Optimal. Leaf size=137 \[ -\frac {a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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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} \[ -\frac {a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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 )^{7/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b^7 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^7} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac {\left (3 b^7 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname {Subst}\left (\int \left (\frac {a^2}{b^9 (a+b x)^7}-\frac {2 a}{b^9 (a+b x)^6}+\frac {1}{b^9 (a+b x)^5}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac {a^2}{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}}}+\frac {6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.42 \[ \frac {-a^2-6 a b \sqrt [3]{x}-15 b^2 x^{2/3}}{20 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 209, normalized size = 1.53 \[ -\frac {280 \, a^{2} b^{12} x^{4} - 1400 \, a^{5} b^{9} x^{3} + 735 \, a^{8} b^{6} x^{2} - 14 \, a^{11} b^{3} x + a^{14} + 3 \, {\left (5 \, b^{14} x^{4} - 210 \, a^{3} b^{11} x^{3} + 483 \, a^{6} b^{8} x^{2} - 112 \, a^{9} b^{5} x\right )} x^{\frac {2}{3}} - 3 \, {\left (28 \, a b^{13} x^{4} - 357 \, a^{4} b^{10} x^{3} + 390 \, a^{7} b^{7} x^{2} - 35 \, a^{10} b^{4} x\right )} x^{\frac {1}{3}}}{20 \, {\left (b^{21} x^{6} + 6 \, a^{3} b^{18} x^{5} + 15 \, a^{6} b^{15} x^{4} + 20 \, a^{9} b^{12} x^{3} + 15 \, a^{12} b^{9} x^{2} + 6 \, a^{15} b^{6} x + a^{18} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 43, normalized size = 0.31 \[ -\frac {15 \, b^{2} x^{\frac {2}{3}} + 6 \, a b x^{\frac {1}{3}} + a^{2}}{20 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
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 \[ -\frac {\sqrt {b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}}\, \left (15 b^{2} x^{\frac {2}{3}}+6 a b \,x^{\frac {1}{3}}+a^{2}\right )}{20 \left (b \,x^{\frac {1}{3}}+a \right )^{7} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 53, normalized size = 0.39 \[ -\frac {3}{4 \, b^{7} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{4}} + \frac {6 \, a}{5 \, b^{8} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{5}} - \frac {a^{2}}{2 \, b^{9} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.23, size = 53, normalized size = 0.39 \[ -\frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^2+15\,b^2\,x^{2/3}+6\,a\,b\,x^{1/3}\right )}{20\,b^3\,{\left (a+b\,x^{1/3}\right )}^7} \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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