Optimal. Leaf size=238 \[ \frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Rubi [A]
time = 1.20, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {45, 5355, 12,
6853, 6874, 733, 430, 946, 174, 552, 551} \begin {gather*} \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}} \Pi \left (2;\text {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}} F\left (\text {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}} \end {gather*}
Antiderivative was successfully verified.
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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*} \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 {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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} F\left (\sin ^{-1}\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} \Pi \left (2;\sin ^{-1}\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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 11.22, size = 226, normalized size = 0.95 \begin {gather*} \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 (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 \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 \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 282, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \mathrm {arccsc}\left (c x \right )-\frac {\mathrm {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(282\) |
default | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \mathrm {arccsc}\left (c x \right )-\frac {\mathrm {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(282\) |
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]
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 {acsc}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, 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 {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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