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

Optimal. Leaf size=39 \[ \frac{x^2}{b \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0121325, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ \frac{x^2}{b \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0079885, size = 27, normalized size = 0.69 \[ \frac{x (b x-a \log (a+b x))}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 27, normalized size = 0.7 \begin{align*} -{\frac{x \left ( a\ln \left ( bx+a \right ) -bx \right ) }{{b}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.79, size = 62, normalized size = 1.59 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08518, size = 68, normalized size = 1.74 \begin{align*} \frac{a \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{2} \sqrt{c}} + \frac{\sqrt{c x^{2}}}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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