3.1137 \(\int (a+a x)^{5/2} (c-c x)^{5/2} \, dx\)

Optimal. Leaf size=126 \[ \frac{5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )+\frac{5}{16} a^2 c^2 x \sqrt{a x+a} \sqrt{c-c x}+\frac{5}{24} a c x (a x+a)^{3/2} (c-c x)^{3/2}+\frac{1}{6} x (a x+a)^{5/2} (c-c x)^{5/2} \]

[Out]

(5*a^2*c^2*x*Sqrt[a + a*x]*Sqrt[c - c*x])/16 + (5*a*c*x*(a + a*x)^(3/2)*(c - c*x
)^(3/2))/24 + (x*(a + a*x)^(5/2)*(c - c*x)^(5/2))/6 + (5*a^(5/2)*c^(5/2)*ArcTan[
(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])])/8

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Rubi [A]  time = 0.127767, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )+\frac{5}{16} a^2 c^2 x \sqrt{a x+a} \sqrt{c-c x}+\frac{5}{24} a c x (a x+a)^{3/2} (c-c x)^{3/2}+\frac{1}{6} x (a x+a)^{5/2} (c-c x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + a*x)^(5/2)*(c - c*x)^(5/2),x]

[Out]

(5*a^2*c^2*x*Sqrt[a + a*x]*Sqrt[c - c*x])/16 + (5*a*c*x*(a + a*x)^(3/2)*(c - c*x
)^(3/2))/24 + (x*(a + a*x)^(5/2)*(c - c*x)^(5/2))/6 + (5*a^(5/2)*c^(5/2)*ArcTan[
(Sqrt[c]*Sqrt[a + a*x])/(Sqrt[a]*Sqrt[c - c*x])])/8

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Rubi in Sympy [A]  time = 19.1528, size = 116, normalized size = 0.92 \[ - \frac{5 a^{\frac{5}{2}} c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{- c x + c}}{\sqrt{c} \sqrt{a x + a}} \right )}}{8} + \frac{5 a^{2} c^{2} x \sqrt{a x + a} \sqrt{- c x + c}}{16} + \frac{5 a c x \left (a x + a\right )^{\frac{3}{2}} \left (- c x + c\right )^{\frac{3}{2}}}{24} + \frac{x \left (a x + a\right )^{\frac{5}{2}} \left (- c x + c\right )^{\frac{5}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+a)**(5/2)*(-c*x+c)**(5/2),x)

[Out]

-5*a**(5/2)*c**(5/2)*atan(sqrt(a)*sqrt(-c*x + c)/(sqrt(c)*sqrt(a*x + a)))/8 + 5*
a**2*c**2*x*sqrt(a*x + a)*sqrt(-c*x + c)/16 + 5*a*c*x*(a*x + a)**(3/2)*(-c*x + c
)**(3/2)/24 + x*(a*x + a)**(5/2)*(-c*x + c)**(5/2)/6

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Mathematica [A]  time = 0.180811, size = 91, normalized size = 0.72 \[ \frac{c^2 (a (x+1))^{5/2} \left (x \sqrt{x+1} \left (8 x^4-26 x^2+33\right ) \sqrt{c-c x}-30 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{x+1} \sqrt{c-c x}}{\sqrt{c} (x-1)}\right )\right )}{48 (x+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + a*x)^(5/2)*(c - c*x)^(5/2),x]

[Out]

(c^2*(a*(1 + x))^(5/2)*(x*Sqrt[1 + x]*Sqrt[c - c*x]*(33 - 26*x^2 + 8*x^4) - 30*S
qrt[c]*ArcTan[(Sqrt[1 + x]*Sqrt[c - c*x])/(Sqrt[c]*(-1 + x))]))/(48*(1 + x)^(5/2
))

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Maple [B]  time = 0.02, size = 193, normalized size = 1.5 \[ -{\frac{1}{6\,c} \left ( ax+a \right ) ^{{\frac{5}{2}}} \left ( -cx+c \right ) ^{{\frac{7}{2}}}}-{\frac{a}{6\,c} \left ( ax+a \right ) ^{{\frac{3}{2}}} \left ( -cx+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}}{8\,c}\sqrt{ax+a} \left ( -cx+c \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{24} \left ( -cx+c \right ) ^{{\frac{5}{2}}}\sqrt{ax+a}}+{\frac{5\,{a}^{2}c}{48} \left ( -cx+c \right ) ^{{\frac{3}{2}}}\sqrt{ax+a}}+{\frac{5\,{a}^{2}{c}^{2}}{16}\sqrt{ax+a}\sqrt{-cx+c}}+{\frac{5\,{a}^{3}{c}^{3}}{16}\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+a)^(5/2)*(-c*x+c)^(5/2),x)

