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 = 0.540582, antiderivative size = 332, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt [6]{d} \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac{\sqrt [6]{d} \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{2 b^{7/6}}+\frac{\sqrt{3} \sqrt [6]{d} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac{\sqrt{3} \sqrt [6]{d} \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{7/6}}+\frac{2 \sqrt [6]{d} \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{b^{7/6}}-\frac{6 \sqrt [6]{c+d x}}{b \sqrt [6]{a+b x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 331
Rule 296
Rule 634
Rule 618
Rule 204
Rule 628
Rule 208
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*}
Mathematica [C] time = 0.0264023, size = 71, normalized size = 0.21 \[ -\frac{6 \sqrt [6]{c+d x} \, _2F_1\left (-\frac{1}{6},-\frac{1}{6};\frac{5}{6};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt [6]{a+b x} \sqrt [6]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [6]{dx+c} \left ( bx+a \right ) ^{-{\frac{7}{6}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{6}}}{{\left (b x + a\right )}^{\frac{7}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88771, size = 1704, normalized size = 5.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [6]{c + d x}}{\left (a + b x\right )^{\frac{7}{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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