Integrand size = 19, antiderivative size = 236 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d (b c-a d)^{5/4} \sqrt {a+b x}}+\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d (b c-a d)^{5/4} \sqrt {a+b x}} \]
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Time = 0.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d \sqrt {a+b x} (b c-a d)^{5/4}}-\frac {12 b^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d \sqrt {a+b x} (b c-a d)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 \sqrt [4]{c+d x} (b c-a d)^2}+\frac {4 \sqrt {a+b x}}{5 (c+d x)^{5/4} (b c-a d)} \]
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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 {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {(3 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/4}} \, dx}{5 (b c-a d)} \\ & = \frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{5 (b c-a d)^2} \\ & = \frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {\left (12 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 d (b c-a d)^2} \\ & = \frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (12 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 d (b c-a d)^{3/2}}-\frac {\left (12 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 d (b c-a d)^{3/2}} \\ & = \frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}+\frac {\left (12 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 d (b c-a d)^{3/2} \sqrt {a+b x}}-\frac {\left (12 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 d (b c-a d)^{3/2} \sqrt {a+b x}} \\ & = \frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}+\frac {12 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 d (b c-a d)^{5/4} \sqrt {a+b x}}-\frac {\left (12 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 d (b c-a d)^{3/2} \sqrt {a+b x}} \\ & = \frac {4 \sqrt {a+b x}}{5 (b c-a d) (c+d x)^{5/4}}+\frac {12 b \sqrt {a+b x}}{5 (b c-a d)^2 \sqrt [4]{c+d x}}-\frac {12 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 d (b c-a d)^{5/4} \sqrt {a+b x}}+\frac {12 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 d (b c-a d)^{5/4} \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 = 0.30 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{9/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{9/4}} \]
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\[\int \frac {1}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {9}{4}}}d x\]
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\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {9}{4}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {9}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{9/4}} \, dx=\int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{9/4}} \,d x \]
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