3.2818 \(\int \frac {1}{(c (a+b x)^3)^{5/2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \[ -\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a + b*x)^3)^(-5/2),x]

[Out]

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

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 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]

Rubi steps

\begin {align*} \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (c x^3\right )^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x)^{3/2} \operatorname {Subst}\left (\int \frac {1}{x^{15/2}} \, dx,x,a+b x\right )}{b c^2 \sqrt {c (a+b x)^3}}\\ &=-\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.83 \[ -\frac {2 (a+b x)}{13 b \left (c (a+b x)^3\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(a + b*x)^3)^(-5/2),x]

[Out]

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

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fricas [B]  time = 0.67, size = 151, normalized size = 5.03 \[ -\frac {2 \, \sqrt {b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{13 \, {\left (b^{9} c^{3} x^{8} + 8 \, a b^{8} c^{3} x^{7} + 28 \, a^{2} b^{7} c^{3} x^{6} + 56 \, a^{3} b^{6} c^{3} x^{5} + 70 \, a^{4} b^{5} c^{3} x^{4} + 56 \, a^{5} b^{4} c^{3} x^{3} + 28 \, a^{6} b^{3} c^{3} x^{2} + 8 \, a^{7} b^{2} c^{3} x + a^{8} b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*(b*x+a)^3)^(5/2),x, algorithm="fricas")

[Out]

-2/13*sqrt(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)/(b^9*c^3*x^8 + 8*a*b^8*c^3*x^7 + 28*a^2*b^7*c^3*x^
6 + 56*a^3*b^6*c^3*x^5 + 70*a^4*b^5*c^3*x^4 + 56*a^5*b^4*c^3*x^3 + 28*a^6*b^3*c^3*x^2 + 8*a^7*b^2*c^3*x + a^8*
b*c^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*(b*x+a)^3)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.00, size = 22, normalized size = 0.73 \[ -\frac {2 \left (b x +a \right )}{13 \left (\left (b x +a \right )^{3} c \right )^{\frac {5}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x+a)^3*c)^(5/2),x)

[Out]

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

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maxima [B]  time = 0.79, size = 85, normalized size = 2.83 \[ -\frac {2 \, \sqrt {c}}{13 \, {\left (b^{6} c^{3} x^{5} + 5 \, a b^{5} c^{3} x^{4} + 10 \, a^{2} b^{4} c^{3} x^{3} + 10 \, a^{3} b^{3} c^{3} x^{2} + 5 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )} {\left (b x + a\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*(b*x+a)^3)^(5/2),x, algorithm="maxima")

[Out]

-2/13*sqrt(c)/((b^6*c^3*x^5 + 5*a*b^5*c^3*x^4 + 10*a^2*b^4*c^3*x^3 + 10*a^3*b^3*c^3*x^2 + 5*a^4*b^2*c^3*x + a^
5*b*c^3)*(b*x + a)^(3/2))

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mupad [B]  time = 1.32, size = 26, normalized size = 0.87 \[ -\frac {2\,\sqrt {c\,{\left (a+b\,x\right )}^3}}{13\,b\,c^3\,{\left (a+b\,x\right )}^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*(a + b*x)^3)^(5/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \left (a + b x\right )^{3}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*(b*x+a)**3)**(5/2),x)

[Out]

Integral((c*(a + b*x)**3)**(-5/2), x)

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