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

Optimal. Leaf size=87 \[ -\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {\sqrt {c x^2}}{a^2 x^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 87, 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}}{a^2 x (a+b x)}-\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {\sqrt {c x^2}}{a^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^3*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]  time = 0.03, size = 57, normalized size = 0.66 \begin {gather*} -\frac {c (a (a+2 b x)+2 b x \log (x) (a+b x)-2 b x (a+b x) \log (a+b x))}{a^3 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*(a*(a + 2*b*x) + 2*b*x*(a + b*x)*Log[x] - 2*b*x*(a + b*x)*Log[a + b*x]))/(a^3*Sqrt[c*x^2]*(a + b*x)))

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IntegrateAlgebraic [A]  time = 0.07, size = 59, normalized size = 0.68 \begin {gather*} \sqrt {c x^2} \left (-\frac {2 b \log (x)}{a^3 x}+\frac {2 b \log (a+b x)}{a^3 x}+\frac {-a-2 b x}{a^2 x^2 (a+b x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[c*x^2]*((-a - 2*b*x)/(a^2*x^2*(a + b*x)) - (2*b*Log[x])/(a^3*x) + (2*b*Log[a + b*x])/(a^3*x))

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fricas [A]  time = 1.04, size = 60, normalized size = 0.69 \begin {gather*} -\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b x^{3} + a^{4} 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)^2,x, algorithm="fricas")

[Out]

-(2*a*b*x + a^2 - 2*(b^2*x^2 + a*b*x)*log((b*x + a)/x))*sqrt(c*x^2)/(a^3*b*x^3 + a^4*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)^2,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 = 74, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} x^{2} \ln \relax (x )-2 b^{2} x^{2} \ln \left (b x +a \right )+2 a b x \ln \relax (x )-2 a b x \ln \left (b x +a \right )+2 a b x +a^{2}\right )}{\left (b x +a \right ) a^{3} 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)^2,x)

[Out]

-(c*x^2)^(1/2)*(2*b^2*x^2*ln(x)-2*b^2*x^2*ln(b*x+a)+2*a*b*x*ln(x)-2*a*b*x*ln(b*x+a)+2*a*b*x+a^2)/x^2/a^3/(b*x+
a)

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maxima [A]  time = 1.40, size = 58, normalized size = 0.67 \begin {gather*} -\frac {2 \, b \sqrt {c} x + a \sqrt {c}}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \sqrt {c} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \sqrt {c} \log \relax (x)}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2}}{x^3\,{\left (a+b\,x\right )}^2} \,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)^2),x)

[Out]

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

[Out]

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

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