Optimal. Leaf size=404 \[ -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {8 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (3 c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
[Out]
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Rubi [A]
time = 1.68, antiderivative size = 502, normalized size of antiderivative = 1.24, number
of steps used = 24, number of rules used = 15, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.790, Rules
used = {45, 5355, 12, 6853, 6874, 757, 858, 733, 435, 430, 972, 946, 174, 552, 551}
\begin {gather*} -\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {8 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c e^2 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}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {8 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e 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} (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 e 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 x \sqrt {1-\frac {1}{c^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 757
Rule 858
Rule 946
Rule 972
Rule 5355
Rule 6853
Rule 6874
Rubi steps
\begin {align*} \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx &=-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {b \int \frac {2 (d+e x)^{3/2} (-2 d+3 e x)}{15 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {(2 b) \int \frac {(d+e x)^{3/2} (-2 d+3 e x)}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{15 c e^2}\\ &=-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {(d+e x)^{3/2} (-2 d+3 e x)}{x \sqrt {1-c^2 x^2}} \, dx}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (\frac {3 e (d+e x)^{3/2}}{\sqrt {1-c^2 x^2}}-\frac {2 d (d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}\right ) \, dx}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}} \, dx}{15 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 {(d+e x)^{3/2}}{\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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \left (\frac {2 d e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {d^2}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e^2 x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {\frac {1}{2} \left (-3 c^2 d^2-e^2\right )-2 c^2 d e x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^3 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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\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}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 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}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b (c d-e) (c d+e) \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 \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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {\left (4 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}{15 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 {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{15 c e \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}{15 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 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 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (16 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 )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (4 b (c d-e) (c d+e) \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 \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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {16 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {16 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 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 )}{15 c 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 )}{15 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 {\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 )}{15 c^2 e \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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {8 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 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^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 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 )}{15 c e^2 \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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {8 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 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^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^4 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 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 = 10.95, size = 368, normalized size = 0.91 \begin {gather*} \frac {1}{15} \left (\frac {4 b \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{c}+\frac {2 a \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}+\frac {2 b \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right ) \csc ^{-1}(c x)}{e^2}-\frac {4 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (-2 c d (c d-e) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+\left (-c^2 d^2-2 c d e+e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+2 c^2 d^2 \Pi \left (1+\frac {e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )\right )}{c^3 e^2 \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs.
\(2(363)=726\).
time = 0.66, size = 827, normalized size = 2.05
method | result | size |
derivativedivides | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\left (e x +d \right )^{\frac {3}{2}} d}{3}\right )-2 b \left (-\frac {\mathrm {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\mathrm {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\frac {-\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {2 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}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{15}+\frac {4 \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}}\, \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 {4 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}}\, \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}}{15}+\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\frac {4 \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}}\, \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 {4 \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}}\, \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}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{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}}\, \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^{2}}\) | \(827\) |
default | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\left (e x +d \right )^{\frac {3}{2}} d}{3}\right )-2 b \left (-\frac {\mathrm {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\mathrm {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\frac {-\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {2 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}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{15}+\frac {4 \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}}\, \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 {4 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}}\, \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}}{15}+\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\frac {4 \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}}\, \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 {4 \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}}\, \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}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{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}}\, \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^{2}}\) | \(827\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 x \left (a + b \operatorname {acsc}{\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 x\,\left (a+b\,\mathrm {asin}\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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