Optimal. Leaf size=141 \[ \frac{d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{b d x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{f-c f x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.257583, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4673, 4763, 4641, 4677, 8} \[ \frac{d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{b d x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{f-c f x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4673
Rule 4763
Rule 4641
Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{d+c d x} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{f-c f x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(d+c d x) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{d \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{c d x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\right ) \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=\frac{\left (d \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}+\frac{\left (c d \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{\left (b d \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{\sqrt{d+c d x} \sqrt{f-c f x}}\\ &=\frac{b d x \sqrt{1-c^2 x^2}}{\sqrt{d+c d x} \sqrt{f-c f x}}-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d+c d x} \sqrt{f-c f x}}+\frac{d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d+c d x} \sqrt{f-c f x}}\\ \end{align*}
Mathematica [A] time = 0.705165, size = 200, normalized size = 1.42 \[ \frac{\frac{2 \sqrt{c d x+d} \sqrt{f-c f x} \left (b c x-a \sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}}-2 a \sqrt{d} \sqrt{f} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left (c^2 x^2-1\right )}\right )+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}-2 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)}{2 c f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.243, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arcsin \left ( cx \right ) )\sqrt{cdx+d}{\frac{1}{\sqrt{-cfx+f}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{c d x + d} \sqrt{-c f x + f}{\left (b \arcsin \left (c x\right ) + a\right )}}{c f x - f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{\sqrt{- f \left (c x - 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d x + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{\sqrt{-c f x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]