Optimal. Leaf size=137 \[ -\frac {3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A]
time = 0.05, 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 = {1355, 660, 45}
\begin {gather*} -\frac {3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx &=3 \text {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b^{11} \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^{11}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac {\left (3 b^{11} \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \left (\frac {a^2}{b^{13} (a+b x)^{11}}-\frac {2 a}{b^{13} (a+b x)^{10}}+\frac {1}{b^{13} (a+b x)^9}\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}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{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}}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.41 \begin {gather*} \frac {\left (a+b \sqrt [3]{x}\right ) \left (-a^2-10 a b \sqrt [3]{x}-45 b^2 x^{2/3}\right )}{120 b^3 \left (\left (a+b \sqrt [3]{x}\right )^2\right )^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 54, normalized size = 0.39
method | result | size |
derivativedivides | \(-\frac {\left (45 b^{2} x^{\frac {2}{3}}+10 a b \,x^{\frac {1}{3}}+a^{2}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{120 b^{3} \left (\left (a +b \,x^{\frac {1}{3}}\right )^{2}\right )^{\frac {11}{2}}}\) | \(43\) |
default | \(-\frac {\sqrt {a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}}\, \left (45 b^{2} x^{\frac {2}{3}}+10 a b \,x^{\frac {1}{3}}+a^{2}\right )}{120 \left (a +b \,x^{\frac {1}{3}}\right )^{11} b^{3}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 53, normalized size = 0.39 \begin {gather*} -\frac {3}{8 \, b^{11} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{8}} + \frac {2 \, a}{3 \, b^{12} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{9}} - \frac {3 \, a^{2}}{10 \, b^{13} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs.
\(2 (107) = 214\).
time = 0.39, size = 343, normalized size = 2.50 \begin {gather*} \frac {440 \, a b^{21} x^{7} - 25630 \, a^{4} b^{18} x^{6} + 186252 \, a^{7} b^{15} x^{5} - 326150 \, a^{10} b^{12} x^{4} + 154000 \, a^{13} b^{9} x^{3} - 16005 \, a^{16} b^{6} x^{2} + 110 \, a^{19} b^{3} x - a^{22} - 27 \, {\left (88 \, a^{2} b^{20} x^{6} - 2200 \, a^{5} b^{17} x^{5} + 9625 \, a^{8} b^{14} x^{4} - 10910 \, a^{11} b^{11} x^{3} + 3245 \, a^{14} b^{8} x^{2} - 176 \, a^{17} b^{5} x\right )} x^{\frac {2}{3}} - 9 \, {\left (5 \, b^{22} x^{7} - 990 \, a^{3} b^{19} x^{6} + 12705 \, a^{6} b^{16} x^{5} - 34760 \, a^{9} b^{13} x^{4} + 25542 \, a^{12} b^{10} x^{3} - 4620 \, a^{15} b^{7} x^{2} + 110 \, a^{18} b^{4} x\right )} x^{\frac {1}{3}}}{120 \, {\left (b^{33} x^{10} + 10 \, a^{3} b^{30} x^{9} + 45 \, a^{6} b^{27} x^{8} + 120 \, a^{9} b^{24} x^{7} + 210 \, a^{12} b^{21} x^{6} + 252 \, a^{15} b^{18} x^{5} + 210 \, a^{18} b^{15} x^{4} + 120 \, a^{21} b^{12} x^{3} + 45 \, a^{24} b^{9} x^{2} + 10 \, a^{27} b^{6} x + a^{30} b^{3}\right )}} \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 \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {11}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.21, size = 43, normalized size = 0.31 \begin {gather*} -\frac {45 \, b^{2} x^{\frac {2}{3}} + 10 \, a b x^{\frac {1}{3}} + a^{2}}{120 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.34, size = 53, normalized size = 0.39 \begin {gather*} -\frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^2+45\,b^2\,x^{2/3}+10\,a\,b\,x^{1/3}\right )}{120\,b^3\,{\left (a+b\,x^{1/3}\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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