3.9.57 \(\int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx\) [857]

Optimal. Leaf size=42 \[ \frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \begin {gather*} \frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^2*(a + b*x)),x]

[Out]

(Sqrt[c*x^2]*Log[x])/(a*x) - (Sqrt[c*x^2]*Log[a + b*x])/(a*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.62 \begin {gather*} \frac {c x (\log (x)-\log (a+b x))}{a \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 26, normalized size = 0.62

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{a x}\)	\(26\)
risch	\(\frac {\sqrt {c \,x^{2}}\, \ln \left (-x \right )}{x a}-\frac {\ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a x}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 24, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {c} \log \left (b x + a\right )}{a} + \frac {\sqrt {c} \log \left (x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.42, size = 64, normalized size = 1.52 \begin {gather*} \left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(c*x^2)*log(x/(b*x + a))/(a*x), 2*sqrt(-c)*arctan(sqrt(c*x^2)*(2*b*x + a)*sqrt(-c)/(a*c*x))/a]

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

[Out]

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

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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^2/(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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

[Out]

int((c*x^2)^(1/2)/(x^2*(a + b*x)), x)

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