Integrand size = 19, antiderivative size = 403 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}+\frac {7 \sqrt [6]{b} (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}+\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}} \]
[Out]
Time = 0.37 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {49, 52, 65, 246, 216, 648, 632, 210, 642, 214} \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\frac {7 \sqrt [6]{b} (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \arctan \left (\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}+\frac {7 \sqrt [6]{b} (b c-a d) \log \left (-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 d^{13/6}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}} \]
[In]
[Out]
Rule 49
Rule 52
Rule 65
Rule 210
Rule 214
Rule 216
Rule 246
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {(7 b) \int \frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}} \, dx}{d} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {(7 b (b c-a d)) \int \frac {1}{(a+b x)^{5/6} \sqrt [6]{c+d x}} \, dx}{6 d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {(7 (b c-a d)) \text {Subst}\left (\int \frac {1}{\sqrt [6]{c-\frac {a d}{b}+\frac {d x^6}{b}}} \, dx,x,\sqrt [6]{a+b x}\right )}{d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {(7 (b c-a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^6}{b}} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b}-\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{3 d^2}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b}+\frac {\sqrt [6]{d} x}{2}}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{3 d^2}-\frac {\left (7 \sqrt [3]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{3 d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {7 \sqrt [6]{b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}+\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {-\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt [6]{b} \sqrt [6]{d}+2 \sqrt [3]{d} x}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}}-\frac {\left (7 \sqrt [3]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{4 d^2}-\frac {\left (7 \sqrt [3]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [6]{b} \sqrt [6]{d} x+\sqrt [3]{d} x^2} \, dx,x,\frac {\sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{4 d^2} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}-\frac {7 \sqrt [6]{b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}+\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}}-\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 d^{13/6}}+\frac {\left (7 \sqrt [6]{b} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 d^{13/6}} \\ & = -\frac {6 (a+b x)^{7/6}}{d \sqrt [6]{c+d x}}+\frac {7 b \sqrt [6]{a+b x} (c+d x)^{5/6}}{d^2}+\frac {7 \sqrt [6]{b} (b c-a d) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}}{\sqrt {3}}\right )}{2 \sqrt {3} d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 d^{13/6}}+\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}}-\frac {7 \sqrt [6]{b} (b c-a d) \log \left (\sqrt [3]{b}+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}\right )}{12 d^{13/6}} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\frac {\frac {6 \sqrt [6]{d} \sqrt [6]{a+b x} (7 b c-6 a d+b d x)}{\sqrt [6]{c+d x}}-7 \sqrt {3} \sqrt [6]{b} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{-2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )+7 \sqrt {3} \sqrt [6]{b} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{c+d x}}{2 \sqrt [6]{d} \sqrt [6]{a+b x}+\sqrt [6]{b} \sqrt [6]{c+d x}}\right )-14 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )-7 \sqrt [6]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}+\frac {\sqrt [6]{b} \sqrt [6]{c+d x}}{\sqrt [6]{d} \sqrt [6]{a+b x}}\right )}{6 d^{13/6}} \]
[In]
[Out]
\[\int \frac {\left (b x +a \right )^{\frac {7}{6}}}{\left (d x +c \right )^{\frac {7}{6}}}d x\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1562 vs. \(2 (301) = 602\).
Time = 0.26 (sec) , antiderivative size = 1562, normalized size of antiderivative = 3.88 \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{6}}}{\left (c + d x\right )^{\frac {7}{6}}}\, dx \]
[In]
[Out]
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{6}}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^{7/6}}{(c+d x)^{7/6}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/6}}{{\left (c+d\,x\right )}^{7/6}} \,d x \]
[In]
[Out]