Integrand size = 19, antiderivative size = 344 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Time = 1.20 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {45, 5355, 12, 6853, 6874, 733, 435, 958, 430, 946, 174, 552, 551} \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {8 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}} \]
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Rule 12
Rule 45
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 946
Rule 958
Rule 5355
Rule 6853
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {b \int \frac {2 (-2 d+e x) \sqrt {d+e x}}{3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {(2 b) \int \frac {(-2 d+e x) \sqrt {d+e x}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e^2} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {(-2 d+e x) \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 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 \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}} \, dx}{3 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 {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d^2 \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 )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d^2 \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 )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {2 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b d \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 )}{3 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \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 )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.11 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {2 \left (a (-2 d+e x) (d+e x)+b (-2 d+e x) (d+e x) \csc ^{-1}(c x)-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d-e}} \sqrt {\frac {e-c e x}{c d+e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{c (-1+c x) \sqrt {\frac {e (1+c x)}{-c d+e}}}-\frac {4 b c d^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-c^2 x^2}}\right )}{3 e^2 \sqrt {d+e x}} \]
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Time = 8.05 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+d \sqrt {e x +d}\right )-2 b \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -2 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\) | \(410\) |
default | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+d \sqrt {e x +d}\right )-2 b \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -2 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\) | \(410\) |
parts | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-d \sqrt {e x +d}\right )}{e^{2}}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}-\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}-\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -2 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\) | \(415\) |
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]
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Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \]
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