Optimal. Leaf size=136 \[ \frac {7776 b^3 (a+b x)^{11/6}}{124729 (c+d x)^{11/6} (b c-a d)^4}+\frac {1296 b^2 (a+b x)^{11/6}}{11339 (c+d x)^{17/6} (b c-a d)^3}+\frac {108 b (a+b x)^{11/6}}{667 (c+d x)^{23/6} (b c-a d)^2}+\frac {6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {7776 b^3 (a+b x)^{11/6}}{124729 (c+d x)^{11/6} (b c-a d)^4}+\frac {1296 b^2 (a+b x)^{11/6}}{11339 (c+d x)^{17/6} (b c-a d)^3}+\frac {108 b (a+b x)^{11/6}}{667 (c+d x)^{23/6} (b c-a d)^2}+\frac {6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (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)^{5/6}}{(c+d x)^{35/6}} \, dx &=\frac {6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac {(18 b) \int \frac {(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx}{29 (b c-a d)}\\ &=\frac {6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac {108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac {\left (216 b^2\right ) \int \frac {(a+b x)^{5/6}}{(c+d x)^{23/6}} \, dx}{667 (b c-a d)^2}\\ &=\frac {6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac {108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac {1296 b^2 (a+b x)^{11/6}}{11339 (b c-a d)^3 (c+d x)^{17/6}}+\frac {\left (1296 b^3\right ) \int \frac {(a+b x)^{5/6}}{(c+d x)^{17/6}} \, dx}{11339 (b c-a d)^3}\\ &=\frac {6 (a+b x)^{11/6}}{29 (b c-a d) (c+d x)^{29/6}}+\frac {108 b (a+b x)^{11/6}}{667 (b c-a d)^2 (c+d x)^{23/6}}+\frac {1296 b^2 (a+b x)^{11/6}}{11339 (b c-a d)^3 (c+d x)^{17/6}}+\frac {7776 b^3 (a+b x)^{11/6}}{124729 (b c-a d)^4 (c+d x)^{11/6}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 118, normalized size = 0.87 \begin {gather*} \frac {6 (a+b x)^{11/6} \left (-4301 a^3 d^3+561 a^2 b d^2 (29 c+6 d x)-33 a b^2 d \left (667 c^2+348 c d x+72 d^2 x^2\right )+b^3 \left (11339 c^3+12006 c^2 d x+6264 c d^2 x^2+1296 d^3 x^3\right )\right )}{124729 (c+d x)^{29/6} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 95, normalized size = 0.70 \begin {gather*} \frac {6 (a+b x)^{29/6} \left (\frac {11339 b^3 (c+d x)^3}{(a+b x)^3}-\frac {22011 b^2 d (c+d x)^2}{(a+b x)^2}+\frac {16269 b d^2 (c+d x)}{a+b x}-4301 d^3\right )}{124729 (c+d x)^{29/6} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.18, size = 533, normalized size = 3.92 \begin {gather*} \frac {6 \, {\left (1296 \, b^{4} d^{3} x^{4} + 11339 \, a b^{3} c^{3} - 22011 \, a^{2} b^{2} c^{2} d + 16269 \, a^{3} b c d^{2} - 4301 \, a^{4} d^{3} + 216 \, {\left (29 \, b^{4} c d^{2} - 5 \, a b^{3} d^{3}\right )} x^{3} + 18 \, {\left (667 \, b^{4} c^{2} d - 290 \, a b^{3} c d^{2} + 55 \, a^{2} b^{2} d^{3}\right )} x^{2} + {\left (11339 \, b^{4} c^{3} - 10005 \, a b^{3} c^{2} d + 4785 \, a^{2} b^{2} c d^{2} - 935 \, a^{3} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{124729 \, {\left (b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4} + {\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} x^{5} + 5 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} x^{4} + 10 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} x^{3} + 10 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} x^{2} + 5 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\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 {5}{6}}}{{\left (d x + c\right )}^{\frac {35}{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 = 171, normalized size = 1.26 \begin {gather*} -\frac {6 \left (b x +a \right )^{\frac {11}{6}} \left (-1296 b^{3} d^{3} x^{3}+2376 a \,b^{2} d^{3} x^{2}-6264 b^{3} c \,d^{2} x^{2}-3366 a^{2} b \,d^{3} x +11484 a \,b^{2} c \,d^{2} x -12006 b^{3} c^{2} d x +4301 a^{3} d^{3}-16269 a^{2} b c \,d^{2}+22011 a \,b^{2} c^{2} d -11339 b^{3} c^{3}\right )}{124729 \left (d x +c \right )^{\frac {29}{6}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\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 {5}{6}}}{{\left (d x + c\right )}^{\frac {35}{6}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 303, normalized size = 2.23 \begin {gather*} \frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {7776\,b^4\,x^4\,{\left (a+b\,x\right )}^{5/6}}{124729\,d^2\,{\left (a\,d-b\,c\right )}^4}-\frac {{\left (a+b\,x\right )}^{5/6}\,\left (25806\,a^4\,d^3-97614\,a^3\,b\,c\,d^2+132066\,a^2\,b^2\,c^2\,d-68034\,a\,b^3\,c^3\right )}{124729\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,{\left (a+b\,x\right )}^{5/6}\,\left (-5610\,a^3\,b\,d^3+28710\,a^2\,b^2\,c\,d^2-60030\,a\,b^3\,c^2\,d+68034\,b^4\,c^3\right )}{124729\,d^5\,{\left (a\,d-b\,c\right )}^4}+\frac {108\,b^2\,x^2\,{\left (a+b\,x\right )}^{5/6}\,\left (55\,a^2\,d^2-290\,a\,b\,c\,d+667\,b^2\,c^2\right )}{124729\,d^4\,{\left (a\,d-b\,c\right )}^4}-\frac {1296\,b^3\,x^3\,\left (5\,a\,d-29\,b\,c\right )\,{\left (a+b\,x\right )}^{5/6}}{124729\,d^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^5+\frac {c^5}{d^5}+\frac {5\,c\,x^4}{d}+\frac {5\,c^4\,x}{d^4}+\frac {10\,c^2\,x^3}{d^2}+\frac {10\,c^3\,x^2}{d^3}} \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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