3.1144 \(\int \frac{1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{16 x}{35 a^4 c^4 \sqrt{a x+a} \sqrt{c-c x}}+\frac{8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac{x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0276369, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {40, 39} \[ \frac{16 x}{35 a^4 c^4 \sqrt{a x+a} \sqrt{c-c x}}+\frac{8 x}{35 a^3 c^3 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{6 x}{35 a^2 c^2 (a x+a)^{5/2} (c-c x)^{5/2}}+\frac{x}{7 a c (a x+a)^{7/2} (c-c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0394339, size = 54, normalized size = 0.45 \[ \frac{x \left (16 x^6-56 x^4+70 x^2-35\right )}{35 a^4 c^4 \left (x^2-1\right )^3 \sqrt{a (x+1)} \sqrt{c-c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-35 + 70*x^2 - 56*x^4 + 16*x^6))/(35*a^4*c^4*Sqrt[a*(1 + x)]*Sqrt[c - c*x]*(-1 + x^2)^3)

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Maple [A]  time = 0.003, size = 42, normalized size = 0.4 \begin{align*}{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 16\,{x}^{6}-56\,{x}^{4}+70\,{x}^{2}-35 \right ) }{35} \left ( ax+a \right ) ^{-{\frac{9}{2}}} \left ( -cx+c \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.999581, size = 120, normalized size = 0.99 \begin{align*} \frac{x}{7 \,{\left (-a c x^{2} + a c\right )}^{\frac{7}{2}} a c} + \frac{6 \, x}{35 \,{\left (-a c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{2}} + \frac{8 \, x}{35 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{3}} + \frac{16 \, x}{35 \, \sqrt{-a c x^{2} + a c} a^{4} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59296, size = 192, normalized size = 1.59 \begin{align*} -\frac{{\left (16 \, x^{7} - 56 \, x^{5} + 70 \, x^{3} - 35 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{35 \,{\left (a^{5} c^{5} x^{8} - 4 \, a^{5} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{5} c^{5} x^{2} + a^{5} c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [B]  time = 1.81705, size = 590, normalized size = 4.88 \begin{align*} -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (a x + a\right )}{\left ({\left (a x + a\right )}{\left (\frac{256 \,{\left (a x + a\right )}{\left | a \right |}}{a^{2} c} - \frac{1617 \,{\left | a \right |}}{a c}\right )} + \frac{3430 \,{\left | a \right |}}{c}\right )} - \frac{2450 \, a{\left | a \right |}}{c}\right )} \sqrt{a x + a}}{1120 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{4}} + \frac{16384 \, a^{12} c^{6} - 51744 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{10} c^{5} + 66416 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{8} c^{4} - 43120 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{6} c^{3} + 14280 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8} a^{4} c^{2} - 2450 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{10} a^{2} c + 175 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{12}}{280 \,{\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 )}^{7} \sqrt{-a c} a c^{3}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1120*sqrt(-(a*x + a)*a*c + 2*a^2*c)*((a*x + a)*((a*x + a)*(256*(a*x + a)*abs(a)/(a^2*c) - 1617*abs(a)/(a*c)
) + 3430*abs(a)/c) - 2450*a*abs(a)/c)*sqrt(a*x + a)/((a*x + a)*a*c - 2*a^2*c)^4 + 1/280*(16384*a^12*c^6 - 5174
4*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2*a^10*c^5 + 66416*(sqrt(-a*c)*sqrt(a*x + a) - s
qrt(-(a*x + a)*a*c + 2*a^2*c))^4*a^8*c^4 - 43120*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^6
*a^6*c^3 + 14280*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^8*a^4*c^2 - 2450*(sqrt(-a*c)*sqrt
(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^10*a^2*c + 175*(sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2
*a^2*c))^12)/((2*a^2*c - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)^7*sqrt(-a*c)*a*c^3*abs
(a))