3.1138 \(\int (a+a x)^{3/2} (c-c x)^{3/2} \, dx\)

Optimal. Leaf size=96 \[ \frac{3}{4} a^{3/2} c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )+\frac{3}{8} a c x \sqrt{a x+a} \sqrt{c-c x}+\frac{1}{4} x (a x+a)^{3/2} (c-c x)^{3/2} \]

[Out]

(3*a*c*x*Sqrt[a + a*x]*Sqrt[c - c*x])/8 + (x*(a + a*x)^(3/2)*(c - c*x)^(3/2))/4
+ (3*a^(3/2)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])])/4

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Rubi [A]  time = 0.0880958, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{3}{4} a^{3/2} c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )+\frac{3}{8} a c x \sqrt{a x+a} \sqrt{c-c x}+\frac{1}{4} x (a x+a)^{3/2} (c-c x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + a*x)^(3/2)*(c - c*x)^(3/2),x]

[Out]

(3*a*c*x*Sqrt[a + a*x]*Sqrt[c - c*x])/8 + (x*(a + a*x)^(3/2)*(c - c*x)^(3/2))/4
+ (3*a^(3/2)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])])/4

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Rubi in Sympy [A]  time = 13.8433, size = 87, normalized size = 0.91 \[ \frac{3 a^{\frac{3}{2}} c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{a x + a}}{\sqrt{a} \sqrt{- c x + c}} \right )}}{4} + \frac{3 a c x \sqrt{a x + a} \sqrt{- c x + c}}{8} + \frac{x \left (a x + a\right )^{\frac{3}{2}} \left (- c x + c\right )^{\frac{3}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+a)**(3/2)*(-c*x+c)**(3/2),x)

[Out]

3*a**(3/2)*c**(3/2)*atan(sqrt(c)*sqrt(a*x + a)/(sqrt(a)*sqrt(-c*x + c)))/4 + 3*a
*c*x*sqrt(a*x + a)*sqrt(-c*x + c)/8 + x*(a*x + a)**(3/2)*(-c*x + c)**(3/2)/4

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Mathematica [A]  time = 0.111019, size = 84, normalized size = 0.88 \[ -\frac{c (a (x+1))^{3/2} \left (x \sqrt{x+1} \left (2 x^2-5\right ) \sqrt{c-c x}+6 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{x+1} \sqrt{c-c x}}{\sqrt{c} (x-1)}\right )\right )}{8 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + a*x)^(3/2)*(c - c*x)^(3/2),x]

[Out]

-(c*(a*(1 + x))^(3/2)*(x*Sqrt[1 + x]*Sqrt[c - c*x]*(-5 + 2*x^2) + 6*Sqrt[c]*ArcT
an[(Sqrt[1 + x]*Sqrt[c - c*x])/(Sqrt[c]*(-1 + x))]))/(8*(1 + x)^(3/2))

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Maple [B]  time = 0.009, size = 143, normalized size = 1.5 \[ -{\frac{1}{4\,c} \left ( ax+a \right ) ^{{\frac{3}{2}}} \left ( -cx+c \right ) ^{{\frac{5}{2}}}}-{\frac{a}{4\,c}\sqrt{ax+a} \left ( -cx+c \right ) ^{{\frac{5}{2}}}}+{\frac{a}{8}\sqrt{ax+a} \left ( -cx+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ac}{8}\sqrt{ax+a}\sqrt{-cx+c}}+{\frac{3\,{a}^{2}{c}^{2}}{8}\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+a)^(3/2)*(-c*x+c)^(3/2),x)

[Out]

-1/4/c*(a*x+a)^(3/2)*(-c*x+c)^(5/2)-1/4*a/c*(a*x+a)^(1/2)*(-c*x+c)^(5/2)+1/8*(a*
x+a)^(1/2)*(-c*x+c)^(3/2)*a+3/8*a*c*(-c*x+c)^(1/2)*(a*x+a)^(1/2)+3/8*a^2*c^2*((-
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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + a)^(3/2)*(-c*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219897, size = 1, normalized size = 0.01 \[ \left [\frac{3}{16} \, \sqrt{-a c} a c \log \left (2 \, a c x^{2} + 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right ) - \frac{1}{8} \,{\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt{a x + a} \sqrt{-c x + c}, \frac{3}{8} \, \sqrt{a c} a c \arctan \left (\frac{a c x}{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c}}\right ) - \frac{1}{8} \,{\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt{a x + a} \sqrt{-c x + c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + a)^(3/2)*(-c*x + c)^(3/2),x, algorithm="fricas")

[Out]

[3/16*sqrt(-a*c)*a*c*log(2*a*c*x^2 + 2*sqrt(-a*c)*sqrt(a*x + a)*sqrt(-c*x + c)*x
- a*c) - 1/8*(2*a*c*x^3 - 5*a*c*x)*sqrt(a*x + a)*sqrt(-c*x + c), 3/8*sqrt(a*c)*
a*c*arctan(a*c*x/(sqrt(a*c)*sqrt(a*x + a)*sqrt(-c*x + c))) - 1/8*(2*a*c*x^3 - 5*
a*c*x)*sqrt(a*x + a)*sqrt(-c*x + c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a \left (x + 1\right )\right )^{\frac{3}{2}} \left (- c \left (x - 1\right )\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+a)**(3/2)*(-c*x+c)**(3/2),x)

[Out]

Integral((a*(x + 1))**(3/2)*(-c*(x - 1))**(3/2), x)

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GIAC/XCAS [A]  time = 0.296639, size = 277, normalized size = 2.89 \[ -\frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a} a x\right )} c{\left | a \right |}}{2 \, a^{2}} + \frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (a x + a\right )}{\left (2 \,{\left (a x + a\right )}{\left (\frac{a x + a}{a^{2}} - \frac{3}{a}\right )} + 5\right )} - a\right )} \sqrt{a x + a}\right )} c{\left | a \right |}}{8 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + a)^(3/2)*(-c*x + c)^(3/2),x, algorithm="giac")

[Out]

-1/2*(2*a^3*c*ln(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)*c*abs(a)/a^2 +
1/8*(2*a^3*c*ln(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)*((a*x + a)*(2*(a*x + a)*((a*x + a)/
a^2 - 3/a) + 5) - a)*sqrt(a*x + a))*c*abs(a)/a^2