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

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

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Rubi [A]  time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 0.78 \begin {gather*} \frac {2 \left (-b^2+4 b c x+8 c^2 x^2\right )}{3 b^3 x \sqrt {x (b+c x)}} \end {gather*}

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)])

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IntegrateAlgebraic [A]  time = 0.27, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-b^2+4 b c x+8 c^2 x^2\right )}{3 b^3 x^2 (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 50, normalized size = 0.98 \begin {gather*} \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 {gather*}

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.04, size = 39, normalized size = 0.76 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-8 c^{2} x^{2}-4 b c x +b^{2}\right )}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3}} \end {gather*}

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)

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maxima [A]  time = 1.33, size = 57, normalized size = 1.12 \begin {gather*} \frac {16 \, c^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {8 \, c}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2}{3 \, \sqrt {c x^{2} + b x} b x} \end {gather*}

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]

16/3*c^2*x/(sqrt(c*x^2 + b*x)*b^3) + 8/3*c/(sqrt(c*x^2 + b*x)*b^2) - 2/3/(sqrt(c*x^2 + b*x)*b*x)

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mupad [B]  time = 0.24, size = 45, normalized size = 0.88 \begin {gather*} \frac {2\,\sqrt {c\,x^2+b\,x}\,\left (-b^2+4\,b\,c\,x+8\,c^2\,x^2\right )}{3\,b^3\,x^2\,\left (b+c\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x + c*x^2)^(3/2)),x)

[Out]

(2*(b*x + c*x^2)^(1/2)*(8*c^2*x^2 - b^2 + 4*b*c*x))/(3*b^3*x^2*(b + c*x))

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

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)

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