\(\int \frac {1}{(\frac {c}{(a+b x)^2})^{5/2}} \, dx\) [2830]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [B] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [B] (verification not implemented)
   Giac [B] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {(a+b x)^5}{6 b c^2 \sqrt {\frac {c}{(a+b x)^2}}} \]

[Out]

1/6*(b*x+a)^5/b/c^2/(c/(b*x+a)^2)^(1/2)

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} \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {(a+b x)^5}{6 b c^2 \sqrt {\frac {c}{(a+b x)^2}}} \]

[In]

Int[(c/(a + b*x)^2)^(-5/2),x]

[Out]

(a + b*x)^5/(6*b*c^2*Sqrt[c/(a + b*x)^2])

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 \frac {1}{\left (\frac {c}{x^2}\right )^{5/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int x^5 \, dx,x,a+b x\right )}{b c^2 \sqrt {\frac {c}{(a+b x)^2}} (a+b x)} \\ & = \frac {(a+b x)^5}{6 b c^2 \sqrt {\frac {c}{(a+b x)^2}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {a+b x}{6 b \left (\frac {c}{(a+b x)^2}\right )^{5/2}} \]

[In]

Integrate[(c/(a + b*x)^2)^(-5/2),x]

[Out]

(a + b*x)/(6*b*(c/(a + b*x)^2)^(5/2))

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73

method result size
default \(\frac {b x +a}{6 \left (\frac {c}{\left (b x +a \right )^{2}}\right )^{\frac {5}{2}} b}\) \(22\)
risch \(\frac {\left (b x +a \right )^{5}}{6 b \,c^{2} \sqrt {\frac {c}{\left (b x +a \right )^{2}}}}\) \(27\)
gosper \(\frac {x \left (b^{5} x^{5}+6 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} b^{2} x^{2}+15 a^{4} b x +6 a^{5}\right )}{6 \left (b x +a \right )^{5} \left (\frac {c}{\left (b x +a \right )^{2}}\right )^{\frac {5}{2}}}\) \(73\)
trager \(\frac {\left (b x +a \right ) \left (b^{5} x^{5}+6 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} b^{2} x^{2}+15 a^{4} b x +6 a^{5}\right ) x \sqrt {\frac {c}{b^{2} x^{2}+2 a b x +a^{2}}}}{6 c^{3}}\) \(85\)

[In]

int(1/(c/(b*x+a)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6/(c/(b*x+a)^2)^(5/2)*(b*x+a)/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (26) = 52\).

Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.07 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {{\left (b^{6} x^{7} + 7 \, a b^{5} x^{6} + 21 \, a^{2} b^{4} x^{5} + 35 \, a^{3} b^{3} x^{4} + 35 \, a^{4} b^{2} x^{3} + 21 \, a^{5} b x^{2} + 6 \, a^{6} x\right )} \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, c^{3}} \]

[In]

integrate(1/(c/(b*x+a)^2)^(5/2),x, algorithm="fricas")

[Out]

1/6*(b^6*x^7 + 7*a*b^5*x^6 + 21*a^2*b^4*x^5 + 35*a^3*b^3*x^4 + 35*a^4*b^2*x^3 + 21*a^5*b*x^2 + 6*a^6*x)*sqrt(c
/(b^2*x^2 + 2*a*b*x + a^2))/c^3

Sympy [A] (verification not implemented)

Time = 1.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\begin {cases} \frac {\frac {a}{b} + x}{6 \left (\frac {c}{\left (a + b x\right )^{2}}\right )^{\frac {5}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\frac {c}{a^{2}}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(c/(b*x+a)**2)**(5/2),x)

[Out]

Piecewise(((a/b + x)/(6*(c/(a + b*x)**2)**(5/2)), Ne(b, 0)), (x/(c/a**2)**(5/2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2} + 6 \, a^{5} x}{6 \, c^{\frac {5}{2}}} \]

[In]

integrate(1/(c/(b*x+a)^2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(b^5*x^6 + 6*a*b^4*x^5 + 15*a^2*b^3*x^4 + 20*a^3*b^2*x^3 + 15*a^4*b*x^2 + 6*a^5*x)/c^(5/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.83 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {b^{5} \sqrt {c} x^{6} + 6 \, a b^{4} \sqrt {c} x^{5} + 15 \, a^{2} b^{3} \sqrt {c} x^{4} + 20 \, a^{3} b^{2} \sqrt {c} x^{3} + 15 \, a^{4} b \sqrt {c} x^{2} + 6 \, a^{5} \sqrt {c} x}{6 \, c^{3} \mathrm {sgn}\left (b x + a\right )} \]

[In]

integrate(1/(c/(b*x+a)^2)^(5/2),x, algorithm="giac")

[Out]

1/6*(b^5*sqrt(c)*x^6 + 6*a*b^4*sqrt(c)*x^5 + 15*a^2*b^3*sqrt(c)*x^4 + 20*a^3*b^2*sqrt(c)*x^3 + 15*a^4*b*sqrt(c
)*x^2 + 6*a^5*sqrt(c)*x)/(c^3*sgn(b*x + a))

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.27 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\sqrt {\frac {c}{{\left (a+b\,x\right )}^2}}\,\left (\frac {a^6\,x}{c^3}+\frac {b^6\,x^7}{6\,c^3}+\frac {7\,a^5\,b\,x^2}{2\,c^3}+\frac {7\,a\,b^5\,x^6}{6\,c^3}+\frac {35\,a^4\,b^2\,x^3}{6\,c^3}+\frac {35\,a^3\,b^3\,x^4}{6\,c^3}+\frac {7\,a^2\,b^4\,x^5}{2\,c^3}\right ) \]

[In]

int(1/(c/(a + b*x)^2)^(5/2),x)

[Out]

(c/(a + b*x)^2)^(1/2)*((a^6*x)/c^3 + (b^6*x^7)/(6*c^3) + (7*a^5*b*x^2)/(2*c^3) + (7*a*b^5*x^6)/(6*c^3) + (35*a
^4*b^2*x^3)/(6*c^3) + (35*a^3*b^3*x^4)/(6*c^3) + (7*a^2*b^4*x^5)/(2*c^3))