Optimal. Leaf size=66 \[ \frac {36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx &=\frac {6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac {(6 b) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{7 (b c-a d)}\\ &=\frac {6 \sqrt [6]{a+b x}}{7 (b c-a d) (c+d x)^{7/6}}+\frac {36 b \sqrt [6]{a+b x}}{7 (b c-a d)^2 \sqrt [6]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {6 \sqrt [6]{a+b x} (-a d+7 b c+6 b d x)}{7 (c+d x)^{7/6} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 57, normalized size = 0.86 \begin {gather*} \frac {6 \left (\frac {7 b \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}-\frac {d (a+b x)^{7/6}}{(c+d x)^{7/6}}\right )}{7 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.44, size = 118, normalized size = 1.79 \begin {gather*} \frac {6 \, {\left (6 \, 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}}}{7 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\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 {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {13}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.80 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (-6 b d x +a d -7 b c \right )}{7 \left (d x +c \right )^{\frac {7}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \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 {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {13}{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 = -0.02 \begin {gather*} \int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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