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

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

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Rubi [A]  time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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+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*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 0.69 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.13, size = 101, normalized size = 1.51 \begin {gather*} \frac {(a+b x)^{3/2} \left (\frac {9 c^2 (a c-b c x)}{a+b x}-\frac {9 c (a c-b c x)^2}{(a+b x)^2}-\frac {(a c-b c x)^3}{(a+b x)^3}+c^3\right )}{24 a^4 b c^4 (a c-b c x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.23, size = 72, normalized size = 1.07 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 2.38, size = 251, normalized size = 3.75 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.00, size = 45, normalized size = 0.67 \begin {gather*} \frac {\left (-b x +a \right ) \left (-2 b^{2} x^{2}+3 a^{2}\right ) x}{3 \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}} a^{4}} \end {gather*}

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 = 1.43, size = 53, normalized size = 0.79 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.58, size = 80, normalized size = 1.19 \begin {gather*} -\frac {3\,a^2\,x\,\sqrt {a\,c-b\,c\,x}-2\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^2\,\left (3\,a^4\,\left (a\,c-b\,c\,x\right )-6\,a^5\,c\right )\,\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a^2*x*(a*c - b*c*x)^(1/2) - 2*b^2*x^3*(a*c - b*c*x)^(1/2))/((a*c - b*c*x)^2*(3*a^4*(a*c - b*c*x) - 6*a^5*c
)*(a + b*x)^(1/2))

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sympy [C]  time = 15.85, size = 94, normalized size = 1.40 \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 {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 {gather*}

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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