\(\int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx\) [883]

Optimal result

Integrand size = 20, antiderivative size = 77 \[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx=\frac {b}{a^2 \sqrt {c x^2}}-\frac {1}{2 a x \sqrt {c x^2}}+\frac {b^2 x \log (x)}{a^3 \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 46} \[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx=\frac {b^2 x \log (x)}{a^3 \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 \sqrt {c x^2}}+\frac {b}{a^2 \sqrt {c x^2}}-\frac {1}{2 a x \sqrt {c x^2}} \]

[In]

Int[1/(x^2*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

b/(a^2*Sqrt[c*x^2]) - 1/(2*a*x*Sqrt[c*x^2]) + (b^2*x*Log[x])/(a^3*Sqrt[c*x^2]) - (b^2*x*Log[a + b*x])/(a^3*Sqr
t[c*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x^3 (a+b x)} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx}{\sqrt {c x^2}} \\ & = \frac {b}{a^2 \sqrt {c x^2}}-\frac {1}{2 a x \sqrt {c x^2}}+\frac {b^2 x \log (x)}{a^3 \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 \sqrt {c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx=-\frac {c \left (a x (a-2 b x)-2 b^2 x^3 \log (x)+2 b^2 x^3 \log (a+b x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

-1/2*(c*(a*x*(a - 2*b*x) - 2*b^2*x^3*Log[x] + 2*b^2*x^3*Log[a + b*x]))/(a^3*(c*x^2)^(3/2))

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}}{2 x \sqrt {c \,x^{2}}\, a^{3}}\)	\(51\)
risch	\(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{\sqrt {c \,x^{2}}\, x}+\frac {x \,b^{2} \ln \left (-x \right )}{\sqrt {c \,x^{2}}\, a^{3}}-\frac {b^{2} x \ln \left (b x +a \right )}{a^{3} \sqrt {c \,x^{2}}}\)	\(66\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx=\frac {{\left (2 \, b^{2} x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, a^{3} c x^{3}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx=-\frac {b^{2} \log \left (b x + a\right )}{a^{3} \sqrt {c}} + \frac {b^{2} \log \left (x\right )}{a^{3} \sqrt {c}} + \frac {2 \, b \sqrt {c} x - a \sqrt {c}}{2 \, a^{2} c x^{2}} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 \sqrt {c x^2} (a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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