Integrand size = 19, antiderivative size = 795 \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=-\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {48 d^2 (a+b x)^{3/4}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {48 \sqrt {b} d^{3/2} \sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{5 (b c-a d)^4 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )}+\frac {24 \sqrt {2} \sqrt [4]{b} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{5 (b c-a d)^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {12 \sqrt {2} \sqrt [4]{b} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right ),\frac {1}{2}\right )}{5 (b c-a d)^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}} \]
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Time = 0.62 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {53, 64, 637, 311, 226, 1210} \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\frac {48 (a+b x)^{3/4} d^2}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {48 \sqrt {b} \sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2} d^{3/2}}{5 (b c-a d)^4 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )}+\frac {24 \sqrt {2} \sqrt [4]{b} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right ) d^{5/4}}{5 (b c-a d)^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {12 \sqrt {2} \sqrt [4]{b} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right ),\frac {1}{2}\right ) d^{5/4}}{5 (b c-a d)^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}-\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}} \]
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Rule 53
Rule 64
Rule 226
Rule 311
Rule 637
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}-\frac {(6 d) \int \frac {1}{(a+b x)^{5/4} (c+d x)^{5/4}} \, dx}{5 (b c-a d)} \\ & = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {\left (12 d^2\right ) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{5/4}} \, dx}{5 (b c-a d)^2} \\ & = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {48 d^2 (a+b x)^{3/4}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {\left (24 b d^2\right ) \int \frac {1}{\sqrt [4]{a+b x} \sqrt [4]{c+d x}} \, dx}{5 (b c-a d)^3} \\ & = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {48 d^2 (a+b x)^{3/4}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {\left (24 b d^2 \sqrt [4]{(a+b x) (c+d x)}\right ) \int \frac {1}{\sqrt [4]{a c+(b c+a d) x+b d x^2}} \, dx}{5 (b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x}} \\ & = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {48 d^2 (a+b x)^{3/4}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {\left (96 b d^2 \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{5 (b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)} \\ & = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {48 d^2 (a+b x)^{3/4}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {\left (48 \sqrt {b} d^{3/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)}+\frac {\left (48 \sqrt {b} d^{3/2} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {1-\frac {2 \sqrt {b} \sqrt {d} x^2}{b c-a d}}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x)} \\ & = -\frac {4}{5 (b c-a d) (a+b x)^{5/4} \sqrt [4]{c+d x}}+\frac {24 d}{5 (b c-a d)^2 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}+\frac {48 d^2 (a+b x)^{3/4}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {48 \sqrt {b} d^{3/2} \sqrt {(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{5 (b c-a d)^4 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )}+\frac {24 \sqrt {2} \sqrt [4]{b} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{5 (b c-a d)^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}}-\frac {12 \sqrt {2} \sqrt [4]{b} d^{5/4} \sqrt [4]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt {\frac {(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac {2 \sqrt {b} \sqrt {d} \sqrt {(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt {b c-a d}}\right )|\frac {1}{2}\right )}{5 (b c-a d)^{3/2} \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c+a d+2 b d x) \sqrt {(a d+b (c+2 d x))^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.09 \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=-\frac {4 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {5}{4},-\frac {1}{4},\frac {d (a+b x)}{-b c+a d}\right )}{5 b (a+b x)^{5/4} (c+d x)^{5/4}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {9}{4}} \left (d x +c \right )^{\frac {5}{4}}}d x\]
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\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {9}{4}} \left (c + d x\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {9}{4}} {\left (d x + c\right )}^{\frac {5}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{9/4} (c+d x)^{5/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{9/4}\,{\left (c+d\,x\right )}^{5/4}} \,d x \]
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