Optimal. Leaf size=257 \[ -\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{x}-\frac {i c \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c \sqrt {d+c d x} \sqrt {e-c e x} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.42, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4823, 4781,
4721, 3798, 2221, 2317, 2438, 4737} \begin {gather*} -\frac {c \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{3 b \sqrt {1-c^2 x^2}}-\frac {i c \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {c d x+d} \sqrt {e-c e x} \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{x}-\frac {i b^2 c \sqrt {c d x+d} \sqrt {e-c e x} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4737
Rule 4781
Rule 4823
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^2} \, 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^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (c^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}+\frac {\left (2 b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {i c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}-\frac {\left (4 i b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {i c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \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 ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {i c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (i b^2 c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {i c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c \sqrt {d+c d x} \sqrt {e-c e x} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 374, normalized size = 1.46 \begin {gather*} \frac {-3 a^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-3 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (-i a c x+b c x-i b \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2-b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3+3 a^2 c \sqrt {d} \sqrt {e} x \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) \left (-a \sqrt {1-c^2 x^2}+b c x \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )+6 a b c x \sqrt {d+c d x} \sqrt {e-c e x} \log (c x)-3 i b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 x \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c d x +d}\, \sqrt {-c e x +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}{x^{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{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 {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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