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

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

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Rubi [A]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {40, 39} \begin {gather*} \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}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 42, normalized size = 0.69 \begin {gather*} \frac {x (x+1) \left (2 x^2-3\right )}{3 c^2 (x-1) (a (x+1))^{5/2} \sqrt {c-c x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.12, size = 93, normalized size = 1.52 \begin {gather*} \frac {(a x+a)^{3/2} \left (-\frac {a^3 (c-c x)^3}{(a x+a)^3}-\frac {9 a^2 c (c-c x)^2}{(a x+a)^2}+\frac {9 a c^2 (c-c x)}{a x+a}+c^3\right )}{24 a^4 c^4 (c-c x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + a*x)^(3/2)*(c^3 + (9*a*c^2*(c - c*x))/(a + a*x) - (9*a^2*c*(c - c*x)^2)/(a + a*x)^2 - (a^3*(c - c*x)^3)/
(a + a*x)^3))/(24*a^4*c^4*(c - c*x)^(3/2))

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fricas [A]  time = 1.08, size = 57, normalized size = 0.93 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 0.81, size = 237, normalized size = 3.89 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.00, size = 32, normalized size = 0.52 \begin {gather*} \frac {\left (x +1\right ) \left (x -1\right ) \left (2 x^{2}-3\right ) x}{3 \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {5}{2}}} \end {gather*}

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

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maxima [A]  time = 1.35, size = 45, normalized size = 0.74 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.41, size = 62, normalized size = 1.02 \begin {gather*} -\frac {3\,x\,\sqrt {c-c\,x}-2\,x^3\,\sqrt {c-c\,x}}{\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^2\,\left (3\,a^2\,\left (c-c\,x\right )-6\,a^2\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x*(c - c*x)^(1/2) - 2*x^3*(c - c*x)^(1/2))/((a + a*x)^(1/2)*(c - c*x)^2*(3*a^2*(c - c*x) - 6*a^2*c))

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sympy [C]  time = 13.69, size = 82, normalized size = 1.34 \begin {gather*} \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 {gather*}

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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