Optimal. Leaf size=474 \[ -\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {4 b c \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {8 b c d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Rubi [A]
time = 1.20, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {45, 6445, 12,
6853, 6874, 733, 435, 958, 430, 946, 174, 552, 551} \begin {gather*} -\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {8 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {8 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}} \end {gather*}
Antiderivative was successfully verified.
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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 6445
Rule 6853
Rule 6874
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx &=-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-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 {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {\left (4 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-\sqrt {-c^2} x} \sqrt {1+\sqrt {-c^2} x} \sqrt {d+e x}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (8 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {8 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 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}{\sqrt {-c^2}}-\frac {e x^2}{\sqrt {-c^2}}}} \, dx,x,\sqrt {1-\sqrt {-c^2} 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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {8 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 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} \sqrt {1+\frac {e \left (-1+\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{\sqrt {-c^2} \left (d+\frac {e}{\sqrt {-c^2}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c^2} 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 \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {8 b \sqrt {-c^2} d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {1+c^2 x^2} \sqrt {1-\frac {e \left (1-\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.83, size = 343, normalized size = 0.72 \begin {gather*} \frac {2 \left (a (-2 d+e x) \sqrt {d+e x}+b (-2 d+e x) \sqrt {d+e x} \text {csch}^{-1}(c x)+\frac {2 b \sqrt {-\frac {e (-i+c x)}{c d+i e}} \sqrt {-\frac {e (i+c x)}{c d-i e}} \left ((i c d-e) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+(i c d+e) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )-2 i c d \Pi \left (1-\frac {i e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )\right )}{c^2 \sqrt {-\frac {c}{c d-i e}} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{3 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.82, size = 868, normalized size = 1.83
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}} \mathrm {arccsch}\left (c x \right )}{3}+\mathrm {arccsch}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (2 i \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -2 i \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+2 \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \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 {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{2}}\) | \(868\) |
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}} \mathrm {arccsch}\left (c x \right )}{3}+\mathrm {arccsch}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (2 i \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -2 i \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+2 \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \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 {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{2}}\) | \(868\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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