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

Optimal. Leaf size=65 \[ -\frac {a^2 \sqrt {c x^2}}{b^3 x (a+b x)}-\frac {2 a \sqrt {c x^2} \log (a+b x)}{b^3 x}+\frac {\sqrt {c x^2}}{b^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \begin {gather*} -\frac {a^2 \sqrt {c x^2}}{b^3 x (a+b x)}-\frac {2 a \sqrt {c x^2} \log (a+b x)}{b^3 x}+\frac {\sqrt {c x^2}}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[c*x^2])/(a + b*x)^2,x]

[Out]

Sqrt[c*x^2]/b^2 - (a^2*Sqrt[c*x^2])/(b^3*x*(a + b*x)) - (2*a*Sqrt[c*x^2]*Log[a + b*x])/(b^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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 53, normalized size = 0.82 \begin {gather*} \frac {c x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.06, size = 57, normalized size = 0.88 \begin {gather*} \sqrt {c x^2} \left (\frac {-a^2+a b x+b^2 x^2}{b^3 x (a+b x)}-\frac {2 a \log (a+b x)}{b^3 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 57, normalized size = 0.88 \begin {gather*} \frac {{\left (b^{2} x^{2} + a b x - a^{2} - 2 \, {\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{4} x^{2} + a b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*x^2 + a*b*x - a^2 - 2*(a*b*x + a^2)*log(b*x + a))*sqrt(c*x^2)/(b^4*x^2 + a*b^3*x)

________________________________________________________________________________________

giac [A]  time = 0.92, size = 58, normalized size = 0.89 \begin {gather*} \sqrt {c} {\left (\frac {x \mathrm {sgn}\relax (x)}{b^{2}} - \frac {2 \, a \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} + \frac {{\left (2 \, a \log \left ({\left | a \right |}\right ) + a\right )} \mathrm {sgn}\relax (x)}{b^{3}} - \frac {a^{2} \mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 62, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {c \,x^{2}}\, \left (2 a b x \ln \left (b x +a \right )-b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-a b x +a^{2}\right )}{\left (b x +a \right ) b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.51, size = 96, normalized size = 1.48 \begin {gather*} \frac {\sqrt {c x^{2}} a}{b^{3} x + a b^{2}} - \frac {2 \, \left (-1\right )^{\frac {2 \, c x}{b}} a \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{3}} - \frac {2 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3}} + \frac {\sqrt {c x^{2}}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*a/(b^3*x + a*b^2) - 2*(-1)^(2*c*x/b)*a*sqrt(c)*log(2*c*x/b)/b^3 - 2*(-1)^(2*a*c*x/b)*a*sqrt(c)*log
(-2*a*c*x/(b*abs(b*x + a)))/b^3 + sqrt(c*x^2)/b^2

________________________________________________________________________________________

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

[Out]

int((x*(c*x^2)^(1/2))/(a + b*x)^2, x)

________________________________________________________________________________________

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 )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________