Integrand size = 21, antiderivative size = 551 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=-\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {32 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {8 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
[Out]
Time = 1.81 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {45, 5355, 12, 6853, 6874, 733, 430, 946, 174, 552, 551, 858, 435, 945, 1598} \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}-\frac {64 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {32 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \left (1-c^2 x^2\right ) \sqrt {d+e x}}{15 c^3 e^2 x \sqrt {1-\frac {1}{c^2 x^2}}} \]
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Rule 12
Rule 45
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 858
Rule 945
Rule 946
Rule 1598
Rule 5355
Rule 6853
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {b \int \frac {2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )}{5 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c} \\ & = \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {(2 b) \int \frac {16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{5 c e^4} \\ & = \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (\frac {8 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {16 d^3}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}-\frac {2 d e^2 x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e^3 x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 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 {x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{5 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {e x-2 c^2 d x^2}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (32 b d^2 \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 )}{5 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}-\frac {32 b d^2 \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 )}{5 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (64 b d^3 \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 )}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {e-2 c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b d \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 )}{5 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (8 b d^2 \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 )}{5 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {8 b d \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 )}{5 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {8 b d^2 \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^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (2 c^2 d^2+e^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (64 b d^3 \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 )}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {8 b d \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 )}{5 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {8 b d^2 \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^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {64 b d^3 \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 )}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d \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 )}{15 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (4 b \left (2 c^2 d^2+e^2\right ) \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 )}{15 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {32 b d \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 )}{15 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {8 b d^2 \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^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \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 )}{15 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {64 b d^3 \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 )}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 34.38 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.48 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {a d^4 \left (1+\frac {e x}{d}\right )^{3/2} B_{-\frac {e x}{d}}\left (4,-\frac {1}{2}\right )}{e^4 (d+e x)^{3/2}}+\frac {b \left (-\frac {c^2 \left (e+\frac {d}{x}\right )^2 x^2 \left (\frac {32 c d \sqrt {1-\frac {1}{c^2 x^2}}}{15 e^3}-\frac {32 c^2 d^2 \csc ^{-1}(c x)}{5 e^4}+\frac {2 c^2 d^2 \csc ^{-1}(c x)}{e^3 \left (e+\frac {d}{x}\right )}-\frac {2 c^2 x^2 \csc ^{-1}(c x)}{5 e^2}-\frac {2 c x \left (2 e \sqrt {1-\frac {1}{c^2 x^2}}-9 c d \csc ^{-1}(c x)\right )}{15 e^3}\right )}{(d+e x)^{3/2}}-\frac {2 \left (e+\frac {d}{x}\right )^{3/2} (c x)^{3/2} \left (\frac {2 \left (32 c^2 d^2 e+e^3\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (48 c^3 d^3+8 c d e^2\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {16 c d e \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c 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 )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c 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 )\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{15 e^4 (d+e x)^{3/2}}\right )}{c^4} \]
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Time = 10.08 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\left (e x +d \right )^{\frac {3}{2}} d -3 d^{2} \sqrt {e x +d}-\frac {d^{3}}{\sqrt {e x +d}}\right )-2 b \left (-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d -3 \,\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{3}}{\sqrt {e x +d}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+24 d^{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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}+8 \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}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-48 d^{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}}\, \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^{2}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-8 \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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +8 \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}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+\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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \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^{4}}\) | \(880\) |
default | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\left (e x +d \right )^{\frac {3}{2}} d -3 d^{2} \sqrt {e x +d}-\frac {d^{3}}{\sqrt {e x +d}}\right )-2 b \left (-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d -3 \,\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{3}}{\sqrt {e x +d}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+24 d^{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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}+8 \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}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-48 d^{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}}\, \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^{2}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-8 \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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +8 \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}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+\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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \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^{4}}\) | \(880\) |
parts | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\left (e x +d \right )^{\frac {3}{2}} d +3 d^{2} \sqrt {e x +d}+\frac {d^{3}}{\sqrt {e x +d}}\right )}{e^{4}}+\frac {2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d +3 \,\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\operatorname {arccsc}\left (c x \right ) d^{3}}{\sqrt {e x +d}}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}+\frac {16 d^{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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{5}+\frac {16 \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}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}}{15}-\frac {32 d^{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}}\, \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^{2}}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}-\frac {16 \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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}+\frac {16 \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}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}+\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}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \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^{4}}\) | \(893\) |
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\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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