3.1142 \(\int \frac{1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 x}{3 a^2 c^2 \sqrt{a x+a} \sqrt{c-c x}}+\frac{x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0098129, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {40, 39} \[ \frac{2 x}{3 a^2 c^2 \sqrt{a x+a} \sqrt{c-c x}}+\frac{x}{3 a c (a x+a)^{3/2} (c-c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*x)^(5/2)*(c - c*x)^(5/2)),x]

[Out]

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

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m +
1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; F
reeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx &=\frac{x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{2 \int \frac{1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{3 a c}\\ &=\frac{x}{3 a c (a+a x)^{3/2} (c-c x)^{3/2}}+\frac{2 x}{3 a^2 c^2 \sqrt{a+a x} \sqrt{c-c x}}\\ \end{align*}

Mathematica [A]  time = 0.0272902, size = 42, normalized size = 0.69 \[ \frac{x (x+1) \left (2 x^2-3\right )}{3 c^2 (x-1) (a (x+1))^{5/2} \sqrt{c-c x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*x)^(5/2)*(c - c*x)^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.001, size = 32, normalized size = 0.5 \begin{align*}{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 2\,{x}^{2}-3 \right ) }{3} \left ( ax+a \right ) ^{-{\frac{5}{2}}} \left ( -cx+c \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x)

[Out]

1/3*(1+x)*(-1+x)*x*(2*x^2-3)/(a*x+a)^(5/2)/(-c*x+c)^(5/2)

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Maxima [A]  time = 0.98581, size = 61, normalized size = 1. \begin{align*} \frac{x}{3 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a c} + \frac{2 \, x}{3 \, \sqrt{-a c x^{2} + a c} a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53267, size = 120, normalized size = 1.97 \begin{align*} -\frac{{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{3 \,{\left (a^{3} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{3} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(2*x^3 - 3*x)*sqrt(a*x + a)*sqrt(-c*x + c)/(a^3*c^3*x^4 - 2*a^3*c^3*x^2 + a^3*c^3)

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Sympy [C]  time = 53.7394, size = 82, normalized size = 1.34 \begin{align*} \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{1}{2}, \frac{5}{2}, 3 \\\frac{5}{4}, \frac{7}{4}, 2, \frac{5}{2}, 3 & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{\frac{5}{2}} c^{\frac{5}{2}}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{2}, \frac{3}{4}, \frac{5}{4}, 1 & \\\frac{3}{4}, \frac{5}{4} & - \frac{1}{2}, 0, 2, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{\frac{5}{2}} c^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)**(5/2)/(-c*x+c)**(5/2),x)

[Out]

I*meijerg(((5/4, 7/4, 1), (1/2, 5/2, 3)), ((5/4, 7/4, 2, 5/2, 3), (0,)), x**(-2))/(3*pi**(3/2)*a**(5/2)*c**(5/
2)) + meijerg(((-1/2, 0, 1/2, 3/4, 5/4, 1), ()), ((3/4, 5/4), (-1/2, 0, 2, 0)), exp_polar(-2*I*pi)/x**2)/(3*pi
**(3/2)*a**(5/2)*c**(5/2))

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Giac [B]  time = 1.2169, size = 320, normalized size = 5.25 \begin{align*} -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}{\left (\frac{4 \,{\left (a x + a\right )}{\left | a \right |}}{a^{2} c} - \frac{9 \,{\left | a \right |}}{a c}\right )}}{12 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{2}} - \frac{16 \, \sqrt{-a c} a^{4} c^{2} - 18 \, \sqrt{-a c}{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{2} c + 3 \, \sqrt{-a c}{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4}}{3 \,{\left (2 \, a^{2} c -{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{3} c^{2}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+a)^(5/2)/(-c*x+c)^(5/2),x, algorithm="giac")

[Out]

-1/12*sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*(4*(a*x + a)*abs(a)/(a^2*c) - 9*abs(a)/(a*c))/((a*x + a)*a*
c - 2*a^2*c)^2 - 1/3*(16*sqrt(-a*c)*a^4*c^2 - 18*sqrt(-a*c)*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c +
2*a^2*c))^2*a^2*c + 3*sqrt(-a*c)*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^4)/((2*a^2*c - (s
qrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^3*c^2*abs(a))