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

Optimal. Leaf size=22 \[ \frac {\sqrt {c x^2} \log (a+b x)}{b x} \]

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 31} \begin {gather*} \frac {\sqrt {c x^2} \log (a+b x)}{b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x*(a + b*x)),x]

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.95 \begin {gather*} \frac {c x \log (a+b x)}{b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Log[a + b*x])/(b*Sqrt[c*x^2])

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IntegrateAlgebraic [A]  time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {\sqrt {c x^2} \log (a+b x)}{b x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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fricas [A]  time = 1.17, size = 20, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c x^{2}} \log \left (b x + a\right )}{b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*x)

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giac [A]  time = 0.95, size = 28, normalized size = 1.27 \begin {gather*} \sqrt {c} {\left (\frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b} - \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 21, normalized size = 0.95 \begin {gather*} \frac {\sqrt {c \,x^{2}}\, \ln \left (b x +a \right )}{b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)/(x*(a + b*x)),x)

[Out]

int((c*x^2)^(1/2)/(x*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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