\(\int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx\) [48]

Optimal result

Integrand size = 17, antiderivative size = 48 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {b x+c x^2}}{3 b x^2}+\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {4 c \sqrt {b x+c x^2}}{3 b^2 x}-\frac {2 \sqrt {b x+c x^2}}{3 b x^2} \]

[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 664

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 672

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*} \text {integral}& = -\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] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x (b+c x)} (-b+2 c x)}{3 b^2 x^2} \]

[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)

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.50

method	result	size
pseudoelliptic	\(-\frac {2 \left (-2 c x +b \right ) \sqrt {x \left (c x +b \right )}}{3 b^{2} x^{2}}\)	\(24\)
trager	\(-\frac {2 \left (-2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 b^{2} x^{2}}\)	\(26\)
risch	\(-\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 b^{2} x \sqrt {x \left (c x +b \right )}}\)	\(29\)
gosper	\(-\frac {2 \left (c x +b \right ) \left (-2 c x +b \right )}{3 x \,b^{2} \sqrt {c \,x^{2}+b x}}\)	\(31\)
default	\(-\frac {2 \sqrt {c \,x^{2}+b x}}{3 b \,x^{2}}+\frac {4 c \sqrt {c \,x^{2}+b x}}{3 b^{2} x}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x - b\right )}}{3 \, b^{2} x^{2}} \]

[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)

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\int \frac {1}{x^{2} \sqrt {x \left (b + c x\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {4 \, \sqrt {c x^{2} + b x} c}{3 \, b^{2} x} - \frac {2 \, \sqrt {c x^{2} + b x}}{3 \, b x^{2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 9.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx=-\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (b-2\,c\,x\right )}{3\,b^2\,x^2} \]

[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)