Optimal. Leaf size=66 \[ \frac {36 b (a+b x)^{13/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)} \]
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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 (a+b x)^{13/6}}{247 (c+d x)^{13/6} (b c-a d)^2}+\frac {6 (a+b x)^{13/6}}{19 (c+d x)^{19/6} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^{7/6}}{(c+d x)^{25/6}} \, dx &=\frac {6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {(6 b) \int \frac {(a+b x)^{7/6}}{(c+d x)^{19/6}} \, dx}{19 (b c-a d)}\\ &=\frac {6 (a+b x)^{13/6}}{19 (b c-a d) (c+d x)^{19/6}}+\frac {36 b (a+b x)^{13/6}}{247 (b c-a d)^2 (c+d x)^{13/6}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 0.70 \begin {gather*} \frac {6 (a+b x)^{13/6} (-13 a d+19 b c+6 b d x)}{247 (c+d x)^{19/6} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 51, normalized size = 0.77 \begin {gather*} \frac {6 (a+b x)^{13/6} \left (19 b-\frac {13 d (a+b x)}{c+d x}\right )}{247 (c+d x)^{13/6} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.34, size = 235, normalized size = 3.56 \begin {gather*} \frac {6 \, {\left (6 \, b^{3} d x^{3} + 19 \, a^{2} b c - 13 \, a^{3} d + {\left (19 \, b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (19 \, a b^{2} c - 10 \, a^{2} b d\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{247 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{4} + 4 \, {\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x^{3} + 6 \, {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{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 {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {25}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {13}{6}} \left (-6 b d x +13 a d -19 b c \right )}{247 \left (d x +c \right )^{\frac {19}{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 {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {25}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 189, normalized size = 2.86 \begin {gather*} -\frac {{\left (c+d\,x\right )}^{5/6}\,\left (\frac {\left (78\,a^3\,d-114\,a^2\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{247\,d^4\,{\left (a\,d-b\,c\right )}^2}-\frac {36\,b^3\,x^3\,{\left (a+b\,x\right )}^{1/6}}{247\,d^3\,{\left (a\,d-b\,c\right )}^2}-\frac {x^2\,\left (114\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (a+b\,x\right )}^{1/6}}{247\,d^4\,{\left (a\,d-b\,c\right )}^2}+\frac {12\,a\,b\,x\,\left (10\,a\,d-19\,b\,c\right )\,{\left (a+b\,x\right )}^{1/6}}{247\,d^4\,{\left (a\,d-b\,c\right )}^2}\right )}{x^4+\frac {c^4}{d^4}+\frac {4\,c\,x^3}{d}+\frac {4\,c^3\,x}{d^3}+\frac {6\,c^2\,x^2}{d^2}} \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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