Optimal. Leaf size=134 \[ -\frac{7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac{1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac{108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \]
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Rubi [A] time = 0.0327841, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac{1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac{108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac{(18 d) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx}{b c-a d}\\ &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac{108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac{(216 b d) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx}{17 (b c-a d)^2}\\ &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac{108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac{1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac{\left (1296 b^2 d\right ) \int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{187 (b c-a d)^3}\\ &=-\frac{6}{(b c-a d) \sqrt [6]{a+b x} (c+d x)^{17/6}}-\frac{108 d (a+b x)^{5/6}}{17 (b c-a d)^2 (c+d x)^{17/6}}-\frac{1296 b d (a+b x)^{5/6}}{187 (b c-a d)^3 (c+d x)^{11/6}}-\frac{7776 b^2 d (a+b x)^{5/6}}{935 (b c-a d)^4 (c+d x)^{5/6}}\\ \end{align*}
Mathematica [A] time = 0.0467412, size = 118, normalized size = 0.88 \[ -\frac{6 \left (-15 a^2 b d^2 (17 c+6 d x)+55 a^3 d^3+3 a b^2 d \left (187 c^2+204 c d x+72 d^2 x^2\right )+b^3 \left (3366 c^2 d x+935 c^3+3672 c d^2 x^2+1296 d^3 x^3\right )\right )}{935 \sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 171, normalized size = 1.3 \begin{align*} -{\frac{7776\,{x}^{3}{b}^{3}{d}^{3}+1296\,a{b}^{2}{d}^{3}{x}^{2}+22032\,{b}^{3}c{d}^{2}{x}^{2}-540\,{a}^{2}b{d}^{3}x+3672\,a{b}^{2}c{d}^{2}x+20196\,{b}^{3}{c}^{2}dx+330\,{a}^{3}{d}^{3}-1530\,{a}^{2}cb{d}^{2}+3366\,a{b}^{2}{c}^{2}d+5610\,{b}^{3}{c}^{3}}{935\,{a}^{4}{d}^{4}-3740\,{a}^{3}bc{d}^{3}+5610\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-3740\,a{b}^{3}{c}^{3}d+935\,{b}^{4}{c}^{4}}{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{17}{6}}}} \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{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67966, size = 946, normalized size = 7.06 \begin{align*} -\frac{6 \,{\left (1296 \, b^{3} d^{3} x^{3} + 935 \, b^{3} c^{3} + 561 \, a b^{2} c^{2} d - 255 \, a^{2} b c d^{2} + 55 \, a^{3} d^{3} + 216 \,{\left (17 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 18 \,{\left (187 \, b^{3} c^{2} d + 34 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{1}{6}}}{935 \,{\left (a b^{4} c^{7} - 4 \, a^{2} b^{3} c^{6} d + 6 \, a^{3} b^{2} c^{5} d^{2} - 4 \, a^{4} b c^{4} d^{3} + a^{5} c^{3} d^{4} +{\left (b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{4} +{\left (3 \, b^{5} c^{5} d^{2} - 11 \, a b^{4} c^{4} d^{3} + 14 \, a^{2} b^{3} c^{3} d^{4} - 6 \, a^{3} b^{2} c^{2} d^{5} - a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{3} + 3 \,{\left (b^{5} c^{6} d - 3 \, a b^{4} c^{5} d^{2} + 2 \, a^{2} b^{3} c^{4} d^{3} + 2 \, a^{3} b^{2} c^{3} d^{4} - 3 \, a^{4} b c^{2} d^{5} + a^{5} c d^{6}\right )} x^{2} +{\left (b^{5} c^{7} - a b^{4} c^{6} d - 6 \, a^{2} b^{3} c^{5} d^{2} + 14 \, a^{3} b^{2} c^{4} d^{3} - 11 \, a^{4} b c^{3} d^{4} + 3 \, a^{5} c^{2} d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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