a^{7} x^{\frac {7}{3}} - 14 \, a^{6} b x^{2} + 84 \, a^{5} b^{2} x^{\frac {5}{3}} + 556 \, a^{4} b^{3} x^{\frac {4}{3}} + 544 \, a^{3} b^{4} x - 444 \, a^{2} b^{5} x^{\frac {2}{3}} - 856 \, a b^{6} x^{\frac {1}{3}} - 319 \, b^{7}}{4 \, {\left (a^{12} x^{\frac {4}{3}} + 4 \, a^{11} b x + 6 \, a^{10} b^{2} x^{\frac {2}{3}} + 4 \, a^{9} b^{3} x^{\frac {1}{3}} + a^{8} b^{4}\right )}} - \frac {105 \, b^{3} \log \left (a x^{\frac {1}{3}} + b\right )}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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