Optimal. Leaf size=134 \[ -\frac {b c x^2 \sqrt {d+c d x} \sqrt {f-c f x}}{4 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {f-c f x} (a+b \text {ArcSin}(c x))+\frac {\sqrt {d+c d x} \sqrt {f-c f x} (a+b \text {ArcSin}(c x))^2}{4 b c \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4763, 4741,
4737, 30} \begin {gather*} \frac {\sqrt {c d x+d} \sqrt {f-c f x} (a+b \text {ArcSin}(c x))^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {c d x+d} \sqrt {f-c f x} (a+b \text {ArcSin}(c x))-\frac {b c x^2 \sqrt {c d x+d} \sqrt {f-c f x}}{4 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4737
Rule 4741
Rule 4763
Rubi steps
\begin {align*} \int \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} x \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (\sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c x^2 \sqrt {d+c d x} \sqrt {f-c f x}}{4 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 207, normalized size = 1.54 \begin {gather*} \frac {1}{2} a x \sqrt {-f (-1+c x)} \sqrt {d (1+c x)}-\frac {a \sqrt {d} \sqrt {f} \text {ArcTan}\left (\frac {c x \sqrt {-f (-1+c x)} \sqrt {d (1+c x)}}{\sqrt {d} \sqrt {f} (-1+c x) (1+c x)}\right )}{2 c}+\frac {b \sqrt {d+c d x} \sqrt {f-c f x} \sqrt {-d f \left (1-c^2 x^2\right )} (\cos (2 \text {ArcSin}(c x))+2 \text {ArcSin}(c x) (\text {ArcSin}(c x)+\sin (2 \text {ArcSin}(c x))))}{8 c \sqrt {(-d-c d x) (f-c f x)} \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \sqrt {c d x +d}\, \left (a +b \arcsin \left (c x \right )\right ) \sqrt {-c f x +f}\, dx\]
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 \sqrt {d \left (c x + 1\right )} \sqrt {- f \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\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 \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d+c\,d\,x}\,\sqrt {f-c\,f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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