3.1150 \(\int \frac{1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.011025, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {40, 39} \[ \frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(3*a^2*c*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (2*x)/(3*a^4*c^2*Sqrt[a + b*x]*Sqrt[a*c - b*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+b x)^{5/2} (a c-b c x)^{5/2}} \, dx &=\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{2 \int \frac{1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{3 a^2 c}\\ &=\frac{x}{3 a^2 c (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{2 x}{3 a^4 c^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}

Mathematica [A]  time = 0.0237397, size = 46, normalized size = 0.69 \[ \frac{3 a^2 x-2 b^2 x^3}{3 a^4 c (a+b x)^{3/2} (c (a-b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 45, normalized size = 0.7 \begin{align*}{\frac{ \left ( -bx+a \right ) x \left ( -2\,{b}^{2}{x}^{2}+3\,{a}^{2} \right ) }{3\,{a}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-b*x+a)*x*(-2*b^2*x^2+3*a^2)/(b*x+a)^(3/2)/a^4/(-b*c*x+a*c)^(5/2)

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Maxima [A]  time = 0.98127, size = 72, normalized size = 1.07 \begin{align*} \frac{x}{3 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{3}{2}} a^{2} c} + \frac{2 \, x}{3 \, \sqrt{-b^{2} c x^{2} + a^{2} c} a^{4} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.64699, size = 147, normalized size = 2.19 \begin{align*} -\frac{{\left (2 \, b^{2} x^{3} - 3 \, a^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{3 \,{\left (a^{4} b^{4} c^{3} x^{4} - 2 \, a^{6} b^{2} c^{3} x^{2} + a^{8} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*b^2*x^3 - 3*a^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^4*b^4*c^3*x^4 - 2*a^6*b^2*c^3*x^2 + a^8*c^3)

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Sympy [C]  time = 47.5316, size = 94, normalized size = 1.4 \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{a^{2}}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{4} b 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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{3 \pi ^{\frac{3}{2}} a^{4} b c^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B]  time = 1.25254, size = 339, normalized size = 5.06 \begin{align*} -\frac{\sqrt{-b c x + a c}{\left (\frac{9 \,{\left | c \right |}}{a^{3} b c} + \frac{4 \,{\left (b c x - a c\right )}{\left | c \right |}}{a^{4} b c^{2}}\right )}}{12 \,{\left (2 \, a c^{2} +{\left (b c x - a c\right )} c\right )}^{\frac{3}{2}}} + \frac{16 \, a^{2} \sqrt{-c} c^{4} - 18 \, a{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2} \sqrt{-c} c^{2} + 3 \,{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{4} \sqrt{-c}}{3 \,{\left (2 \, a c^{2} -{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}\right )}^{3} a^{3} b{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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