Integrand size = 19, antiderivative size = 178 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {5 d^2 \sqrt {a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac {5 b^{3/4} d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{(b c-a d)^{11/4} \sqrt {a+b x}} \]
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Time = 0.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {53, 65, 230, 227} \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\frac {5 b^{3/4} d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt {a+b x} (b c-a d)^{11/4}}+\frac {5 d^2 \sqrt {a+b x}}{(c+d x)^{3/4} (b c-a d)^3}+\frac {3 d}{\sqrt {a+b x} (c+d x)^{3/4} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)} \]
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Rule 53
Rule 65
Rule 227
Rule 230
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}-\frac {(3 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{7/4}} \, dx}{2 (b c-a d)} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {\left (15 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{7/4}} \, dx}{4 (b c-a d)^2} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {5 d^2 \sqrt {a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac {\left (5 b d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{4 (b c-a d)^3} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {5 d^2 \sqrt {a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac {(5 b d) \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 )}{(b c-a d)^3} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {5 d^2 \sqrt {a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac {\left (5 b d \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 )}{(b c-a d)^3 \sqrt {a+b x}} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac {3 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}+\frac {5 d^2 \sqrt {a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac {5 b^{3/4} d \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 )}{(b c-a d)^{11/4} \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 = 73, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{7/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {7}{4},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} (c+d x)^{7/4}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {7}{4}}}d x\]
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\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {7}{4}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {7}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{7/4}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{7/4}} \,d x \]
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