Integrand size = 19, antiderivative size = 152 \[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {5 \sqrt [4]{b} (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac {5 \sqrt [4]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}} \]
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Time = 0.07 (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, 246, 218, 214, 211} \[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=-\frac {5 \sqrt [4]{b} (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}-\frac {5 \sqrt [4]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 d^{9/4}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}} \]
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Rule 49
Rule 52
Rule 65
Rule 211
Rule 214
Rule 218
Rule 246
Rubi steps \begin{align*} \text {integral}& = -\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {(5 b) \int \frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx}{d} \\ & = -\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {(5 b (b c-a d)) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 d^2} \\ & = -\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{d^2} \\ & = -\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {(5 (b c-a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{d^2} \\ & = -\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {\left (5 \sqrt {b} (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 d^2}-\frac {\left (5 \sqrt {b} (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 d^2} \\ & = -\frac {4 (a+b x)^{5/4}}{d \sqrt [4]{c+d x}}+\frac {5 b \sqrt [4]{a+b x} (c+d x)^{3/4}}{d^2}-\frac {5 \sqrt [4]{b} (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 d^{9/4}}-\frac {5 \sqrt [4]{b} (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 d^{9/4}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=\frac {\sqrt [4]{a+b x} (5 b c-4 a d+b d x)}{d^2 \sqrt [4]{c+d x}}+\frac {5 \sqrt [4]{b} (b c-a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 d^{9/4}}-\frac {5 \sqrt [4]{b} (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 d^{9/4}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {5}{4}}}{\left (d x +c \right )^{\frac {5}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 754, normalized size of antiderivative = 4.96 \[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=-\frac {5 \, {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 5 \, {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (d^{3} x + c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 5 \, {\left (-i \, d^{3} x - i \, c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (i \, d^{3} x + i \, c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 5 \, {\left (i \, d^{3} x + i \, c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {5 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (-i \, d^{3} x - i \, c d^{2}\right )} \left (\frac {b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{d^{9}}\right )^{\frac {1}{4}}\right )}}{d x + c}\right ) - 4 \, {\left (b d x + 5 \, b c - 4 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, {\left (d^{3} x + c d^{2}\right )}} \]
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\[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{4}}}{\left (c + d x\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {5}{4}}}{{\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{5/4}}{(c+d x)^{5/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/4}}{{\left (c+d\,x\right )}^{5/4}} \,d x \]
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