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

Optimal. Leaf size=59 \[ -\frac {x \log (a+b x)}{a^2 \sqrt {c x^2}}+\frac {x \log (x)}{a^2 \sqrt {c x^2}}+\frac {x}{a \sqrt {c x^2} (a+b x)} \]

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Rubi [A]  time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 44} \begin {gather*} -\frac {x \log (a+b x)}{a^2 \sqrt {c x^2}}+\frac {x \log (x)}{a^2 \sqrt {c x^2}}+\frac {x}{a \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

x/(a*Sqrt[c*x^2]*(a + b*x)) + (x*Log[x])/(a^2*Sqrt[c*x^2]) - (x*Log[a + b*x])/(a^2*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}{\sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{x (a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {x}{a \sqrt {c x^2} (a+b x)}+\frac {x \log (x)}{a^2 \sqrt {c x^2}}-\frac {x \log (a+b x)}{a^2 \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 0.75 \begin {gather*} \frac {x (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 57, normalized size = 0.97 \begin {gather*} \sqrt {c x^2} \left (-\frac {\log (a+b x)}{a^2 c x}+\frac {\log (x)}{a^2 c x}+\frac {1}{a c x (a+b x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 44, normalized size = 0.75 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (\frac {x}{b x + a}\right ) + a\right )}}{a^{2} b c x^{2} + a^{3} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.08, size = 86, normalized size = 1.46 \begin {gather*} -\frac {\log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} \sqrt {c} \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 \sqrt {c} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 50, normalized size = 0.85 \begin {gather*} \frac {\left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \ln \relax (x )-a \ln \left (b x +a \right )+a \right ) x}{\sqrt {c \,x^{2}}\, \left (b x +a \right ) a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.46, size = 61, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {c x^{2}} b}{a^{2} b c x + a^{3} c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{2} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x^2)^(1/2)*(a + b*x)^2),x)

[Out]

int(1/((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 \begin {gather*} \int \frac {1}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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