Optimal. Leaf size=332 \[ -\frac {\sqrt [6]{d} \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac {\sqrt [6]{d} \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \]
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Rubi [A] time = 0.54, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {47, 63, 331, 296, 634, 618, 204, 628, 208} \begin {gather*} -\frac {\sqrt [6]{d} \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^{7/6}}+\frac {\sqrt [6]{d} \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^{7/6}}+\frac {\sqrt {3} \sqrt [6]{d} \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^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \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^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 204
Rule 208
Rule 296
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\sqrt [6]{c+d x}}{(a+b x)^{7/6}} \, dx &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {d \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{b}\\ &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {(6 d) \operatorname {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^2}\\ &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {(6 d) \operatorname {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^2}\\ &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\left (2 \sqrt [3]{d}\right ) \operatorname {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 )}{b^{7/6}}+\frac {\left (2 \sqrt [3]{d}\right ) \operatorname {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 )}{b^{7/6}}+\frac {\left (2 \sqrt [3]{d}\right ) \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}\\ &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \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^{7/6}}+\frac {\sqrt [6]{d} \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^{7/6}}-\frac {\left (3 \sqrt [3]{d}\right ) \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}-\frac {\left (3 \sqrt [3]{d}\right ) \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}\\ &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \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^{7/6}}+\frac {\sqrt [6]{d} \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^{7/6}}-\frac {\left (3 \sqrt [6]{d}\right ) \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^{7/6}}+\frac {\left (3 \sqrt [6]{d}\right ) \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^{7/6}}\\ &=-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}}+\frac {\sqrt {3} \sqrt [6]{d} \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^{7/6}}-\frac {\sqrt {3} \sqrt [6]{d} \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^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac {\sqrt [6]{d} \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^{7/6}}+\frac {\sqrt [6]{d} \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^{7/6}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 71, normalized size = 0.21 \begin {gather*} -\frac {6 \sqrt [6]{c+d x} \, _2F_1\left (-\frac {1}{6},-\frac {1}{6};\frac {5}{6};\frac {d (a+b x)}{a d-b c}\right )}{b \sqrt [6]{a+b x} \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 256, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt {3} \sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{b^{7/6}}+\frac {\sqrt {3} \sqrt [6]{d} \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt {3} \sqrt [6]{d} \sqrt [6]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{7/6}}+\frac {2 \sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{b^{7/6}}+\frac {\sqrt [6]{d} \tanh ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{c+d x}}{\sqrt [6]{a+b x} \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\sqrt [3]{d}\right )}\right )}{b^{7/6}}-\frac {6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.58, size = 663, normalized size = 2.00 \begin {gather*} -\frac {4 \, \sqrt {3} {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\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^{6} \left (\frac {d}{b^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{7} x + a b^{6}\right )} \sqrt {\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\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 {d}{b^{7}}\right )^{\frac {5}{6}} + \sqrt {3} {\left (b d x + a d\right )}}{3 \, {\left (b d x + a d\right )}}\right ) + 4 \, \sqrt {3} {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\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^{6} \left (\frac {d}{b^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (b^{7} x + a b^{6}\right )} \sqrt {-\frac {{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\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 {d}{b^{7}}\right )^{\frac {5}{6}} - \sqrt {3} {\left (b d x + a d\right )}}{3 \, {\left (b d x + a d\right )}}\right ) - {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\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}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\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 (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\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}} b \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b^{3} x + a b^{2}\right )} \left (\frac {d}{b^{7}}\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 ) - 2 \, {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + 2 \, {\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (b^{2} x + a b\right )} \left (\frac {d}{b^{7}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{b x + a}\right ) + 12 \, {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{2 \, {\left (b^{2} x + a b\right )}} \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*} \int \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (d x +c \right )^{\frac {1}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}\, dx \end {gather*}
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*} \int \frac {{\left (d x + c\right )}^{\frac {1}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}}\,{d x} \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 {{\left (c+d\,x\right )}^{1/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \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 [6]{c + d x}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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