3.189 \(\int \frac{1}{x^4 (a+b x^2)^3} \, dx\)

Optimal. Leaf size=87 \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{7}{8 a^2 x^3 \left (a+b x^2\right )}+\frac{35 b}{8 a^4 x}-\frac{35}{24 a^3 x^3}+\frac{1}{4 a x^3 \left (a+b x^2\right )^2} \]

[Out]

-35/(24*a^3*x^3) + (35*b)/(8*a^4*x) + 1/(4*a*x^3*(a + b*x^2)^2) + 7/(8*a^2*x^3*(a + b*x^2)) + (35*b^(3/2)*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))

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Rubi [A]  time = 0.0336019, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{7}{8 a^2 x^3 \left (a+b x^2\right )}+\frac{35 b}{8 a^4 x}-\frac{35}{24 a^3 x^3}+\frac{1}{4 a x^3 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^3),x]

[Out]

-35/(24*a^3*x^3) + (35*b)/(8*a^4*x) + 1/(4*a*x^3*(a + b*x^2)^2) + 7/(8*a^2*x^3*(a + b*x^2)) + (35*b^(3/2)*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^3} \, dx &=\frac{1}{4 a x^3 \left (a+b x^2\right )^2}+\frac{7 \int \frac{1}{x^4 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{1}{4 a x^3 \left (a+b x^2\right )^2}+\frac{7}{8 a^2 x^3 \left (a+b x^2\right )}+\frac{35 \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac{35}{24 a^3 x^3}+\frac{1}{4 a x^3 \left (a+b x^2\right )^2}+\frac{7}{8 a^2 x^3 \left (a+b x^2\right )}-\frac{(35 b) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{8 a^3}\\ &=-\frac{35}{24 a^3 x^3}+\frac{35 b}{8 a^4 x}+\frac{1}{4 a x^3 \left (a+b x^2\right )^2}+\frac{7}{8 a^2 x^3 \left (a+b x^2\right )}+\frac{\left (35 b^2\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^4}\\ &=-\frac{35}{24 a^3 x^3}+\frac{35 b}{8 a^4 x}+\frac{1}{4 a x^3 \left (a+b x^2\right )^2}+\frac{7}{8 a^2 x^3 \left (a+b x^2\right )}+\frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0408913, size = 79, normalized size = 0.91 \[ \frac{56 a^2 b x^2-8 a^3+175 a b^2 x^4+105 b^3 x^6}{24 a^4 x^3 \left (a+b x^2\right )^2}+\frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^3),x]

[Out]

(-8*a^3 + 56*a^2*b*x^2 + 175*a*b^2*x^4 + 105*b^3*x^6)/(24*a^4*x^3*(a + b*x^2)^2) + (35*b^(3/2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(8*a^(9/2))

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Maple [A]  time = 0.012, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{a}^{3}{x}^{3}}}+3\,{\frac{b}{{a}^{4}x}}+{\frac{11\,{b}^{3}{x}^{3}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}x}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{b}^{2}}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^3,x)

[Out]

-1/3/a^3/x^3+3*b/a^4/x+11/8/a^4*b^3/(b*x^2+a)^2*x^3+13/8/a^3*b^2/(b*x^2+a)^2*x+35/8/a^4*b^2/(a*b)^(1/2)*arctan
(b*x/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24974, size = 504, normalized size = 5.79 \begin{align*} \left [\frac{210 \, b^{3} x^{6} + 350 \, a b^{2} x^{4} + 112 \, a^{2} b x^{2} - 16 \, a^{3} + 105 \,{\left (b^{3} x^{7} + 2 \, a b^{2} x^{5} + a^{2} b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{48 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}, \frac{105 \, b^{3} x^{6} + 175 \, a b^{2} x^{4} + 56 \, a^{2} b x^{2} - 8 \, a^{3} + 105 \,{\left (b^{3} x^{7} + 2 \, a b^{2} x^{5} + a^{2} b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{24 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/48*(210*b^3*x^6 + 350*a*b^2*x^4 + 112*a^2*b*x^2 - 16*a^3 + 105*(b^3*x^7 + 2*a*b^2*x^5 + a^2*b*x^3)*sqrt(-b/
a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3), 1/24*(105*b^3*x^6 +
175*a*b^2*x^4 + 56*a^2*b*x^2 - 8*a^3 + 105*(b^3*x^7 + 2*a*b^2*x^5 + a^2*b*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)))
/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3)]

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Sympy [A]  time = 0.852244, size = 138, normalized size = 1.59 \begin{align*} - \frac{35 \sqrt{- \frac{b^{3}}{a^{9}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}}}{b^{2}} + x \right )}}{16} + \frac{35 \sqrt{- \frac{b^{3}}{a^{9}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}}}{b^{2}} + x \right )}}{16} + \frac{- 8 a^{3} + 56 a^{2} b x^{2} + 175 a b^{2} x^{4} + 105 b^{3} x^{6}}{24 a^{6} x^{3} + 48 a^{5} b x^{5} + 24 a^{4} b^{2} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**3,x)

[Out]

-35*sqrt(-b**3/a**9)*log(-a**5*sqrt(-b**3/a**9)/b**2 + x)/16 + 35*sqrt(-b**3/a**9)*log(a**5*sqrt(-b**3/a**9)/b
**2 + x)/16 + (-8*a**3 + 56*a**2*b*x**2 + 175*a*b**2*x**4 + 105*b**3*x**6)/(24*a**6*x**3 + 48*a**5*b*x**5 + 24
*a**4*b**2*x**7)

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Giac [A]  time = 2.12686, size = 96, normalized size = 1.1 \begin{align*} \frac{35 \, b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{4}} + \frac{11 \, b^{3} x^{3} + 13 \, a b^{2} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{4}} + \frac{9 \, b x^{2} - a}{3 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^3,x, algorithm="giac")

[Out]

35/8*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/8*(11*b^3*x^3 + 13*a*b^2*x)/((b*x^2 + a)^2*a^4) + 1/3*(9*b*
x^2 - a)/(a^4*x^3)