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

Optimal. Leaf size=85 \[ \frac{a^3 \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{2 a \sqrt{c x^2}}{b^3}+\frac{x \sqrt{c x^2}}{2 b^2} \]

[Out]

(-2*a*Sqrt[c*x^2])/b^3 + (x*Sqrt[c*x^2])/(2*b^2) + (a^3*Sqrt[c*x^2])/(b^4*x*(a + b*x)) + (3*a^2*Sqrt[c*x^2]*Lo
g[a + b*x])/(b^4*x)

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Rubi [A]  time = 0.032175, antiderivative size = 85, 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} \[ \frac{a^3 \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{2 a \sqrt{c x^2}}{b^3}+\frac{x \sqrt{c x^2}}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*Sqrt[c*x^2])/(a + b*x)^2,x]

[Out]

(-2*a*Sqrt[c*x^2])/b^3 + (x*Sqrt[c*x^2])/(2*b^2) + (a^3*Sqrt[c*x^2])/(b^4*x*(a + b*x)) + (3*a^2*Sqrt[c*x^2]*Lo
g[a + b*x])/(b^4*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^2 \sqrt{c x^2}}{(a+b x)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{x^3}{(a+b x)^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{2 a \sqrt{c x^2}}{b^3}+\frac{x \sqrt{c x^2}}{2 b^2}+\frac{a^3 \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}\\ \end{align*}

Mathematica [A]  time = 0.020186, size = 70, normalized size = 0.82 \[ \frac{c x \left (-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)+2 a^3-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(2*a^3 - 4*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 6*a^2*(a + b*x)*Log[a + b*x]))/(2*b^4*Sqrt[c*x^2]*(a + b*x))

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Maple [A]  time = 0.009, size = 76, normalized size = 0.9 \begin{align*}{\frac{{b}^{3}{x}^{3}+6\,\ln \left ( bx+a \right ) x{a}^{2}b-3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -4\,{a}^{2}bx+2\,{a}^{3}}{2\,x{b}^{4} \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2)^(1/2)*(b^3*x^3+6*ln(b*x+a)*x*a^2*b-3*a*b^2*x^2+6*a^3*ln(b*x+a)-4*a^2*b*x+2*a^3)/x/b^4/(b*x+a)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53493, size = 154, normalized size = 1.81 \begin{align*} \frac{{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \,{\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (b^{5} x^{2} + a b^{4} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*x^3 - 3*a*b^2*x^2 - 4*a^2*b*x + 2*a^3 + 6*(a^2*b*x + a^3)*log(b*x + a))*sqrt(c*x^2)/(b^5*x^2 + a*b^4*
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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.04986, size = 108, normalized size = 1.27 \begin{align*} \frac{1}{2} \, \sqrt{c}{\left (\frac{6 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{4}} + \frac{2 \, a^{3} \mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b^{4}} - \frac{2 \,{\left (3 \, a^{2} \log \left ({\left | a \right |}\right ) + a^{2}\right )} \mathrm{sgn}\left (x\right )}{b^{4}} + \frac{b^{2} x^{2} \mathrm{sgn}\left (x\right ) - 4 \, a b x \mathrm{sgn}\left (x\right )}{b^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c)*(6*a^2*log(abs(b*x + a))*sgn(x)/b^4 + 2*a^3*sgn(x)/((b*x + a)*b^4) - 2*(3*a^2*log(abs(a)) + a^2)*s
gn(x)/b^4 + (b^2*x^2*sgn(x) - 4*a*b*x*sgn(x))/b^4)