Optimal. Leaf size=177 \[ \frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}} \]
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Rubi [A]
time = 0.26, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4823, 4767,
4715, 267} \begin {gather*} -\frac {\left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {c d x+d} \sqrt {e-c e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 4767
Rule 4823
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 150, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (2 a b c x \sqrt {1-c^2 x^2}+a^2 \left (-1+c^2 x^2\right )-2 b^2 \left (-1+c^2 x^2\right )+2 b \left (b c x \sqrt {1-c^2 x^2}+a \left (-1+c^2 x^2\right )\right ) \text {ArcSin}(c x)+b^2 \left (-1+c^2 x^2\right ) \text {ArcSin}(c x)^2\right )}{c^2 d e (-1+c x) (1+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \sqrt {-c e x +e}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 157, normalized size = 0.89 \begin {gather*} 2 \, {\left (\frac {x \arcsin \left (c x\right ) e^{\left (-\frac {1}{2}\right )}}{c \sqrt {d}} + \frac {\sqrt {-c^{2} x^{2} + 1} e^{\left (-\frac {1}{2}\right )}}{c^{2} \sqrt {d}}\right )} b^{2} + \frac {2 \, a b x e^{\left (-\frac {1}{2}\right )}}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} e + d e} b^{2} \arcsin \left (c x\right )^{2} e^{\left (-1\right )}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} e + d e} a b \arcsin \left (c x\right ) e^{\left (-1\right )}}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} e + d e} a^{2} e^{\left (-1\right )}}{c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.31, size = 153, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {c d x + d} {\left (2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {-{\left (c x - 1\right )} e} + {\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-{\left (c x - 1\right )} e}\right )} e^{\left (-1\right )}}{c^{4} d x^{2} - c^{2} d} \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 {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )}}\, 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 = -0.01 \begin {gather*} \int \frac {x\,{\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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