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

Optimal. Leaf size=20 \[ \frac {x \log (a+b x)}{b \sqrt {c x^2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 31} \begin {gather*} \frac {x \log (a+b x)}{b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

(x*Log[a + b*x])/(b*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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \log (a+b x)}{b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[a + b*x])/(b*Sqrt[c*x^2])

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IntegrateAlgebraic [A]  time = 0.02, size = 25, normalized size = 1.25 \begin {gather*} \frac {\sqrt {c x^2} \log (a+b x)}{b c x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*c*x)

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fricas [A]  time = 0.63, size = 23, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c x^{2}} \log \left (b x + a\right )}{b c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*c*x)

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giac [A]  time = 0.90, size = 36, normalized size = 1.80 \begin {gather*} -\frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} \frac {x \ln \left (b x +a \right )}{\sqrt {c \,x^{2}}\, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.46, size = 46, normalized size = 2.30 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b \sqrt {c}} + \frac {\log \left (b x\right )}{b \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((c*x^2)^(1/2)*(a + b*x)),x)

[Out]

int(x/((c*x^2)^(1/2)*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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