3.1152 \(\int \frac{1}{(a+b x)^{9/2} (a c-b c x)^{9/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac{16 x}{35 a^8 c^4 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}} \]

[Out]

x/(7*a^2*c*(a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)) + (6*x)/(35*a^4*c^2*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)) + (8*
x)/(35*a^6*c^3*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (16*x)/(35*a^8*c^4*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.033947, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {40, 39} \[ \frac{16 x}{35 a^8 c^4 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(9/2)*(a*c - b*c*x)^(9/2)),x]

[Out]

x/(7*a^2*c*(a + b*x)^(7/2)*(a*c - b*c*x)^(7/2)) + (6*x)/(35*a^4*c^2*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2)) + (8*
x)/(35*a^6*c^3*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2)) + (16*x)/(35*a^8*c^4*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)^{9/2} (a c-b c x)^{9/2}} \, dx &=\frac{x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac{6 \int \frac{1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx}{7 a^2 c}\\ &=\frac{x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac{6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{24 \int \frac{1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{35 a^4 c^2}\\ &=\frac{x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac{6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{16 \int \frac{1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{35 a^6 c^3}\\ &=\frac{x}{7 a^2 c (a+b x)^{7/2} (a c-b c x)^{7/2}}+\frac{6 x}{35 a^4 c^2 (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac{8 x}{35 a^6 c^3 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{16 x}{35 a^8 c^4 \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}

Mathematica [A]  time = 0.0412714, size = 76, normalized size = 0.57 \[ \frac{x \left (-70 a^4 b^2 x^2+56 a^2 b^4 x^4+35 a^6-16 b^6 x^6\right ) \sqrt{c (a-b x)}}{35 a^8 c^5 (a-b x)^4 (a+b x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(9/2)*(a*c - b*c*x)^(9/2)),x]

[Out]

(x*Sqrt[c*(a - b*x)]*(35*a^6 - 70*a^4*b^2*x^2 + 56*a^2*b^4*x^4 - 16*b^6*x^6))/(35*a^8*c^5*(a - b*x)^4*(a + b*x
)^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 67, normalized size = 0.5 \begin{align*}{\frac{ \left ( -bx+a \right ) x \left ( -16\,{b}^{6}{x}^{6}+56\,{b}^{4}{x}^{4}{a}^{2}-70\,{b}^{2}{x}^{2}{a}^{4}+35\,{a}^{6} \right ) }{35\,{a}^{8}} \left ( bx+a \right ) ^{-{\frac{7}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x)

[Out]

1/35*(-b*x+a)*x*(-16*b^6*x^6+56*a^2*b^4*x^4-70*a^4*b^2*x^2+35*a^6)/(b*x+a)^(7/2)/a^8/(-b*c*x+a*c)^(9/2)

________________________________________________________________________________________

Maxima [A]  time = 0.971443, size = 142, normalized size = 1.07 \begin{align*} \frac{x}{7 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{7}{2}} a^{2} c} + \frac{6 \, x}{35 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{5}{2}} a^{4} c^{2}} + \frac{8 \, x}{35 \,{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac{3}{2}} a^{6} c^{3}} + \frac{16 \, x}{35 \, \sqrt{-b^{2} c x^{2} + a^{2} c} a^{8} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x, algorithm="maxima")

[Out]

1/7*x/((-b^2*c*x^2 + a^2*c)^(7/2)*a^2*c) + 6/35*x/((-b^2*c*x^2 + a^2*c)^(5/2)*a^4*c^2) + 8/35*x/((-b^2*c*x^2 +
a^2*c)^(3/2)*a^6*c^3) + 16/35*x/(sqrt(-b^2*c*x^2 + a^2*c)*a^8*c^4)

________________________________________________________________________________________

Fricas [A]  time = 1.87974, size = 257, normalized size = 1.93 \begin{align*} -\frac{{\left (16 \, b^{6} x^{7} - 56 \, a^{2} b^{4} x^{5} + 70 \, a^{4} b^{2} x^{3} - 35 \, a^{6} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{35 \,{\left (a^{8} b^{8} c^{5} x^{8} - 4 \, a^{10} b^{6} c^{5} x^{6} + 6 \, a^{12} b^{4} c^{5} x^{4} - 4 \, a^{14} b^{2} c^{5} x^{2} + a^{16} c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(16*b^6*x^7 - 56*a^2*b^4*x^5 + 70*a^4*b^2*x^3 - 35*a^6*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)/(a^8*b^8*c^5*
x^8 - 4*a^10*b^6*c^5*x^6 + 6*a^12*b^4*c^5*x^4 - 4*a^14*b^2*c^5*x^2 + a^16*c^5)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(9/2)/(-b*c*x+a*c)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.05113, size = 657, normalized size = 4.94 \begin{align*} -\frac{\sqrt{-b c x + a c}{\left ({\left (b c x - a c\right )}{\left ({\left (b c x - a c\right )}{\left (\frac{1617 \,{\left | c \right |}}{a^{7} b c} + \frac{256 \,{\left (b c x - a c\right )}{\left | c \right |}}{a^{8} b c^{2}}\right )} + \frac{3430 \,{\left | c \right |}}{a^{6} b}\right )} + \frac{2450 \, c{\left | c \right |}}{a^{5} b}\right )}}{1120 \,{\left (2 \, a c^{2} +{\left (b c x - a c\right )} c\right )}^{\frac{7}{2}}} - \frac{16384 \, a^{6} c^{12} - 51744 \, a^{5}{\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^{10} + 66416 \, a^{4}{\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^{8} - 43120 \, 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 )}^{6} c^{6} + 14280 \, 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 )}^{8} c^{4} - 2450 \, a{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{10} c^{2} + 175 \,{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{12}}{280 \,{\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 )}^{7} a^{7} b \sqrt{-c} c{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(9/2)/(-b*c*x+a*c)^(9/2),x, algorithm="giac")

[Out]

-1/1120*sqrt(-b*c*x + a*c)*((b*c*x - a*c)*((b*c*x - a*c)*(1617*abs(c)/(a^7*b*c) + 256*(b*c*x - a*c)*abs(c)/(a^
8*b*c^2)) + 3430*abs(c)/(a^6*b)) + 2450*c*abs(c)/(a^5*b))/(2*a*c^2 + (b*c*x - a*c)*c)^(7/2) - 1/280*(16384*a^6
*c^12 - 51744*a^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*c^10 + 66416*a^4*(sqrt(-b*
c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*c^8 - 43120*a^3*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2
*a*c^2 + (b*c*x - a*c)*c))^6*c^6 + 14280*a^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^8
*c^4 - 2450*a*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^10*c^2 + 175*(sqrt(-b*c*x + a*c)
*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^12)/((2*a*c^2 - (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*
c*x - a*c)*c))^2)^7*a^7*b*sqrt(-c)*c*abs(c))