Optimal. Leaf size=432 \[ -2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2-\frac {2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.48, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4823, 4783,
4803, 4268, 2611, 2320, 6724, 4715, 267} \begin {gather*} \frac {2 i b \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {c d x+d} \sqrt {e-c e x} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}+\sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2-\frac {2 a b c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_3\left (-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_3\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \text {ArcSin}(c x) \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-2 b^2 \sqrt {c d x+d} \sqrt {e-c e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2320
Rule 2611
Rule 4268
Rule 4715
Rule 4783
Rule 4803
Rule 4823
Rule 6724
Rubi steps
\begin {align*} \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=-2 b^2 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.27, size = 434, normalized size = 1.00 \begin {gather*} a^2 \sqrt {d+c d x} \sqrt {e-c e x}+a^2 \sqrt {d} \sqrt {e} \log (c x)-a^2 \sqrt {d} \sqrt {e} \log \left (d e+\sqrt {d} \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x}\right )-\frac {2 a b \sqrt {d+c d x} \sqrt {e-c e x} \left (c x-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)-\text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )+\text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )-i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )+i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (2 \sqrt {1-c^2 x^2}+2 c x \text {ArcSin}(c x)-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2-\text {ArcSin}(c x)^2 \log \left (1-e^{i \text {ArcSin}(c x)}\right )+\text {ArcSin}(c x)^2 \log \left (1+e^{i \text {ArcSin}(c x)}\right )-2 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )+2 i \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )+2 \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )-2 \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c d x +d}\, \sqrt {-c e x +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}{x}\, dx\]
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(-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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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 {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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