3.1151 \(\int \frac{1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0199559, antiderivative size = 100, normalized size of antiderivative = 1., 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} \[ \frac{8 x}{15 a^6 c^3 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac{x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]

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 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)^{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 = 0.0302893, size = 57, normalized size = 0.57 \[ \frac{-20 a^2 b^2 x^3+15 a^4 x+8 b^4 x^5}{15 a^6 c (a+b x)^{5/2} (c (a-b x))^{5/2}} \]

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))

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Maple [A]  time = 0.004, size = 56, normalized size = 0.6 \begin{align*}{\frac{ \left ( -bx+a \right ) x \left ( 8\,{b}^{4}{x}^{4}-20\,{x}^{2}{a}^{2}{b}^{2}+15\,{a}^{4} \right ) }{15\,{a}^{6}} \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( -bcx+ac \right ) ^{-{\frac{7}{2}}}} \end{align*}

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)

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Maxima [A]  time = 0.996784, size = 107, normalized size = 1.07 \begin{align*} \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{align*}

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)

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Fricas [A]  time = 1.75966, size = 203, normalized size = 2.03 \begin{align*} -\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{align*}

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)

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

[Out]

Timed out

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Giac [B]  time = 1.42209, size = 494, normalized size = 4.94 \begin{align*} -\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{align*}

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))