Optimal. Leaf size=113 \[ \frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac {4 b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}-\frac {4 b}{a^2 \sqrt [3]{a+b x}}-\frac {1}{a x \sqrt [3]{a+b x}} \]
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Rubi [A] time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {51, 55, 617, 204, 31} \begin {gather*} -\frac {4 (a+b x)^{2/3}}{a^2 x}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac {4 b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {3}{a x \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx &=\frac {3}{a x \sqrt [3]{a+b x}}+\frac {4 \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx}{a}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}-\frac {(4 b) \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{3 a^2}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}+\frac {2 b \log (x)}{3 a^{7/3}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{a^2}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{7/3}}\\ &=\frac {3}{a x \sqrt [3]{a+b x}}-\frac {4 (a+b x)^{2/3}}{a^2 x}-\frac {4 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.27 \begin {gather*} -\frac {3 b \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {b x}{a}+1\right )}{a^2 \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 138, normalized size = 1.22 \begin {gather*} -\frac {4 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3}}+\frac {2 b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{3 a^{7/3}}-\frac {4 b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a^{7/3}}+\frac {3 a-4 (a+b x)}{a^2 x \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 407, normalized size = 3.60 \begin {gather*} \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (4 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - 2 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.40, size = 120, normalized size = 1.06 \begin {gather*} -\frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, b \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x + a\right )} b - 3 \, a b}{{\left ({\left (b x + a\right )}^{\frac {4}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 108, normalized size = 0.96 \begin {gather*} -\frac {4 \sqrt {3}\, b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {7}{3}}}-\frac {4 b \ln \left (-a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {1}{3}}\right )}{3 a^{\frac {7}{3}}}+\frac {2 b \ln \left (a^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {2}{3}}\right )}{3 a^{\frac {7}{3}}}-\frac {3 b}{\left (b x +a \right )^{\frac {1}{3}} a^{2}}-\frac {\left (b x +a \right )^{\frac {2}{3}}}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 122, normalized size = 1.08 \begin {gather*} -\frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x + a\right )} b - 3 \, a b}{{\left (b x + a\right )}^{\frac {4}{3}} a^{2} - {\left (b x + a\right )}^{\frac {1}{3}} a^{3}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {7}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 173, normalized size = 1.53 \begin {gather*} -\frac {\frac {3\,b}{a}-\frac {4\,b\,\left (a+b\,x\right )}{a^2}}{a\,{\left (a+b\,x\right )}^{1/3}-{\left (a+b\,x\right )}^{4/3}}+\frac {\ln \left (a^{7/3}\,{\left (2\,b-\sqrt {3}\,b\,2{}\mathrm {i}\right )}^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (2\,b-\sqrt {3}\,b\,2{}\mathrm {i}\right )}{3\,a^{7/3}}+\frac {\ln \left (a^{7/3}\,{\left (2\,b+\sqrt {3}\,b\,2{}\mathrm {i}\right )}^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (2\,b+\sqrt {3}\,b\,2{}\mathrm {i}\right )}{3\,a^{7/3}}-\frac {4\,b\,\ln \left (16\,a^{7/3}\,b^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )}{3\,a^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.52, size = 857, normalized size = 7.58
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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