### 3.48 $$\int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=48 $\frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2}$

[Out]

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

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Rubi [A]  time = 0.0163138, antiderivative size = 48, 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, 650} $\frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^2*Sqrt[b*x + c*x^2]),x]

[Out]

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

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 650

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))/((p + 1)*(2*c*d - b*e)), 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] && EqQ[m + 2*p + 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0130494, size = 29, normalized size = 0.6 $\frac{2 \sqrt{x (b+c x)} (2 c x-b)}{3 b^2 x^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 31, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+b \right ) }{3\,{b}^{2}x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01151, size = 61, normalized size = 1.27 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x - b\right )}}{3 \, b^{2} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22839, size = 66, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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