Optimal. Leaf size=64 \[ -\frac {36 d (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)^2}-\frac {6}{\sqrt [6]{a+b x} (c+d x)^{5/6} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 64, 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 d (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)^2}-\frac {6}{\sqrt [6]{a+b x} (c+d x)^{5/6} (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)^{7/6} (c+d x)^{11/6}} \, dx &=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}-\frac {(6 d) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{b c-a d}\\ &=-\frac {6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{5/6}}-\frac {36 d (a+b x)^{5/6}}{5 (b c-a d)^2 (c+d x)^{5/6}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.70 \begin {gather*} -\frac {6 (a d+5 b c+6 b d x)}{5 \sqrt [6]{a+b x} (c+d x)^{5/6} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 49, normalized size = 0.77 \begin {gather*} -\frac {6 (a+b x)^{5/6} \left (\frac {5 b (c+d x)}{a+b x}+d\right )}{5 (c+d x)^{5/6} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.33, size = 126, normalized size = 1.97 \begin {gather*} -\frac {6 \, {\left (6 \, b d x + 5 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{5 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} 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 {7}{6}} {\left (d x + c\right )}^{\frac {11}{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 = 0.83 \begin {gather*} -\frac {6 \left (6 b d x +a d +5 b c \right )}{5 \left (b x +a \right )^{\frac {1}{6}} \left (d x +c \right )^{\frac {5}{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 {7}{6}} {\left (d x + c\right )}^{\frac {11}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 72, normalized size = 1.12 \begin {gather*} -\frac {\left (\frac {36\,b\,x}{5\,{\left (a\,d-b\,c\right )}^2}+\frac {6\,a\,d+30\,b\,c}{5\,d\,{\left (a\,d-b\,c\right )}^2}\right )\,{\left (c+d\,x\right )}^{1/6}}{x\,{\left (a+b\,x\right )}^{1/6}+\frac {c\,{\left (a+b\,x\right )}^{1/6}}{d}} \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 {1}{\left (a + b x\right )^{\frac {7}{6}} \left (c + d x\right )^{\frac {11}{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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