3.125 \(\int \frac{A+B x}{x (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{2 (b+2 c x) (3 b B-4 A c)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2 A}{3 b x \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.032367, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {792, 613} \[ -\frac{2 (b+2 c x) (3 b B-4 A c)}{3 b^3 \sqrt{b x+c x^2}}-\frac{2 A}{3 b x \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x*(b*x + c*x^2)^(3/2)),x]

[Out]

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

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 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{A+B x}{x \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 A}{3 b x \sqrt{b x+c x^2}}+\frac{\left (2 \left (b B-A c+\frac{1}{2} (b B-2 A c)\right )\right ) \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{2 A}{3 b x \sqrt{b x+c x^2}}-\frac{2 (3 b B-4 A c) (b+2 c x)}{3 b^3 \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.020922, size = 52, normalized size = 0.87 \[ -\frac{2 \left (A \left (b^2-4 b c x-8 c^2 x^2\right )+3 b B x (b+2 c x)\right )}{3 b^3 x \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 58, normalized size = 1. \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -8\,A{c}^{2}{x}^{2}+6\,B{x}^{2}bc-4\,Abcx+3\,{b}^{2}Bx+A{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((B*x+A)/x/(c*x^2+b*x)^(3/2),x)

[Out]

-2/3*(c*x+b)*(-8*A*c^2*x^2+6*B*b*c*x^2-4*A*b*c*x+3*B*b^2*x+A*b^2)/b^3/(c*x^2+b*x)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.87582, size = 143, normalized size = 2.38 \begin{align*} -\frac{2 \,{\left (A b^{2} + 2 \,{\left (3 \, B b c - 4 \, A c^{2}\right )} x^{2} +{\left (3 \, B b^{2} - 4 \, A b c\right )} x\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((B*x+A)/x/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{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((B*x+A)/x/(c*x**2+b*x)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\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((B*x+A)/x/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

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