3.68 \(\int \frac{A+B x^2}{x^6 (a+b x^2)} \, dx\)

Optimal. Leaf size=80 \[ -\frac{b^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B)}{a^3 x}-\frac{A}{5 a x^5} \]

[Out]

-A/(5*a*x^5) + (A*b - a*B)/(3*a^2*x^3) - (b*(A*b - a*B))/(a^3*x) - (b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqr
t[a]])/a^(7/2)

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Rubi [A]  time = 0.0500374, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {453, 325, 205} \[ -\frac{b^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B)}{a^3 x}-\frac{A}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^6*(a + b*x^2)),x]

[Out]

-A/(5*a*x^5) + (A*b - a*B)/(3*a^2*x^3) - (b*(A*b - a*B))/(a^3*x) - (b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqr
t[a]])/a^(7/2)

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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{A+B x^2}{x^6 \left (a+b x^2\right )} \, dx &=-\frac{A}{5 a x^5}-\frac{(5 A b-5 a B) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{5 a}\\ &=-\frac{A}{5 a x^5}+\frac{A b-a B}{3 a^2 x^3}+\frac{(b (A b-a B)) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac{A}{5 a x^5}+\frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B)}{a^3 x}-\frac{\left (b^2 (A b-a B)\right ) \int \frac{1}{a+b x^2} \, dx}{a^3}\\ &=-\frac{A}{5 a x^5}+\frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B)}{a^3 x}-\frac{b^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0518063, size = 78, normalized size = 0.98 \[ \frac{b^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{A b-a B}{3 a^2 x^3}+\frac{b (a B-A b)}{a^3 x}-\frac{A}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^6*(a + b*x^2)),x]

[Out]

-A/(5*a*x^5) + (A*b - a*B)/(3*a^2*x^3) + (b*(-(A*b) + a*B))/(a^3*x) + (b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*
x)/Sqrt[a]])/a^(7/2)

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Maple [A]  time = 0.006, size = 96, normalized size = 1.2 \begin{align*} -{\frac{A}{5\,a{x}^{5}}}+{\frac{Ab}{3\,{a}^{2}{x}^{3}}}-{\frac{B}{3\,a{x}^{3}}}-{\frac{{b}^{2}A}{{a}^{3}x}}+{\frac{bB}{{a}^{2}x}}-{\frac{A{b}^{3}}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}B}{{a}^{2}}\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((B*x^2+A)/x^6/(b*x^2+a),x)

[Out]

-1/5*A/a/x^5+1/3/a^2/x^3*A*b-1/3/a/x^3*B-1/a^3*b^2/x*A+1/a^2*b/x*B-b^3/a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
*A+b^2/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*B

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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((B*x^2+A)/x^6/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.18793, size = 398, normalized size = 4.97 \begin{align*} \left [-\frac{15 \,{\left (B a b - A b^{2}\right )} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 30 \,{\left (B a b - A b^{2}\right )} x^{4} + 6 \, A a^{2} + 10 \,{\left (B a^{2} - A a b\right )} x^{2}}{30 \, a^{3} x^{5}}, \frac{15 \,{\left (B a b - A b^{2}\right )} x^{5} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 15 \,{\left (B a b - A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \,{\left (B a^{2} - A a b\right )} x^{2}}{15 \, a^{3} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/30*(15*(B*a*b - A*b^2)*x^5*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 30*(B*a*b - A*b^2)
*x^4 + 6*A*a^2 + 10*(B*a^2 - A*a*b)*x^2)/(a^3*x^5), 1/15*(15*(B*a*b - A*b^2)*x^5*sqrt(b/a)*arctan(x*sqrt(b/a))
+ 15*(B*a*b - A*b^2)*x^4 - 3*A*a^2 - 5*(B*a^2 - A*a*b)*x^2)/(a^3*x^5)]

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Sympy [B]  time = 0.689464, size = 163, normalized size = 2.04 \begin{align*} - \frac{\sqrt{- \frac{b^{3}}{a^{7}}} \left (- A b + B a\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{3}}{a^{7}}} \left (- A b + B a\right )}{- A b^{3} + B a b^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{7}}} \left (- A b + B a\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{b^{3}}{a^{7}}} \left (- A b + B a\right )}{- A b^{3} + B a b^{2}} + x \right )}}{2} + \frac{- 3 A a^{2} + x^{4} \left (- 15 A b^{2} + 15 B a b\right ) + x^{2} \left (5 A a b - 5 B a^{2}\right )}{15 a^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**6/(b*x**2+a),x)

[Out]

-sqrt(-b**3/a**7)*(-A*b + B*a)*log(-a**4*sqrt(-b**3/a**7)*(-A*b + B*a)/(-A*b**3 + B*a*b**2) + x)/2 + sqrt(-b**
3/a**7)*(-A*b + B*a)*log(a**4*sqrt(-b**3/a**7)*(-A*b + B*a)/(-A*b**3 + B*a*b**2) + x)/2 + (-3*A*a**2 + x**4*(-
15*A*b**2 + 15*B*a*b) + x**2*(5*A*a*b - 5*B*a**2))/(15*a**3*x**5)

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Giac [A]  time = 1.16381, size = 109, normalized size = 1.36 \begin{align*} \frac{{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{15 \, B a b x^{4} - 15 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 5 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a),x, algorithm="giac")

[Out]

(B*a*b^2 - A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) + 1/15*(15*B*a*b*x^4 - 15*A*b^2*x^4 - 5*B*a^2*x^2 + 5*
A*a*b*x^2 - 3*A*a^2)/(a^3*x^5)