3.11.82 \(\int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx\)

Optimal. Leaf size=100 \[ \frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)),x]

[Out]

x/(5*a^2*c*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)) + (4*x)/(15*a^4*c^2*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (8*
x)/(15*a^6*c^3*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)^{7/2} (a c-b c x)^{7/2}} \, dx &=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{5 a^2 c}\\ &=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{15 a^4 c^2}\\ &=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 57, normalized size = 0.57 \begin {gather*} \frac {15 a^4 x-20 a^2 b^2 x^3+8 b^4 x^5}{15 a^6 c (a+b x)^{5/2} (c (a-b x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)),x]

[Out]

(15*a^4*x - 20*a^2*b^2*x^3 + 8*b^4*x^5)/(15*a^6*c*(c*(a - b*x))^(5/2)*(a + b*x)^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.14, size = 149, normalized size = 1.49 \begin {gather*} \frac {(a+b x)^{5/2} \left (\frac {25 c^4 (a c-b c x)}{a+b x}+\frac {150 c^3 (a c-b c x)^2}{(a+b x)^2}-\frac {150 c^2 (a c-b c x)^3}{(a+b x)^3}-\frac {25 c (a c-b c x)^4}{(a+b x)^4}-\frac {3 (a c-b c x)^5}{(a+b x)^5}+3 c^5\right )}{480 a^6 b c^6 (a c-b c x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)),x]

[Out]

((a + b*x)^(5/2)*(3*c^5 + (25*c^4*(a*c - b*c*x))/(a + b*x) + (150*c^3*(a*c - b*c*x)^2)/(a + b*x)^2 - (150*c^2*
(a*c - b*c*x)^3)/(a + b*x)^3 - (25*c*(a*c - b*c*x)^4)/(a + b*x)^4 - (3*(a*c - b*c*x)^5)/(a + b*x)^5))/(480*a^6
*b*c^6*(a*c - b*c*x)^(5/2))

________________________________________________________________________________________

fricas [A]  time = 1.40, size = 98, normalized size = 0.98 \begin {gather*} -\frac {{\left (8 \, b^{4} x^{5} - 20 \, a^{2} b^{2} x^{3} + 15 \, a^{4} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{15 \, {\left (a^{6} b^{6} c^{4} x^{6} - 3 \, a^{8} b^{4} c^{4} x^{4} + 3 \, a^{10} b^{2} c^{4} x^{2} - a^{12} c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(-b*c*x+a*c)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(8*b^4*x^5 - 20*a^2*b^2*x^3 + 15*a^4*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^6*b^6*c^4*x^6 - 3*a^8*b^4*c^
4*x^4 + 3*a^10*b^2*c^4*x^2 - a^12*c^4)

________________________________________________________________________________________

giac [B]  time = 2.57, size = 366, normalized size = 3.66 \begin {gather*} -\frac {\sqrt {-b c x + a c} {\left ({\left (b c x - a c\right )} {\left (\frac {275 \, c}{a^{5} b {\left | c \right |}} + \frac {64 \, {\left (b c x - a c\right )}}{a^{6} b {\left | c \right |}}\right )} + \frac {300 \, c^{2}}{a^{4} b {\left | c \right |}}\right )}}{240 \, {\left (2 \, a c^{2} + {\left (b c x - a c\right )} c\right )}^{\frac {5}{2}}} - \frac {1024 \, a^{4} c^{8} - 2200 \, a^{3} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{2} c^{6} + 1660 \, a^{2} {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{4} c^{4} - 450 \, a {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{6} c^{2} + 45 \, {\left (\sqrt {-b c x + a c} \sqrt {-c} - \sqrt {2 \, a c^{2} + {\left (b c x - a c\right )} c}\right )}^{8}}{60 \, {\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 )}^{5} a^{5} b \sqrt {-c} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(-b*c*x+a*c)^(7/2),x, algorithm="giac")

[Out]

