Integrand size = 19, antiderivative size = 238 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Time = 1.15 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {45, 5355, 12, 6853, 6874, 733, 430, 946, 174, 552, 551} \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {8 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]
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Rule 12
Rule 45
Rule 174
Rule 430
Rule 551
Rule 552
Rule 733
Rule 946
Rule 5355
Rule 6853
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {b \int \frac {2 (2 d+e x)}{e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {(2 b) \int \frac {2 d+e x}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c e^2} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {2 d+e x}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {2 d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 12.57 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \left (\frac {a (2 d+e x)}{\sqrt {d+e x}}+\frac {b (2 d+e x) \csc ^{-1}(c x)}{\sqrt {d+e x}}-\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-2 \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{e^2} \]
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Time = 6.61 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.18
method | result | size |
parts | \(\frac {2 a \left (\sqrt {e x +d}+\frac {d}{\sqrt {e x +d}}\right )}{e^{2}}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )+\frac {\operatorname {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(280\) |
derivativedivides | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )-\frac {\operatorname {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(282\) |
default | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )-\frac {\operatorname {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(282\) |
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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