Integrand size = 26, antiderivative size = 252 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt [4]{d} (b e-a f) \sqrt {c+d x}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt [4]{d} (b e-a f) \sqrt {c+d x}} \]
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Time = 0.17 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {110, 109, 418, 1232} \[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c+d x} (b e-a f)}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt [4]{d} \sqrt {c+d x} (b e-a f)} \]
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Rule 109
Rule 110
Rule 418
Rule 1232
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-\frac {f (c+d x)}{d e-c f}} \int \frac {1}{(a+b x) (e+f x)^{3/4} \sqrt {-\frac {c f}{d e-c f}-\frac {d f x}{d e-c f}}} \, dx}{\sqrt {c+d x}} \\ & = -\frac {\left (4 \sqrt {-\frac {f (c+d x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (b e-a f-b x^4\right ) \sqrt {\frac {d e}{d e-c f}-\frac {c f}{d e-c f}-\frac {d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{\sqrt {c+d x}} \\ & = -\frac {\left (2 \sqrt {-\frac {f (c+d x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {b e-a f}}\right ) \sqrt {\frac {d e}{d e-c f}-\frac {c f}{d e-c f}-\frac {d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{(b e-a f) \sqrt {c+d x}}-\frac {\left (2 \sqrt {-\frac {f (c+d x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {b e-a f}}\right ) \sqrt {\frac {d e}{d e-c f}-\frac {c f}{d e-c f}-\frac {d x^4}{d e-c f}}} \, dx,x,\sqrt [4]{e+f x}\right )}{(b e-a f) \sqrt {c+d x}} \\ & = -\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt [4]{d} (b e-a f) \sqrt {c+d x}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \Pi \left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt [4]{d} (b e-a f) \sqrt {c+d x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=-\frac {4 \sqrt {\frac {b (c+d x)}{d (a+b x)}} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {3}{4},\frac {9}{4},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{5 b \sqrt {c+d x} (e+f x)^{3/4}} \]
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\[\int \frac {1}{\left (b x +a \right ) \left (f x +e \right )^{\frac {3}{4}} \sqrt {d x +c}}d x\]
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Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt {c + d x} \left (e + f x\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} {\left (f x + e\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} {\left (f x + e\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} (e+f x)^{3/4}} \, dx=\int \frac {1}{{\left (e+f\,x\right )}^{3/4}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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