Optimal. Leaf size=230 \[ -\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}} \]
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Rubi [A]
time = 0.32, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4763, 4847,
4737, 4767, 4715, 267} \begin {gather*} \frac {e \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{\sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \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 4737
Rule 4763
Rule 4767
Rule 4847
Rubi steps
\begin {align*} \int \frac {\sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d+c d x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(e-c e 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 {\sqrt {1-c^2 x^2} \int \left (\frac {e \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {c e x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\left (e \sqrt {1-c^2 x^2}\right ) \int \frac {\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 (c e \sqrt {1-c^2 x^2}\right ) \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 {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b e \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 e \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {2 a b e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 c e \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 e x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 e x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 296, normalized size = 1.29 \begin {gather*} \frac {3 \sqrt {d+c d x} \sqrt {e-c e x} \left (-2 a b c x+a^2 \sqrt {1-c^2 x^2}-2 b^2 \sqrt {1-c^2 x^2}\right )-6 b \sqrt {d+c d x} \sqrt {e-c e x} \left (b c x-a \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)+3 b \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2+b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-3 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 )}{3 c d \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c e x +e}}{\sqrt {c d x +d}}\, dx\]
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 {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {e-c\,e\,x}}{\sqrt {d+c\,d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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