3.189 \(\int \frac{1}{x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{b^2 (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0522904, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 44} \[ -\frac{b^2 (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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

Mathematica [A]  time = 0.0220217, size = 72, normalized size = 0.4 \[ -\frac{(a+b x) \left (a \left (2 a^2-3 a b x+6 b^2 x^2\right )-6 b^3 x^3 \log (a+b x)+6 b^3 x^3 \log (x)\right )}{6 a^4 x^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.181, size = 69, normalized size = 0.4 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,a{b}^{2}{x}^{2}-3\,b{a}^{2}x+2\,{a}^{3} \right ) }{6\,{a}^{4}{x}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(b*x+a)*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*a*b^2*x^2-3*b*a^2*x+2*a^3)/((b*x+a)^2)^(1/2)/a^4/x^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(1/x^4/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73287, size = 126, normalized size = 0.7 \begin{align*} \frac{6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.725751, size = 44, normalized size = 0.24 \begin{align*} - \frac{2 a^{2} - 3 a b x + 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac{b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a**2 - 3*a*b*x + 6*b**2*x**2)/(6*a**3*x**3) + b**3*(-log(x) + log(a/b + x))/a**4

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Giac [A]  time = 1.36114, size = 88, normalized size = 0.49 \begin{align*} \frac{1}{6} \,{\left (\frac{6 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{6 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} - 3 \, a^{2} b x + 2 \, a^{3}}{a^{4} x^{3}}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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