Optimal. Leaf size=351 \[ \frac {b^2 x \sqrt {d+c d x} \sqrt {e-c e x}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.50, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4823, 4783,
4795, 4737, 4723, 327, 222} \begin {gather*} \frac {b x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2}{8 c^2}+\frac {\sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{24 b c^3 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2-\frac {b^2 \text {ArcSin}(c x) \sqrt {c d x+d} \sqrt {e-c e x}}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b^2 x \sqrt {c d x+d} \sqrt {e-c e x}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4783
Rule 4795
Rule 4823
Rubi steps
\begin {align*} \int x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} x^3 \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 {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{4} x^3 \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}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{32} b^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 x \sqrt {d+c d x} \sqrt {e-c e x}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 x \sqrt {d+c d x} \sqrt {e-c e x}}{64 c^2}-\frac {1}{32} b^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}-\frac {b c x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 297, normalized size = 0.85 \begin {gather*} \frac {32 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-96 a^2 \sqrt {d} \sqrt {e} \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 )-12 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) (b \cos (4 \text {ArcSin}(c x))+4 a \sin (4 \text {ArcSin}(c x)))-24 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 (-4 a+b \sin (4 \text {ArcSin}(c x)))+3 \sqrt {d+c d x} \sqrt {e-c e x} \left (32 a^2 c x \sqrt {1-c^2 x^2} \left (-1+2 c^2 x^2\right )-4 a b \cos (4 \text {ArcSin}(c x))+b^2 \sin (4 \text {ArcSin}(c x))\right )}{768 c^3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.72, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {c d x +d}\, \sqrt {-c e x +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]
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 x^{2} \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}\, 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 x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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