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

Optimal. Leaf size=44 \[ \frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0080417, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 36, 29, 31} \[ \frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0071789, size = 27, normalized size = 0.61 \[ \frac{x^3 (\log (x)-\log (a+b x))}{a \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 26, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( \ln \left ( x \right ) -\ln \left ( bx+a \right ) \right ) }{a} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(c*x^2)^(3/2)/(b*x+a),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.54608, size = 153, normalized size = 3.48 \begin{align*} \left [\frac{\sqrt{c x^{2}} \log \left (\frac{x}{b x + a}\right )}{a c^{2} 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 c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.08139, size = 85, normalized size = 1.93 \begin{align*} \frac{\frac{\log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{a \sqrt{c}} - \frac{\log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{a \sqrt{c}}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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