Optimal. Leaf size=309 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}} \]
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Rubi [A]
time = 0.35, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {65, 338, 302,
648, 632, 210, 642, 214} \begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 210
Rule 214
Rule 302
Rule 338
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx &=\frac {6 \text {Subst}\left (\int \frac {x^4}{\left (c-\frac {a d}{b}+\frac {d x^6}{b}\right )^{5/6}} \, dx,x,\sqrt [6]{a+b x}\right )}{b}\\ &=\frac {6 \text {Subst}\left (\int \frac {x^4}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{d^{2/3}}+\frac {2 \text {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}-\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{2/3}}+\frac {2 \text {Subst}\left (\int \frac {-\frac {\sqrt [6]{b}}{2}+\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{2/3}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\text {Subst}\left (\int \frac {\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{2/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{2/3}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{\sqrt [6]{b} d^{5/6}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}}-\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 \sqrt [6]{b} d^{5/6}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 194, normalized size = 0.63 \begin {gather*} \frac {\sqrt {3} \left (-\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}}{\sqrt {3}}\right )+\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}}{\sqrt {3}}\right )\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )+\tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x} \sqrt [6]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [6]{b} d^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {5}{6}}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 620 vs.
\(2 (222) = 444\).
time = 1.34, size = 620, normalized size = 2.01 \begin {gather*} -2 \, \sqrt {3} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b d^{4} \left (\frac {1}{b d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{2} d^{4} x + a b d^{4}\right )} \sqrt {\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} d \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} + {\left (b d^{2} x + a d^{2}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}} \left (\frac {1}{b d^{5}}\right )^{\frac {5}{6}} + \sqrt {3} {\left (b x + a\right )}}{3 \, {\left (b x + a\right )}}\right ) - 2 \, \sqrt {3} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b d^{4} \left (\frac {1}{b d^{5}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{2} d^{4} x + a b d^{4}\right )} \sqrt {-\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} d \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} - {\left (b d^{2} x + a d^{2}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}} \left (\frac {1}{b d^{5}}\right )^{\frac {5}{6}} - \sqrt {3} {\left (b x + a\right )}}{3 \, {\left (b x + a\right )}}\right ) + \frac {1}{2} \, \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} d \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} + {\left (b d^{2} x + a d^{2}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )}}{b x + a}\right ) - \frac {1}{2} \, \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} d \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} - {\left (b d^{2} x + a d^{2}\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )}}{b x + a}\right ) + \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) - \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{5}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\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}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac {5}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{1/6}\,{\left (c+d\,x\right )}^{5/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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