3.9.16 \(\int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx\)

Optimal. Leaf size=61 \[ -\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {\sqrt {c x^2}}{a x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 61, 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} \begin {gather*} -\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {\sqrt {c x^2}}{a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^3*(a + b*x)),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 32, normalized size = 0.52 \begin {gather*} -\frac {c (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x^2]/(x^3*(a + b*x)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 44, normalized size = 0.72 \begin {gather*} \sqrt {c x^2} \left (-\frac {b \log (x)}{a^2 x}+\frac {b \log (a+b x)}{a^2 x}-\frac {1}{a x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[c*x^2]/(x^3*(a + b*x)),x]

[Out]

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

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fricas [A]  time = 0.77, size = 31, normalized size = 0.51 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Sign error (%%%{a,0%%%}+%%%{b,1%%%})

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maple [A]  time = 0.01, size = 33, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {c \,x^{2}}\, \left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \right )}{a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)/x^3/(b*x+a),x)

[Out]

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

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maxima [A]  time = 1.38, size = 37, normalized size = 0.61 \begin {gather*} \frac {b \sqrt {c} \log \left (b x + a\right )}{a^{2}} - \frac {b \sqrt {c} \log \relax (x)}{a^{2}} - \frac {\sqrt {c}}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^3/(b*x+a),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x^2}}{x^3\,\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^3*(a + b*x)),x)

[Out]

int((c*x^2)^(1/2)/(x^3*(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^{3} \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**3/(b*x+a),x)

[Out]

Integral(sqrt(c*x**2)/(x**3*(a + b*x)), x)

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