Optimal. Leaf size=309 \[ -\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{5/6} \sqrt [6]{d}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{5/6} \sqrt [6]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}} \]
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Rubi [A] time = 0.44, 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 = {63, 240, 210, 634, 618, 204, 628, 208} \[ -\frac {\log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{5/6} \sqrt [6]{d}}+\frac {\log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{5/6} \sqrt [6]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{b^{5/6} \sqrt [6]{d}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 208
Rule 210
Rule 240
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx &=\frac {6 \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d x^6}{b}}} \, dx,x,\sqrt [6]{a+b x}\right )}{b}\\ &=\frac {6 \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}-\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 )}{b^{5/6}}+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}+\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 )}{b^{5/6}}+\frac {2 \operatorname {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 )}{b^{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 )}{b^{5/6} \sqrt [6]{d}}+\frac {3 \operatorname {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 b^{2/3}}+\frac {3 \operatorname {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 b^{2/3}}-\frac {\operatorname {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 b^{5/6} \sqrt [6]{d}}+\frac {\operatorname {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 b^{5/6} \sqrt [6]{d}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}}-\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 b^{5/6} \sqrt [6]{d}}+\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 b^{5/6} \sqrt [6]{d}}+\frac {3 \operatorname {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 )}{b^{5/6} \sqrt [6]{d}}-\frac {3 \operatorname {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 )}{b^{5/6} \sqrt [6]{d}}\\ &=-\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 )}{b^{5/6} \sqrt [6]{d}}+\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 )}{b^{5/6} \sqrt [6]{d}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{5/6} \sqrt [6]{d}}-\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 b^{5/6} \sqrt [6]{d}}+\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 b^{5/6} \sqrt [6]{d}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 71, normalized size = 0.23 \[ \frac {6 \sqrt [6]{a+b x} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{6},\frac {1}{6};\frac {7}{6};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt [6]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 620, normalized size = 2.01 \[ -2 \, \sqrt {3} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b^{4} d \left (\frac {1}{b^{5} d}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{4} d^{2} x + b^{4} c d\right )} \sqrt {\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} + {\left (b^{2} d x + b^{2} c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}} \left (\frac {1}{b^{5} d}\right )^{\frac {5}{6}} + \sqrt {3} {\left (d x + c\right )}}{3 \, {\left (d x + c\right )}}\right ) - 2 \, \sqrt {3} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b^{4} d \left (\frac {1}{b^{5} d}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{4} d^{2} x + b^{4} c d\right )} \sqrt {-\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} - {\left (b^{2} d x + b^{2} c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}} \left (\frac {1}{b^{5} d}\right )^{\frac {5}{6}} - \sqrt {3} {\left (d x + c\right )}}{3 \, {\left (d x + c\right )}}\right ) + \frac {1}{2} \, \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} + {\left (b^{2} d x + b^{2} c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )}}{d x + c}\right ) - \frac {1}{2} \, \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (-\frac {4 \, {\left ({\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} - {\left (b^{2} d x + b^{2} c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )}}{d x + c}\right ) + \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) - \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b d x + b c\right )} \left (\frac {1}{b^{5} d}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{d x + c}\right ) \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b x +a \right )^{\frac {5}{6}} \left (d x +c \right )^{\frac {1}{6}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{1/6}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \sqrt [6]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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