Integrand size = 19, antiderivative size = 221 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}+\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}}-\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}} \]
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Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {49, 53, 65, 313, 230, 227, 1214, 1213, 435} \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=-\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{7/4} \sqrt {a+b x} \sqrt [4]{b c-a d}}+\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt {a+b x} \sqrt [4]{b c-a d}}-\frac {d (c+d x)^{3/4}}{b \sqrt {a+b x} (b c-a d)}-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}} \]
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Rule 49
Rule 53
Rule 65
Rule 227
Rule 230
Rule 313
Rule 435
Rule 1213
Rule 1214
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}+\frac {d \int \frac {1}{(a+b x)^{3/2} \sqrt [4]{c+d x}} \, dx}{2 b} \\ & = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}+\frac {d^2 \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{4 b (b c-a d)} \\ & = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}+\frac {d \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 )}{b (b c-a d)} \\ & = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}-\frac {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^{3/2} \sqrt {b c-a d}}+\frac {d \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 )}{b^{3/2} \sqrt {b c-a d}} \\ & = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}-\frac {\left (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^{3/2} \sqrt {b c-a d} \sqrt {a+b x}}+\frac {\left (d \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 )}{b^{3/2} \sqrt {b c-a d} \sqrt {a+b x}} \\ & = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}-\frac {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^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}}+\frac {\left (d \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 )}{b^{3/2} \sqrt {b c-a d} \sqrt {a+b x}} \\ & = -\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}+\frac {d \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 )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}}-\frac {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^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.33 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{4}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/4}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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