3.2831 \(\int (\frac{c}{(a+b x)^3})^{5/2} \, dx\)
Optimal. Leaf size=30 \[ -\frac{2 c^2 \sqrt{\frac{c}{(a+b x)^3}}}{13 b (a+b x)^5} \]
[Out]
(-2*c^2*Sqrt[c/(a + b*x)^3])/(13*b*(a + b*x)^5)
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Rubi [A] time = 0.0092784, antiderivative size = 30, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used =
{247, 15, 30} \[ -\frac{2 c^2 \sqrt{\frac{c}{(a+b x)^3}}}{13 b (a+b x)^5} \]
Antiderivative was successfully verified.
[In]
Int[(c/(a + b*x)^3)^(5/2),x]
[Out]
(-2*c^2*Sqrt[c/(a + b*x)^3])/(13*b*(a + b*x)^5)
Rule 247
Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]
Rule 15
Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] && !IntegerQ[m]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int \left (\frac{c}{(a+b x)^3}\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x^3}\right )^{5/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c^2 \sqrt{\frac{c}{(a+b x)^3}} (a+b x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{15/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 c^2 \sqrt{\frac{c}{(a+b x)^3}}}{13 b (a+b x)^5}\\ \end{align*}
Mathematica [A] time = 0.0123304, size = 25, normalized size = 0.83 \[ -\frac{2 (a+b x) \left (\frac{c}{(a+b x)^3}\right )^{5/2}}{13 b} \]
Antiderivative was successfully verified.
[In]
Integrate[(c/(a + b*x)^3)^(5/2),x]
[Out]
(-2*(c/(a + b*x)^3)^(5/2)*(a + b*x))/(13*b)
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Maple [A] time = 0.001, size = 22, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+2\,a}{13\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{3}}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c/(b*x+a)^3)^(5/2),x)
[Out]
-2/13*(b*x+a)*(c/(b*x+a)^3)^(5/2)/b
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Maxima [A] time = 1.36213, size = 32, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (b c^{\frac{5}{2}} x + a c^{\frac{5}{2}}\right )}}{13 \,{\left (b x + a\right )}^{\frac{15}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c/(b*x+a)^3)^(5/2),x, algorithm="maxima")
[Out]
-2/13*(b*c^(5/2)*x + a*c^(5/2))/((b*x + a)^(15/2)*b)
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Fricas [B] time = 1.3416, size = 190, normalized size = 6.33 \begin{align*} -\frac{2 \, c^{2} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{13 \,{\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c/(b*x+a)^3)^(5/2),x, algorithm="fricas")
[Out]
-2/13*c^2*sqrt(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b
^3*x^2 + 5*a^4*b^2*x + a^5*b)
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Sympy [A] time = 108.255, size = 1294, normalized size = 43.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c/(b*x+a)**3)**(5/2),x)
[Out]
Piecewise((-638*a**6*c**(5/2)*(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(5/2)/(91*a**5*b + 455*a**4
*b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 3828*a**5*b*c**(5/2)*x*(
1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(5/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 +
910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 9570*a**4*b**2*c**(5/2)*x**2*(1/(a**3 + 3*a**2*b*x + 3*
a*b**2*x**2 + b**3*x**3))**(5/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*
a*b**5*x**4 + 91*b**6*x**5) - 12760*a**3*b**3*c**(5/2)*x**3*(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3)
)**(5/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x*
*5) - 1170*a**3*c**(5/2)*(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(3/2)/(91*a**5*b + 455*a**4*b**2
*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 9570*a**2*b**4*c**(5/2)*x**4*
(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(5/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 +
910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 3510*a**2*b*c**(5/2)*x*(1/(a**3 + 3*a**2*b*x + 3*a*b**
2*x**2 + b**3*x**3))**(3/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**
5*x**4 + 91*b**6*x**5) - 3828*a*b**5*c**(5/2)*x**5*(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(5/2)/
(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 351
0*a*b**2*c**(5/2)*x**2*(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(3/2)/(91*a**5*b + 455*a**4*b**2*x
+ 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 638*b**6*c**(5/2)*x**6*(1/(a**3
+ 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))**(5/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 + 910*a**
2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5) - 1170*b**3*c**(5/2)*x**3*(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2
+ b**3*x**3))**(3/2)/(91*a**5*b + 455*a**4*b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4
+ 91*b**6*x**5) + 1794*c**(5/2)*sqrt(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))/(91*a**5*b + 455*a**4*
b**2*x + 910*a**3*b**3*x**2 + 910*a**2*b**4*x**3 + 455*a*b**5*x**4 + 91*b**6*x**5), Ne(b, 0)), (x*(c/a**3)**(5
/2), True))
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Giac [A] time = 1.13138, size = 70, normalized size = 2.33 \begin{align*} -\frac{2 \, c^{9} \mathrm{sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm{sgn}\left (b x + a\right )}{13 \,{\left (b c x + a c\right )}^{\frac{13}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c/(b*x+a)^3)^(5/2),x, algorithm="giac")
[Out]
-2/13*c^9*sgn(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sgn(b*x + a)/((b*c*x + a*c)^(13/2)*b)