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

Optimal. Leaf size=78 \[ -\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}}-\frac {b x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {1}{a^2 \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 44} \[ -\frac {b x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}}-\frac {1}{a^2 \sqrt {c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

-(1/(a^2*Sqrt[c*x^2])) - (b*x)/(a^2*Sqrt[c*x^2]*(a + b*x)) - (2*b*x*Log[x])/(a^3*Sqrt[c*x^2]) + (2*b*x*Log[a +
b*x])/(a^3*Sqrt[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 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 60, normalized size = 0.77 \[ \frac {c x^2 (-a (a+2 b x)-2 b x \log (x) (a+b x)+2 b x (a+b x) \log (a+b x))}{a^3 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 62, normalized size = 0.79 \[ -\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b c x^{3} + a^{4} c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.14, size = 126, normalized size = 1.62 \[ \frac {b {\left (\frac {2 \, \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {1}{{\left (b x + a\right )} a^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {1}{a^{3} {\left (\frac {a}{b x + a} - 1\right )} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}\right )}}{\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 71, normalized size = 0.91 \[ -\frac {2 b^{2} x^{2} \ln \relax (x )-2 b^{2} x^{2} \ln \left (b x +a \right )+2 a b x \ln \relax (x )-2 a b x \ln \left (b x +a \right )+2 a b x +a^{2}}{\sqrt {c \,x^{2}}\, \left (b x +a \right ) a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.43, size = 57, normalized size = 0.73 \[ -\frac {2 \, b x + a}{a^{2} b \sqrt {c} x^{2} + a^{3} \sqrt {c} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3} \sqrt {c}} - \frac {2 \, b \log \relax (x)}{a^{3} \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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