Optimal. Leaf size=149 \[ -\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}}+\frac {14 b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}} \]
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Rubi [A] time = 0.06, antiderivative size = 147, normalized size of antiderivative = 0.99, number of steps used = 7, 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 {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}}+\frac {14 b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {7 (a+b x)^{2/3}}{2 a^2 x^2}+\frac {14 b (a+b x)^{2/3}}{3 a^3 x}+\frac {3}{a x^2 \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^3 (a+b x)^{4/3}} \, dx &=\frac {3}{a x^2 \sqrt [3]{a+b x}}+\frac {7 \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx}{a}\\ &=\frac {3}{a x^2 \sqrt [3]{a+b x}}-\frac {7 (a+b x)^{2/3}}{2 a^2 x^2}-\frac {(14 b) \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx}{3 a^2}\\ &=\frac {3}{a x^2 \sqrt [3]{a+b x}}-\frac {7 (a+b x)^{2/3}}{2 a^2 x^2}+\frac {14 b (a+b x)^{2/3}}{3 a^3 x}+\frac {\left (14 b^2\right ) \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{9 a^3}\\ &=\frac {3}{a x^2 \sqrt [3]{a+b x}}-\frac {7 (a+b x)^{2/3}}{2 a^2 x^2}+\frac {14 b (a+b x)^{2/3}}{3 a^3 x}-\frac {7 b^2 \log (x)}{9 a^{10/3}}-\frac {\left (7 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^{10/3}}+\frac {\left (7 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^3}\\ &=\frac {3}{a x^2 \sqrt [3]{a+b x}}-\frac {7 (a+b x)^{2/3}}{2 a^2 x^2}+\frac {14 b (a+b x)^{2/3}}{3 a^3 x}-\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}}-\frac {\left (14 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{3 a^{10/3}}\\ &=\frac {3}{a x^2 \sqrt [3]{a+b x}}-\frac {7 (a+b x)^{2/3}}{2 a^2 x^2}+\frac {14 b (a+b x)^{2/3}}{3 a^3 x}+\frac {14 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.22 \begin {gather*} \frac {3 b^2 \, _2F_1\left (-\frac {1}{3},3;\frac {2}{3};\frac {b x}{a}+1\right )}{a^3 \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 161, normalized size = 1.08 \begin {gather*} \frac {14 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{10/3}}-\frac {7 b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{9 a^{10/3}}+\frac {14 b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {18 a^2-49 a (a+b x)+28 (a+b x)^2}{6 a^3 x^2 \sqrt [3]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 407, normalized size = 2.73 \begin {gather*} \left [\frac {42 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - 14 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \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 ) + 28 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (28 \, a b^{2} x^{2} + 7 \, a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}, -\frac {14 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \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 ) - 28 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {84 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - 3 \, {\left (28 \, a b^{2} x^{2} + 7 \, a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.59, size = 140, normalized size = 0.94 \begin {gather*} \frac {14 \, \sqrt {3} b^{2} \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 )}{9 \, a^{\frac {10}{3}}} - \frac {7 \, b^{2} \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 )}{9 \, a^{\frac {10}{3}}} + \frac {14 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {10}{3}}} + \frac {3 \, b^{2}}{{\left (b x + a\right )}^{\frac {1}{3}} a^{3}} + \frac {10 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{2} - 13 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{2}}{6 \, a^{3} b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 131, normalized size = 0.88 \begin {gather*} \frac {14 \sqrt {3}\, b^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {10}{3}}}+\frac {14 b^{2} \ln \left (-a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {1}{3}}\right )}{9 a^{\frac {10}{3}}}-\frac {7 b^{2} \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 )}{9 a^{\frac {10}{3}}}+\frac {3 b^{2}}{\left (b x +a \right )^{\frac {1}{3}} a^{3}}-\frac {13 \left (b x +a \right )^{\frac {2}{3}}}{6 a^{2} x^{2}}+\frac {5 \left (b x +a \right )^{\frac {5}{3}}}{3 a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 158, normalized size = 1.06 \begin {gather*} \frac {14 \, \sqrt {3} b^{2} \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 )}{9 \, a^{\frac {10}{3}}} + \frac {28 \, {\left (b x + a\right )}^{2} b^{2} - 49 \, {\left (b x + a\right )} a b^{2} + 18 \, a^{2} b^{2}}{6 \, {\left ({\left (b x + a\right )}^{\frac {7}{3}} a^{3} - 2 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{4} + {\left (b x + a\right )}^{\frac {1}{3}} a^{5}\right )}} - \frac {7 \, b^{2} \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 )}{9 \, a^{\frac {10}{3}}} + \frac {14 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {10}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 221, normalized size = 1.48 \begin {gather*} \frac {\frac {3\,b^2}{a}+\frac {14\,b^2\,{\left (a+b\,x\right )}^2}{3\,a^3}-\frac {49\,b^2\,\left (a+b\,x\right )}{6\,a^2}}{{\left (a+b\,x\right )}^{7/3}-2\,a\,{\left (a+b\,x\right )}^{4/3}+a^2\,{\left (a+b\,x\right )}^{1/3}}+\frac {\ln \left (588\,a^3\,b^4\,{\left (a+b\,x\right )}^{1/3}-3\,a^{10/3}\,{\left (-7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}^2\right )\,\left (-7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}{9\,a^{10/3}}-\frac {\ln \left (588\,a^3\,b^4\,{\left (a+b\,x\right )}^{1/3}-3\,a^{10/3}\,{\left (7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}^2\right )\,\left (7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}{9\,a^{10/3}}+\frac {14\,b^2\,\ln \left (588\,a^3\,b^4\,{\left (a+b\,x\right )}^{1/3}-588\,a^{10/3}\,b^4\right )}{9\,a^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.19, size = 2793, normalized size = 18.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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