[Out]

-1/6/c*(a*x+a)^(5/2)*(-c*x+c)^(7/2)-1/6*a/c*(a*x+a)^(3/2)*(-c*x+c)^(7/2)-1/8*a^2
/c*(a*x+a)^(1/2)*(-c*x+c)^(7/2)+1/24*a^2*(-c*x+c)^(5/2)*(a*x+a)^(1/2)+5/48*a^2*c
*(-c*x+c)^(3/2)*(a*x+a)^(1/2)+5/16*a^2*c^2*(-c*x+c)^(1/2)*(a*x+a)^(1/2)+5/16*a^3
*c^3*((-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)^(5/2)*(-c*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220706, size = 1, normalized size = 0.01 \[ \left [\frac{5}{32} \, \sqrt{-a c} a^{2} c^{2} \log \left (2 \, a c x^{2} + 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right ) + \frac{1}{48} \,{\left (8 \, a^{2} c^{2} x^{5} - 26 \, a^{2} c^{2} x^{3} + 33 \, a^{2} c^{2} x\right )} \sqrt{a x + a} \sqrt{-c x + c}, \frac{5}{16} \, \sqrt{a c} a^{2} c^{2} \arctan \left (\frac{a c x}{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c}}\right ) + \frac{1}{48} \,{\left (8 \, a^{2} c^{2} x^{5} - 26 \, a^{2} c^{2} x^{3} + 33 \, a^{2} c^{2} x\right )} \sqrt{a x + a} \sqrt{-c x + c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + a)^(5/2)*(-c*x + c)^(5/2),x, algorithm="fricas")

[Out]

[5/32*sqrt(-a*c)*a^2*c^2*log(2*a*c*x^2 + 2*sqrt(-a*c)*sqrt(a*x + a)*sqrt(-c*x +
c)*x - a*c) + 1/48*(8*a^2*c^2*x^5 - 26*a^2*c^2*x^3 + 33*a^2*c^2*x)*sqrt(a*x + a)
*sqrt(-c*x + c), 5/16*sqrt(a*c)*a^2*c^2*arctan(a*c*x/(sqrt(a*c)*sqrt(a*x + a)*sq
rt(-c*x + c))) + 1/48*(8*a^2*c^2*x^5 - 26*a^2*c^2*x^3 + 33*a^2*c^2*x)*sqrt(a*x +
a)*sqrt(-c*x + c)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+a)**(5/2)*(-c*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.33746, size = 478, normalized size = 3.79 \[ -\frac{{\left (\frac{6 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (2 \,{\left ({\left (a x + a\right )}{\left (4 \,{\left (a x + a\right )}{\left (\frac{a x + a}{a^{4}} - \frac{5}{a^{3}}\right )} + \frac{39}{a^{2}}\right )} - \frac{37}{a}\right )}{\left (a x + a\right )} + 31\right )}{\left (a x + a\right )} - 3 \, a\right )} \sqrt{a x + a}\right )} c^{2}{\left | a \right |}}{48 \, a} - \frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a} a x\right )} c^{2}{\left | a \right |}}{2 \, a} + \frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (a x + a\right )}{\left (2 \,{\left (a x + a\right )}{\left (\frac{a x + a}{a^{2}} - \frac{3}{a}\right )} + 5\right )} - a\right )} \sqrt{a x + a}\right )} c^{2}{\left | a \right |}}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + a)^(5/2)*(-c*x + c)^(5/2),x, algorithm="giac")

[Out]

-1/48*(6*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)*((2*((a*x + a)*(4*(a*x + a)*((a*x
+ a)/a^4 - 5/a^3) + 39/a^2) - 37/a)*(a*x + a) + 31)*(a*x + a) - 3*a)*sqrt(a*x +
a))*c^2*abs(a)/a - 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^2*abs(a)/a + 1/4*(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^2*abs(a)/a