\(\int (\frac {c}{(a+b x)^3})^{5/2} \, dx\) [2831]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 c^2 \sqrt {\frac {c}{(a+b x)^3}}}{13 b (a+b x)^5} \]

[Out]

-2/13*c^2*(c/(b*x+a)^3)^(1/2)/b/(b*x+a)^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 30} \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 c^2 \sqrt {\frac {c}{(a+b x)^3}}}{13 b (a+b x)^5} \]

[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 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]

Rule 253

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 ) \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \left (\frac {c}{(a+b x)^3}\right )^{5/2} (a+b x)}{13 b} \]

[In]

Integrate[(c/(a + b*x)^3)^(5/2),x]

[Out]

(-2*(c/(a + b*x)^3)^(5/2)*(a + b*x))/(13*b)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73

method	result	size
gosper	\(-\frac {2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}}}{13 b}\)	\(22\)
default	\(-\frac {2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}}}{13 b}\)	\(22\)
trager	\(-\frac {2 c^{2} \sqrt {\frac {c}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}}}{13 b \left (b x +a \right )^{5}}\)	\(49\)

[In]

int((c/(b*x+a)^3)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/13*(b*x+a)*(c/(b*x+a)^3)^(5/2)/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (26) = 52\).

Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.10 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \, c^{2} \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{13 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (27) = 54\).

Time = 27.34 (sec) , antiderivative size = 1243, normalized size of antiderivative = 41.43 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=\begin {cases} - \frac {638 a^{6} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {3828 a^{5} b x \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {9570 a^{4} b^{2} x^{2} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {12760 a^{3} b^{3} x^{3} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {1170 a^{3} c \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {9570 a^{2} b^{4} x^{4} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {3510 a^{2} b c x \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {3828 a b^{5} x^{5} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {3510 a b^{2} c x^{2} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {638 b^{6} x^{6} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} - \frac {1170 b^{3} c x^{3} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {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}} + \frac {1794 c^{2} \sqrt {\frac {c}{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}} & \text {for}\: b \neq 0 \\x \left (\frac {c}{a^{3}}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

integrate((c/(b*x+a)**3)**(5/2),x)

[Out]

Piecewise((-638*a**6*(c/(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*x*(c/(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*x**2*(c/(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*x**3*(c/(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*(c/(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*x**4*(c/(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*x*(c/(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*x**5*(c/(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*b**2*c*x**2*(c/(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*x**6*(c/(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 + 91
0*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*x**3*(c/(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**2*sqrt(c/(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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \, {\left (b c^{\frac {5}{2}} x + a c^{\frac {5}{2}}\right )}}{13 \, {\left (b x + a\right )}^{\frac {15}{2}} b} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \, c^{9} \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )}{13 \, {\left (b c x + a c\right )}^{\frac {13}{2}} b} \]

[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)

Mupad [B] (verification not implemented)

Time = 6.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2\,b^2\,c^2\,\sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}\,\left (x^3+\frac {a^3}{b^3}+\frac {3\,a\,x^2}{b}+\frac {3\,a^2\,x}{b^2}\right )}{13\,{\left (a+b\,x\right )}^8} \]

[In]

int((c/(a + b*x)^3)^(5/2),x)

[Out]

-(2*b^2*c^2*(c/(a + b*x)^3)^(1/2)*(x^3 + a^3/b^3 + (3*a*x^2)/b + (3*a^2*x)/b^2))/(13*(a + b*x)^8)