Optimal. Leaf size=505 \[ \frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.60, antiderivative size = 505, normalized size of antiderivative = 1.00, number of
steps used = 15, number of rules used = 14, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used
= {4823, 4785, 4741, 4737, 4723, 327, 222, 4773, 4721, 3798, 2221, 2317, 2438, 201}
\begin {gather*} -\frac {c d e \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{2 b \sqrt {1-c^2 x^2}}-\frac {i c d e \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{\sqrt {1-c^2 x^2}}+b c d e \sqrt {1-c^2 x^2} \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))-\frac {d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{x}+\frac {2 b c d e \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 {3}{2} c^2 d e x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {3 b c^3 d e x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{2 \sqrt {1-c^2 x^2}}-\frac {i b^2 c d e \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}}-\frac {5 b^2 c d e \text {ArcSin}(c x) \sqrt {c d x+d} \sqrt {e-c e x}}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} b^2 c^2 d e x \sqrt {c d x+d} \sqrt {e-c e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 4723
Rule 4737
Rule 4741
Rule 4773
Rule 4785
Rule 4823
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (3 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d e \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 (3 c^2 d e \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}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {1}{2} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {3 b c^3 d e x^2 \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}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {\left (2 b c d e \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 {\left (b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^4 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{2 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \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}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \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 {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {\left (4 i b c d e \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 {\left (3 b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \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}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \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 {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \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 d e \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 {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \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}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \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 {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \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 d e \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 {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \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}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d e \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 {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \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 d e \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 = 1.39, size = 538, normalized size = 1.07 \begin {gather*} \frac {-8 a^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-4 a^2 c^2 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-4 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3+12 a^2 c d^{3/2} e^{3/2} 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 )-2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x} \cos (2 \text {ArcSin}(c x))+16 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x} \log (c x)-8 i b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )+b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \sin (2 \text {ArcSin}(c x))-2 b d e \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) \left (8 a \sqrt {1-c^2 x^2}+b c x \cos (2 \text {ArcSin}(c x))-8 b c x \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+2 a c x \sin (2 \text {ArcSin}(c x))\right )-2 b d e \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 \left (6 a c x+4 i b c x+4 b \sqrt {1-c^2 x^2}+b c x \sin (2 \text {ArcSin}(c x))\right )}{8 x \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.02, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{x^{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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