Optimal. Leaf size=295 \[ \frac {x (a+b \text {ArcSin}(c x))^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}} \]
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Rubi [A]
time = 0.53, antiderivative size = 295, 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, 4791,
4737, 4765, 3800, 2221, 2317, 2438} \begin {gather*} \frac {x (a+b \text {ArcSin}(c x))^2}{c^2 d e \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c^3 d e \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{c^3 d e \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d e \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d e \sqrt {c d x+d} \sqrt {e-c e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4737
Rule 4765
Rule 4791
Rule 4823
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (4 i b \sqrt {1-c^2 x^2}\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 )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(636\) vs. \(2(295)=590\).
time = 1.51, size = 636, normalized size = 2.16 \begin {gather*} \frac {3 a^2 c \sqrt {d} e x+3 a^2 \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x} \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 )+3 a b \sqrt {d} e \left (2 c x \text {ArcSin}(c x)+\sqrt {1-c^2 x^2} \left (-\text {ArcSin}(c x)^2+2 \left (\log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+\log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )\right )\right )+b^2 \sqrt {d} e \left (6 i \pi \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+3 c x \text {ArcSin}(c x)^2-3 i \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)^3+12 \pi \sqrt {1-c^2 x^2} \log \left (1+e^{-i \text {ArcSin}(c x)}\right )+3 \pi \sqrt {1-c^2 x^2} \log \left (1-i e^{i \text {ArcSin}(c x)}\right )+6 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )-3 \pi \sqrt {1-c^2 x^2} \log \left (1+i e^{i \text {ArcSin}(c x)}\right )+6 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )-12 \pi \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+3 \pi \sqrt {1-c^2 x^2} \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-3 \pi \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )-6 i \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-6 i \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )\right )}{3 c^3 d^{3/2} e^2 \sqrt {d+c d x} \sqrt {e-c e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}}}\, dx\]
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(-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 {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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