3.900 \(\int \frac{\sqrt{c x^2}}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=112 \[ \frac{b^2 \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{3 b^2 \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{2 b \sqrt{c x^2}}{a^3 x^2}-\frac{\sqrt{c x^2}}{2 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0356999, antiderivative size = 112, 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, 44} \[ \frac{b^2 \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{3 b^2 \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{2 b \sqrt{c x^2}}{a^3 x^2}-\frac{\sqrt{c x^2}}{2 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^4*(a + b*x)^2),x]

[Out]

-Sqrt[c*x^2]/(2*a^2*x^3) + (2*b*Sqrt[c*x^2])/(a^3*x^2) + (b^2*Sqrt[c*x^2])/(a^3*x*(a + b*x)) + (3*b^2*Sqrt[c*x
^2]*Log[x])/(a^4*x) - (3*b^2*Sqrt[c*x^2]*Log[a + b*x])/(a^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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{c x^2}}{x^4 (a+b x)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{x^3 (a+b x)^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{\sqrt{c x^2}}{2 a^2 x^3}+\frac{2 b \sqrt{c x^2}}{a^3 x^2}+\frac{b^2 \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{3 b^2 \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 \sqrt{c x^2} \log (a+b x)}{a^4 x}\\ \end{align*}

Mathematica [A]  time = 0.0271562, size = 82, normalized size = 0.73 \[ \frac{\sqrt{c x^2} \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x^2]/(x^4*(a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 95, normalized size = 0.9 \begin{align*}{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,\ln \left ( x \right ){x}^{2}a{b}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx-{a}^{3}}{2\,{x}^{3}{a}^{4} \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)/x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.994223, size = 107, normalized size = 0.96 \begin{align*} \frac{6 \, b^{2} \sqrt{c} x^{2} + 3 \, a b \sqrt{c} x - a^{2} \sqrt{c}}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac{3 \, b^{2} \sqrt{c} \log \left (b x + a\right )}{a^{4}} + \frac{3 \, b^{2} \sqrt{c} \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.34271, size = 154, normalized size = 1.38 \begin{align*} \frac{{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac{x}{b x + a}\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(6*a*b^2*x^2 + 3*a^2*b*x - a^3 + 6*(b^3*x^3 + a*b^2*x^2)*log(x/(b*x + a)))*sqrt(c*x^2)/(a^4*b*x^4 + a^5*x^
3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2}}}{x^{4} \left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(1/2)/x**4/(b*x+a)**2,x)

[Out]

Integral(sqrt(c*x**2)/(x**4*(a + b*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(1/2)/x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError