Integrand size = 19, antiderivative size = 101 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {72 b \sqrt [6]{a+b x}}{91 (b c-a d)^2 (c+d x)^{7/6}}+\frac {432 b^2 \sqrt [6]{a+b x}}{91 (b c-a d)^3 \sqrt [6]{c+d x}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {432 b^2 \sqrt [6]{a+b x}}{91 \sqrt [6]{c+d x} (b c-a d)^3}+\frac {72 b \sqrt [6]{a+b x}}{91 (c+d x)^{7/6} (b c-a d)^2}+\frac {6 \sqrt [6]{a+b x}}{13 (c+d x)^{13/6} (b c-a d)} \]
[In]
[Out]
Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {(12 b) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx}{13 (b c-a d)} \\ & = \frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {72 b \sqrt [6]{a+b x}}{91 (b c-a d)^2 (c+d x)^{7/6}}+\frac {\left (72 b^2\right ) \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx}{91 (b c-a d)^2} \\ & = \frac {6 \sqrt [6]{a+b x}}{13 (b c-a d) (c+d x)^{13/6}}+\frac {72 b \sqrt [6]{a+b x}}{91 (b c-a d)^2 (c+d x)^{7/6}}+\frac {432 b^2 \sqrt [6]{a+b x}}{91 (b c-a d)^3 \sqrt [6]{c+d x}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (7 a^2 d^2-2 a b d (13 c+6 d x)+b^2 \left (91 c^2+156 c d x+72 d^2 x^2\right )\right )}{91 (b c-a d)^3 (c+d x)^{13/6}} \]
[In]
[Out]
Time = 0.94 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {1}{6}} \left (72 d^{2} x^{2} b^{2}-12 x a b \,d^{2}+156 x \,b^{2} c d +7 a^{2} d^{2}-26 a b c d +91 b^{2} c^{2}\right )}{91 \left (d x +c \right )^{\frac {13}{6}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (83) = 166\).
Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.50 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\frac {6 \, {\left (72 \, b^{2} d^{2} x^{2} + 91 \, b^{2} c^{2} - 26 \, a b c d + 7 \, a^{2} d^{2} + 12 \, {\left (13 \, b^{2} c d - a b d^{2}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{91 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{3} + 3 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} + 3 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\right )} x\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {19}{6}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{19/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{19/6}} \,d x \]
[In]
[Out]