Integrand size = 19, antiderivative size = 262 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=-\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}+\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}}-\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}} \]
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Time = 0.19 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {53, 65, 313, 230, 227, 1214, 1213, 435} \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=-\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 \sqrt {a+b x} (b c-a d)^{9/4}}+\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 \sqrt {a+b x} (b c-a d)^{9/4}}-\frac {42 b d \sqrt {a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^3}-\frac {14 d \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} (c+d x)^{5/4} (b c-a d)} \]
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Rule 53
Rule 65
Rule 227
Rule 230
Rule 313
Rule 435
Rule 1213
Rule 1214
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {(7 d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx}{2 (b c-a d)} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {(21 b d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx}{10 (b c-a d)^2} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}+\frac {\left (21 b^2 d\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{10 (b c-a d)^3} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}+\frac {\left (42 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 (b c-a d)^3} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {\left (42 b^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 (b c-a d)^{5/2}}+\frac {\left (42 b^{3/2}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 (b c-a d)^{5/2}} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {\left (42 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 (b c-a d)^{5/2} \sqrt {a+b x}}+\frac {\left (42 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 (b c-a d)^{5/2} \sqrt {a+b x}} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}-\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}}+\frac {\left (42 b^{3/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 (b c-a d)^{5/2} \sqrt {a+b x}} \\ & = -\frac {2}{(b c-a d) \sqrt {a+b x} (c+d x)^{5/4}}-\frac {14 d \sqrt {a+b x}}{5 (b c-a d)^2 (c+d x)^{5/4}}-\frac {42 b d \sqrt {a+b x}}{5 (b c-a d)^3 \sqrt [4]{c+d x}}+\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}}-\frac {42 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 (b c-a d)^{9/4} \sqrt {a+b x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.27 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {9}{4},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} (c+d x)^{9/4}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {9}{4}}}d x\]
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\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {9}{4}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{9/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{9/4}} \,d x \]
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