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

Optimal. Leaf size=51 \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]

[Out]

-2/(3*b*x*Sqrt[b*x + c*x^2]) + (8*c*(b + 2*c*x))/(3*b^3*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0129913, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 613} \[ \frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2}{3 b x \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(b*x + c*x^2)^(3/2)),x]

[Out]

-2/(3*b*x*Sqrt[b*x + c*x^2]) + (8*c*(b + 2*c*x))/(3*b^3*Sqrt[b*x + c*x^2])

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{3 b x \sqrt{b x+c x^2}}-\frac{(4 c) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2}{3 b x \sqrt{b x+c x^2}}+\frac{8 c (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.011478, size = 40, normalized size = 0.78 \[ \frac{2 \left (-b^2+4 b c x+8 c^2 x^2\right )}{3 b^3 x \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(b*x + c*x^2)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 39, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -8\,{c}^{2}{x}^{2}-4\,bcx+{b}^{2} \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(c*x+b)*(-8*c^2*x^2-4*b*c*x+b^2)/b^3/(c*x^2+b*x)^(3/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.90221, size = 99, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (8 \, c^{2} x^{2} + 4 \, b c x - b^{2}\right )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{3} c x^{3} + b^{4} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(8*c^2*x^2 + 4*b*c*x - b^2)*sqrt(c*x^2 + b*x)/(b^3*c*x^3 + b^4*x^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*(x*(b + c*x))**(3/2)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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