Integrand size = 19, antiderivative size = 144 \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=-\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {16 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{7 \sqrt [4]{b} d^3 \sqrt {a+b x}} \]
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Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {52, 65, 230, 227} \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\frac {16 (b c-a d)^{9/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{7 \sqrt [4]{b} d^3 \sqrt {a+b x}}-\frac {8 \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d} \]
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Rule 52
Rule 65
Rule 227
Rule 230
Rubi steps \begin{align*} \text {integral}& = \frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}-\frac {(6 (b c-a d)) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx}{7 d} \\ & = -\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {\left (4 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{7 d^2} \\ & = -\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {\left (16 (b c-a d)^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 )}{7 d^3} \\ & = -\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {\left (16 (b c-a d)^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 )}{7 d^3 \sqrt {a+b x}} \\ & = -\frac {8 (b c-a d) \sqrt {a+b x} \sqrt [4]{c+d x}}{7 d^2}+\frac {4 (a+b x)^{3/2} \sqrt [4]{c+d x}}{7 d}+\frac {16 (b c-a d)^{9/4} \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 )}{7 \sqrt [4]{b} d^3 \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 = 0.51 \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\frac {2 (a+b x)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b (c+d x)^{3/4}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (d x +c \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{\left (c + d x\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{{\left (c+d\,x\right )}^{3/4}} \,d x \]
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