Optimal. Leaf size=246 \[ -\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} c}-\frac {\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{d}}-\frac {\sqrt [3]{a} \log (x)}{2 c}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{d}}+\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 c}+\frac {\sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{d}} \]
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Rubi [A]
time = 0.14, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {457, 85, 59,
631, 210, 31, 60} \begin {gather*} -\frac {\sqrt [3]{b c-a d} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{d}}-\frac {\sqrt [3]{a} \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} c}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{d}}+\frac {\sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{d}}+\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 c}-\frac {\sqrt [3]{a} \log (x)}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 59
Rule 60
Rule 85
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{x \left (c+d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x (c+d x)} \, dx,x,x^3\right )\\ &=\frac {a \text {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{3 c}+\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 c}\\ &=-\frac {\sqrt [3]{a} \log (x)}{2 c}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{d}}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c}-\frac {a^{2/3} \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c}+\frac {\sqrt [3]{b c-a d} \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{d}}+\frac {(b c-a d)^{2/3} \text {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c d^{2/3}}\\ &=-\frac {\sqrt [3]{a} \log (x)}{2 c}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{d}}+\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 c}+\frac {\sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{d}}+\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{c}+\frac {\sqrt [3]{b c-a d} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{c \sqrt [3]{d}}\\ &=-\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} c}-\frac {\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{d}}-\frac {\sqrt [3]{a} \log (x)}{2 c}-\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c \sqrt [3]{d}}+\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 c}+\frac {\sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{d}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 312, normalized size = 1.27 \begin {gather*} -\frac {2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{d} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )-2 \sqrt [3]{a} \sqrt [3]{d} \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+\sqrt [3]{a} \sqrt [3]{d} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+\sqrt [3]{b c-a d} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{6 c \sqrt [3]{d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x \left (d \,x^{3}+c \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 276, normalized size = 1.12 \begin {gather*} -\frac {2 \, \sqrt {3} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 2 \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 2 \, a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}{6 \, c} \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 {\sqrt [3]{a + b x^{3}}}{x \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 311, normalized size = 1.26 \begin {gather*} -\frac {{\left (b c - a d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} - \frac {\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, c} - \frac {a^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, c} + \frac {a^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, c} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, c d} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.74, size = 1607, normalized size = 6.53 \begin {gather*} \ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (6\,a^4\,b^4\,d^5-12\,a^3\,b^5\,c\,d^4+9\,a^2\,b^6\,c^2\,d^3-3\,a\,b^7\,c^3\,d^2\right )-{\left (\frac {a}{27\,c^3}\right )}^{1/3}\,\left (\left (\left (486\,a^3\,b^4\,c^4\,d^5-729\,a^2\,b^5\,c^5\,d^4+243\,a\,b^6\,c^6\,d^3\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}-{\left (b\,x^3+a\right )}^{1/3}\,\left (81\,a\,b^6\,c^5\,d^3-81\,a^2\,b^5\,c^4\,d^4\right )\right )\,{\left (\frac {a}{27\,c^3}\right )}^{2/3}-9\,a\,b^7\,c^4\,d^2+27\,a^2\,b^6\,c^3\,d^3-18\,a^3\,b^5\,c^2\,d^4\right )\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (6\,a^4\,b^4\,d^5-12\,a^3\,b^5\,c\,d^4+9\,a^2\,b^6\,c^2\,d^3-3\,a\,b^7\,c^3\,d^2\right )-\left (\left (\left (486\,a^3\,b^4\,c^4\,d^5-729\,a^2\,b^5\,c^5\,d^4+243\,a\,b^6\,c^6\,d^3\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}-{\left (b\,x^3+a\right )}^{1/3}\,\left (81\,a\,b^6\,c^5\,d^3-81\,a^2\,b^5\,c^4\,d^4\right )\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{2/3}-9\,a\,b^7\,c^4\,d^2+27\,a^2\,b^6\,c^3\,d^3-18\,a^3\,b^5\,c^2\,d^4\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (6\,a^4\,b^4\,d^5-12\,a^3\,b^5\,c\,d^4+9\,a^2\,b^6\,c^2\,d^3-3\,a\,b^7\,c^3\,d^2\right )+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}\,\left ({\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left ({\left (b\,x^3+a\right )}^{1/3}\,\left (81\,a\,b^6\,c^5\,d^3-81\,a^2\,b^5\,c^4\,d^4\right )-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (486\,a^3\,b^4\,c^4\,d^5-729\,a^2\,b^5\,c^5\,d^4+243\,a\,b^6\,c^6\,d^3\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{2/3}+9\,a\,b^7\,c^4\,d^2-27\,a^2\,b^6\,c^3\,d^3+18\,a^3\,b^5\,c^2\,d^4\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}-\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (6\,a^4\,b^4\,d^5-12\,a^3\,b^5\,c\,d^4+9\,a^2\,b^6\,c^2\,d^3-3\,a\,b^7\,c^3\,d^2\right )-\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}\,\left ({\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left ({\left (b\,x^3+a\right )}^{1/3}\,\left (81\,a\,b^6\,c^5\,d^3-81\,a^2\,b^5\,c^4\,d^4\right )+\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (486\,a^3\,b^4\,c^4\,d^5-729\,a^2\,b^5\,c^5\,d^4+243\,a\,b^6\,c^6\,d^3\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{2/3}+9\,a\,b^7\,c^4\,d^2-27\,a^2\,b^6\,c^3\,d^3+18\,a^3\,b^5\,c^2\,d^4\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,d-b\,c}{27\,c^3\,d}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (6\,a^4\,b^4\,d^5-12\,a^3\,b^5\,c\,d^4+9\,a^2\,b^6\,c^2\,d^3-3\,a\,b^7\,c^3\,d^2\right )+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}\,\left ({\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left ({\left (b\,x^3+a\right )}^{1/3}\,\left (81\,a\,b^6\,c^5\,d^3-81\,a^2\,b^5\,c^4\,d^4\right )-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (486\,a^3\,b^4\,c^4\,d^5-729\,a^2\,b^5\,c^5\,d^4+243\,a\,b^6\,c^6\,d^3\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}\right )\,{\left (\frac {a}{27\,c^3}\right )}^{2/3}+9\,a\,b^7\,c^4\,d^2-27\,a^2\,b^6\,c^3\,d^3+18\,a^3\,b^5\,c^2\,d^4\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}-\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (6\,a^4\,b^4\,d^5-12\,a^3\,b^5\,c\,d^4+9\,a^2\,b^6\,c^2\,d^3-3\,a\,b^7\,c^3\,d^2\right )-\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}\,\left ({\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,\left ({\left (b\,x^3+a\right )}^{1/3}\,\left (81\,a\,b^6\,c^5\,d^3-81\,a^2\,b^5\,c^4\,d^4\right )+\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (486\,a^3\,b^4\,c^4\,d^5-729\,a^2\,b^5\,c^5\,d^4+243\,a\,b^6\,c^6\,d^3\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3}\right )\,{\left (\frac {a}{27\,c^3}\right )}^{2/3}+9\,a\,b^7\,c^4\,d^2-27\,a^2\,b^6\,c^3\,d^3+18\,a^3\,b^5\,c^2\,d^4\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{27\,c^3}\right )}^{1/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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