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

Optimal. Leaf size=58 \[ \frac{a^2 \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a \sqrt{c x^2}}{b^2}+\frac{x \sqrt{c x^2}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0182118, antiderivative size = 58, normalized size of antiderivative = 1., 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} \[ \frac{a^2 \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a \sqrt{c x^2}}{b^2}+\frac{x \sqrt{c x^2}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[c*x^2])/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0113913, size = 40, normalized size = 0.69 \[ \frac{c x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 40, normalized size = 0.7 \begin{align*}{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx}{2\,{b}^{3}x}\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60804, size = 89, normalized size = 1.53 \begin{align*} \frac{{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \, b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sqrt{c x^{2}}}{a + b x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(c*x**2)/(a + b*x), x)

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Giac [A]  time = 1.07681, size = 73, normalized size = 1.26 \begin{align*} \frac{1}{2} \, \sqrt{c}{\left (\frac{2 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{3}} - \frac{2 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{3}} + \frac{b x^{2} \mathrm{sgn}\left (x\right ) - 2 \, a x \mathrm{sgn}\left (x\right )}{b^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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