Optimal. Leaf size=403 \[ \frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 b^{13/6}} \]
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Rubi [A]
time = 0.43, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {49, 52, 65,
338, 302, 648, 632, 210, 642, 214} \begin {gather*} -\frac {7 \sqrt [6]{d} (b c-a 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 )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a 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 )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a 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 )}{2 \sqrt {3} b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a 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 )}{2 \sqrt {3} b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}+\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 210
Rule 214
Rule 302
Rule 338
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx &=-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d) \int \frac {\sqrt [6]{c+d x}}{\sqrt [6]{a+b x}} \, dx}{b}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d (b c-a d)) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{5/6}} \, dx}{6 b^2}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d (b c-a d)) \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^3}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {(7 d (b c-a d)) \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^3}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {\left (7 \sqrt [3]{d} (b c-a d)\right ) \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 )}{3 b^{13/6}}+\frac {\left (7 \sqrt [3]{d} (b c-a d)\right ) \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 )}{3 b^{13/6}}+\frac {\left (7 \sqrt [3]{d} (b c-a d)\right ) \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 )}{3 b^2}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}-\frac {\left (7 \sqrt [6]{d} (b c-a d)\right ) \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 )}{12 b^{13/6}}+\frac {\left (7 \sqrt [6]{d} (b c-a d)\right ) \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 )}{12 b^{13/6}}-\frac {\left (7 \sqrt [3]{d} (b c-a d)\right ) \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 )}{4 b^2}-\frac {\left (7 \sqrt [3]{d} (b c-a d)\right ) \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 )}{4 b^2}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a 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 )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a 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 )}{12 b^{13/6}}-\frac {\left (7 \sqrt [6]{d} (b c-a d)\right ) \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 )}{2 b^{13/6}}+\frac {\left (7 \sqrt [6]{d} (b c-a d)\right ) \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 )}{2 b^{13/6}}\\ &=\frac {7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac {6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}}+\frac {7 \sqrt [6]{d} (b c-a 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 )}{2 \sqrt {3} b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a 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 )}{2 \sqrt {3} b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}-\frac {7 \sqrt [6]{d} (b c-a 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 )}{12 b^{13/6}}+\frac {7 \sqrt [6]{d} (b c-a 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 )}{12 b^{13/6}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.05, size = 71, normalized size = 0.18 \begin {gather*} -\frac {6 (c+d x)^{7/6} \, _2F_1\left (-\frac {7}{6},-\frac {1}{6};\frac {5}{6};\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {7}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}\, dx\]
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. 3084 vs.
\(2 (301) = 602\).
time = 0.38, size = 3084, normalized size = 7.65
result too large to display
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 {\left (c + d x\right )^{\frac {7}{6}}}{\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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Giac [F] N/A
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 {{\left (c+d\,x\right )}^{7/6}}{{\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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