3.1.48 \(\int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 48, 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, 650} \begin {gather*} \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} \end {gather*}

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 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]

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^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*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {x (b+c x)} (2 c x-b)}{3 b^2 x^2} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.14, size = 31, normalized size = 0.65 \begin {gather*} \frac {2 (2 c x-b) \sqrt {b x+c x^2}}{3 b^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 27, normalized size = 0.56 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x - b\right )}}{3 \, b^{2} x^{2}} \end {gather*}

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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giac [A]  time = 0.21, size = 49, normalized size = 1.02 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.04, size = 31, normalized size = 0.65 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 \sqrt {c \,x^{2}+b x}\, b^{2} x} \end {gather*}

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 [A]  time = 1.36, size = 40, normalized size = 0.83 \begin {gather*} \frac {4 \, \sqrt {c x^{2} + b x} c}{3 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x}}{3 \, b x^{2}} \end {gather*}

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]

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

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mupad [B]  time = 0.17, size = 25, normalized size = 0.52 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (b-2\,c\,x\right )}{3\,b^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(b*x + c*x^2)^(1/2)),x)

[Out]

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

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

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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