Optimal. Leaf size=358 \[ -\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {\sqrt {3} b^{7/6} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{13/6}}+\frac {b^{7/6} \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{2 d^{13/6}} \]
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Rubi [A]
time = 0.34, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {49, 65, 246,
216, 648, 632, 210, 642, 214} \begin {gather*} -\frac {\sqrt {3} b^{7/6} \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 )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \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 )}{d^{13/6}}-\frac {b^{7/6} \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 d^{13/6}}+\frac {b^{7/6} \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 d^{13/6}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 210
Rule 214
Rule 216
Rule 246
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {(a+b x)^{7/6}}{(c+d x)^{13/6}} \, dx &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}+\frac {b \int \frac {\sqrt [6]{a+b x}}{(c+d x)^{7/6}} \, dx}{d}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {b^2 \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {(6 b) \text {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 )}{d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {(6 b) \text {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 )}{d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {\left (2 b^{7/6}\right ) \text {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 )}{d^2}+\frac {\left (2 b^{7/6}\right ) \text {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 )}{d^2}+\frac {\left (2 b^{4/3}\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 )}{d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \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 d^{13/6}}+\frac {b^{7/6} \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 d^{13/6}}+\frac {\left (3 b^{4/3}\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 )}{2 d^2}+\frac {\left (3 b^{4/3}\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 )}{2 d^2}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \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 d^{13/6}}+\frac {b^{7/6} \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 d^{13/6}}+\frac {\left (3 b^{7/6}\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 )}{d^{13/6}}-\frac {\left (3 b^{7/6}\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 )}{d^{13/6}}\\ &=-\frac {6 (a+b x)^{7/6}}{7 d (c+d x)^{7/6}}-\frac {6 b \sqrt [6]{a+b x}}{d^2 \sqrt [6]{c+d x}}-\frac {\sqrt {3} b^{7/6} \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 )}{d^{13/6}}+\frac {\sqrt {3} b^{7/6} \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 )}{d^{13/6}}+\frac {2 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{d^{13/6}}-\frac {b^{7/6} \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 d^{13/6}}+\frac {b^{7/6} \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 d^{13/6}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 257, normalized size = 0.72 \begin {gather*} \frac {-\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x} (7 b c+a d+8 b d x)}{(c+d x)^{7/6}}-7 \sqrt {3} b^{7/6} \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 )+7 \sqrt {3} b^{7/6} \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 )+14 b^{7/6} \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+7 b^{7/6} \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 )}{7 d^{13/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {7}{6}}}{\left (d x +c \right )^{\frac {13}{6}}}\, dx\]
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. 855 vs.
\(2 (259) = 518\).
time = 0.99, size = 855, normalized size = 2.39 \begin {gather*} -\frac {28 \, \sqrt {3} {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\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 d^{11} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (d^{12} x + c d^{11}\right )} \sqrt {\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b d^{2} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (d^{5} x + c d^{4}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{3}}}{d x + c}} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {5}{6}} + \sqrt {3} {\left (b^{7} d x + b^{7} c\right )}}{3 \, {\left (b^{7} d x + b^{7} c\right )}}\right ) + 28 \, \sqrt {3} {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\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 d^{11} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} {\left (d^{12} x + c d^{11}\right )} \sqrt {-\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b d^{2} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b^{2} - {\left (d^{5} x + c d^{4}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{3}}}{d x + c}} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {5}{6}} - \sqrt {3} {\left (b^{7} d x + b^{7} c\right )}}{3 \, {\left (b^{7} d x + b^{7} c\right )}}\right ) - 7 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\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 d^{2} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (d^{5} x + c d^{4}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{3}}\right )}}{d x + c}\right ) + 7 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\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 d^{2} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b^{2} - {\left (d^{5} x + c d^{4}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{3}}\right )}}{d x + c}\right ) - 14 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b + {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) + 14 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}} b - {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{7}}{d^{13}}\right )^{\frac {1}{6}}}{d x + c}\right ) + 12 \, {\left (8 \, b d x + 7 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{14 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\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 {\left (a + b x\right )^{\frac {7}{6}}}{\left (c + d x\right )^{\frac {13}{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 {{\left (a+b\,x\right )}^{7/6}}{{\left (c+d\,x\right )}^{13/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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