3.1139 \(\int \sqrt{a+a x} \sqrt{c-c x} \, dx\)

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]

(x*Sqrt[a + a*x]*Sqrt[c - c*x])/2 + Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])]

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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]

Int[Sqrt[a + a*x]*Sqrt[c - c*x],x]

[Out]

(x*Sqrt[a + a*x]*Sqrt[c - c*x])/2 + Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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]

Integrate[Sqrt[a + a*x]*Sqrt[c - c*x],x]

[Out]

(Sqrt[a*(1 + x)]*(x*Sqrt[1 + x]*Sqrt[c - c*x] - 2*Sqrt[c]*ArcSin[Sqrt[c - c*x]/(Sqrt[2]*Sqrt[c])]))/(2*Sqrt[1
+ x])

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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]

int((a*x+a)^(1/2)*(-c*x+c)^(1/2),x)

[Out]

-1/2/c*(a*x+a)^(1/2)*(-c*x+c)^(3/2)+1/2*(a*x+a)^(1/2)*(-c*x+c)^(1/2)+1/2*a*c*((-c*x+c)*(a*x+a))^(1/2)/(-c*x+c)
^(1/2)/(a*x+a)^(1/2)/(a*c)^(1/2)*arctan((a*c)^(1/2)*x/(-a*c*x^2+a*c)^(1/2))

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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]

integrate((a*x+a)^(1/2)*(-c*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

integrate((a*x+a)^(1/2)*(-c*x+c)^(1/2),x, algorithm="fricas")

[Out]

[1/2*sqrt(a*x + a)*sqrt(-c*x + c)*x + 1/4*sqrt(-a*c)*log(2*a*c*x^2 + 2*sqrt(-a*c)*sqrt(a*x + a)*sqrt(-c*x + c)
*x - a*c), 1/2*sqrt(a*x + a)*sqrt(-c*x + c)*x - 1/2*sqrt(a*c)*arctan(sqrt(a*c)*sqrt(a*x + a)*sqrt(-c*x + c)*x/
(a*c*x^2 - a*c))]

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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]

integrate((a*x+a)**(1/2)*(-c*x+c)**(1/2),x)

[Out]

Integral(sqrt(a*(x + 1))*sqrt(-c*(x - 1)), x)

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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]

integrate((a*x+a)^(1/2)*(-c*x+c)^(1/2),x, algorithm="giac")

[Out]

-1/2*(2*a^3*c*log(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/sqrt(-a*c) - sqrt(-(a*x + a
)*a*c + 2*a^2*c)*sqrt(a*x + a)*a*x)*abs(a)/a^3