3.1143 \(\int \frac{1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{8 x}{15 a^3 c^3 \sqrt{a x+a} \sqrt{c-c x}}+\frac{4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]

[Out]

x/(5*a*c*(a + a*x)^(5/2)*(c - c*x)^(5/2)) + (4*x)/(15*a^2*c^2*(a + a*x)^(3/2)*(c - c*x)^(3/2)) + (8*x)/(15*a^3
*c^3*Sqrt[a + a*x]*Sqrt[c - c*x])

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Rubi [A]  time = 0.0179005, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {40, 39} \[ \frac{8 x}{15 a^3 c^3 \sqrt{a x+a} \sqrt{c-c x}}+\frac{4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac{x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0362571, size = 49, normalized size = 0.54 \[ \frac{x \left (8 x^4-20 x^2+15\right )}{15 a^3 c^3 \left (x^2-1\right )^2 \sqrt{a (x+1)} \sqrt{c-c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(15 - 20*x^2 + 8*x^4))/(15*a^3*c^3*Sqrt[a*(1 + x)]*Sqrt[c - c*x]*(-1 + x^2)^2)

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Maple [A]  time = 0.004, size = 37, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( 8\,{x}^{4}-20\,{x}^{2}+15 \right ) }{15} \left ( ax+a \right ) ^{-{\frac{7}{2}}} \left ( -cx+c \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981987, size = 90, normalized size = 0.99 \begin{align*} \frac{x}{5 \,{\left (-a c x^{2} + a c\right )}^{\frac{5}{2}} a c} + \frac{4 \, x}{15 \,{\left (-a c x^{2} + a c\right )}^{\frac{3}{2}} a^{2} c^{2}} + \frac{8 \, x}{15 \, \sqrt{-a c x^{2} + a c} a^{3} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60806, size = 157, normalized size = 1.73 \begin{align*} -\frac{{\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \sqrt{a x + a} \sqrt{-c x + c}}{15 \,{\left (a^{4} c^{4} x^{6} - 3 \, a^{4} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{2} - a^{4} c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.57128, size = 450, normalized size = 4.95 \begin{align*} -\frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}{\left ({\left (a x + a\right )}{\left (\frac{64 \,{\left (a x + a\right )}}{c{\left | a \right |}} - \frac{275 \, a}{c{\left | a \right |}}\right )} + \frac{300 \, a^{2}}{c{\left | a \right |}}\right )}}{240 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{3}} + \frac{1024 \, a^{8} c^{4} - 2200 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{6} c^{3} + 1660 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{4} c^{2} - 450 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{2} c + 45 \,{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8}}{60 \,{\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 )}^{5} \sqrt{-a c} c^{2}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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