-1/240*sqrt(-b*c*x + a*c)*((b*c*x - a*c)*(275*c/(a^5*b*abs(c)) + 64*(b*c*x - a*c)/(a^6*b*abs(c))) + 300*c^2/(a
^4*b*abs(c)))/(2*a*c^2 + (b*c*x - a*c)*c)^(5/2) - 1/60*(1024*a^4*c^8 - 2200*a^3*(sqrt(-b*c*x + a*c)*sqrt(-c) -
sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*c^6 + 1660*a^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)
*c))^4*c^4 - 450*a*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*c^2 + 45*(sqrt(-b*c*x + a
*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^8)/((2*a*c^2 - (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (
b*c*x - a*c)*c))^2)^5*a^5*b*sqrt(-c)*abs(c))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 56, normalized size = 0.56 \begin {gather*} \frac {\left (-b x +a \right ) \left (8 b^{4} x^{4}-20 a^{2} b^{2} x^{2}+15 a^{4}\right ) x}{15 \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {7}{2}} a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(7/2)/(-b*c*x+a*c)^(7/2),x)

[Out]

1/15*(-b*x+a)*x*(8*b^4*x^4-20*a^2*b^2*x^2+15*a^4)/(b*x+a)^(5/2)/a^6/(-b*c*x+a*c)^(7/2)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 79, normalized size = 0.79 \begin {gather*} \frac {x}{5 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{2} c} + \frac {4 \, x}{15 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{4} c^{2}} + \frac {8 \, x}{15 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{6} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(7/2)/(-b*c*x+a*c)^(7/2),x, algorithm="maxima")

[Out]

1/5*x/((-b^2*c*x^2 + a^2*c)^(5/2)*a^2*c) + 4/15*x/((-b^2*c*x^2 + a^2*c)^(3/2)*a^4*c^2) + 8/15*x/(sqrt(-b^2*c*x
^2 + a^2*c)*a^6*c^3)

________________________________________________________________________________________

mupad [B]  time = 0.65, size = 111, normalized size = 1.11 \begin {gather*} \frac {15\,a^4\,x\,\sqrt {a\,c-b\,c\,x}+8\,b^4\,x^5\,\sqrt {a\,c-b\,c\,x}-20\,a^2\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^3\,\left (60\,a^8\,c-\left (a\,c-b\,c\,x\right )\,\left (45\,a^7+15\,b\,x\,a^6\right )\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)^(7/2)*(a + b*x)^(7/2)),x)

[Out]

(15*a^4*x*(a*c - b*c*x)^(1/2) + 8*b^4*x^5*(a*c - b*c*x)^(1/2) - 20*a^2*b^2*x^3*(a*c - b*c*x)^(1/2))/((a*c - b*
c*x)^3*(60*a^8*c - (a*c - b*c*x)*(45*a^7 + 15*a^6*b*x))*(a + b*x)^(1/2))

________________________________________________________________________________________

sympy [C]  time = 59.50, size = 97, normalized size = 0.97 \begin {gather*} - \frac {2 i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & \frac {1}{2}, \frac {7}{2}, 4 \\\frac {7}{4}, \frac {9}{4}, 3, \frac {7}{2}, 4 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{6} b c^{\frac {7}{2}}} + \frac {2 {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {5}{4}, \frac {7}{4}, 1 & \\\frac {5}{4}, \frac {7}{4} & - \frac {1}{2}, 0, 3, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{6} b c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(7/2)/(-b*c*x+a*c)**(7/2),x)

[Out]

-2*I*meijerg(((7/4, 9/4, 1), (1/2, 7/2, 4)), ((7/4, 9/4, 3, 7/2, 4), (0,)), a**2/(b**2*x**2))/(15*pi**(3/2)*a*
*6*b*c**(7/2)) + 2*meijerg(((-1/2, 0, 1/2, 5/4, 7/4, 1), ()), ((5/4, 7/4), (-1/2, 0, 3, 0)), a**2*exp_polar(-2
*I*pi)/(b**2*x**2))/(15*pi**(3/2)*a**6*b*c**(7/2))

________________________________________________________________________________________