Optimal. Leaf size=67 \[ \frac{1}{2} x \sqrt{a x+a} \sqrt{c-c x}+\sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0292047, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {38, 63, 217, 203} \[ \frac{1}{2} x \sqrt{a x+a} \sqrt{c-c x}+\sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 38
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \sqrt{a+a x} \sqrt{c-c x} \, dx &=\frac{1}{2} x \sqrt{a+a x} \sqrt{c-c x}+\frac{1}{2} (a c) \int \frac{1}{\sqrt{a+a x} \sqrt{c-c x}} \, dx\\ &=\frac{1}{2} x \sqrt{a+a x} \sqrt{c-c x}+c \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c-\frac{c x^2}{a}}} \, dx,x,\sqrt{a+a x}\right )\\ &=\frac{1}{2} x \sqrt{a+a x} \sqrt{c-c x}+c \operatorname{Subst}\left (\int \frac{1}{1+\frac{c x^2}{a}} \, dx,x,\frac{\sqrt{a+a x}}{\sqrt{c-c x}}\right )\\ &=\frac{1}{2} x \sqrt{a+a x} \sqrt{c-c x}+\sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+a x}}{\sqrt{a} \sqrt{c-c x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0594009, size = 69, normalized size = 1.03 \[ \frac{\sqrt{a (x+1)} \left (x \sqrt{x+1} \sqrt{c-c x}-2 \sqrt{c} \sin ^{-1}\left (\frac{\sqrt{c-c x}}{\sqrt{2} \sqrt{c}}\right )\right )}{2 \sqrt{x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 98, normalized size = 1.5 \begin{align*} -{\frac{1}{2\,c}\sqrt{ax+a} \left ( -cx+c \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{ax+a}\sqrt{-cx+c}}+{\frac{ac}{2}\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \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 [A] time = 1.61877, size = 325, normalized size = 4.85 \begin{align*} \left [\frac{1}{2} \, \sqrt{a x + a} \sqrt{-c x + c} x + \frac{1}{4} \, \sqrt{-a c} \log \left (2 \, a c x^{2} + 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right ), \frac{1}{2} \, \sqrt{a x + a} \sqrt{-c x + c} x - \frac{1}{2} \, \sqrt{a c} \arctan \left (\frac{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c} x}{a c x^{2} - a c}\right )\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 \sqrt{a \left (x + 1\right )} \sqrt{- c \left (x - 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13747, size = 115, normalized size = 1.72 \begin{align*} -\frac{{\left (\frac{2 \, a^{3} c \log \left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a} a x\right )}{\left | a \right |}}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]