Integrand size = 19, antiderivative size = 152 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}-\frac {5 \sqrt [4]{d} (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}} \]
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Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {49, 52, 65, 338, 304, 211, 214} \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=-\frac {5 \sqrt [4]{d} (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}} \]
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Rule 49
Rule 52
Rule 65
Rule 211
Rule 214
Rule 304
Rule 338
Rubi steps \begin{align*} \text {integral}& = -\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d) \int \frac {\sqrt [4]{c+d x}}{\sqrt [4]{a+b x}} \, dx}{b} \\ & = \frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d (b c-a d)) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{4 b^2} \\ & = \frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d (b c-a d)) \text {Subst}\left (\int \frac {x^2}{\left (c-\frac {a d}{b}+\frac {d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{b^3} \\ & = \frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {(5 d (b c-a d)) \text {Subst}\left (\int \frac {x^2}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{b^3} \\ & = \frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}+\frac {\left (5 \sqrt {d} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 b^2}-\frac {\left (5 \sqrt {d} (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 b^2} \\ & = \frac {5 d (a+b x)^{3/4} \sqrt [4]{c+d x}}{b^2}-\frac {4 (c+d x)^{5/4}}{b \sqrt [4]{a+b x}}-\frac {5 \sqrt [4]{d} (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{9/4}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\frac {\sqrt [4]{c+d x} (-4 b c+5 a d+b d x)}{b^2 \sqrt [4]{a+b x}}+\frac {5 \sqrt [4]{d} (b c-a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 b^{9/4}}+\frac {5 \sqrt [4]{d} (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 b^{9/4}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {5}{4}}}{\left (b x +a \right )^{\frac {5}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 754, normalized size of antiderivative = 4.96 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\frac {5 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 5 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} - {\left (b^{3} x + a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 5 \, {\left (i \, b^{3} x + i \, a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (i \, b^{3} x + i \, a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 5 \, {\left (-i \, b^{3} x - i \, a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (-i \, b^{3} x - i \, a b^{2}\right )} \left (\frac {b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}}{b^{9}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 4 \, {\left (b d x - 4 \, b c + 5 \, a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{4 \, {\left (b^{3} x + a b^{2}\right )}} \]
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\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{4}}}{\left (a + b x\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {5}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{5/4}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/4}}{{\left (a+b\,x\right )}^{5/4}} \,d x \]
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