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

Optimal. Leaf size=113 \[ -\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}-\frac {b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b (3 A b-2 a B)}{a^4 x}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{5 a^2 x^5} \]

[Out]

-1/5*A/a^2/x^5+1/3*(2*A*b-B*a)/a^3/x^3-b*(3*A*b-2*B*a)/a^4/x-1/2*b^2*(A*b-B*a)*x/a^4/(b*x^2+a)-1/2*b^(3/2)*(7*
A*b-5*B*a)*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)

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Rubi [A]  time = 0.19, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {456, 1802, 205} \[ -\frac {b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {A}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(5*a^2*x^5) + (2*A*b - a*B)/(3*a^3*x^3) - (b*(3*A*b - 2*a*B))/(a^4*x) - (b^2*(A*b - a*B)*x)/(2*a^4*(a + b*x
^2)) - (b^(3/2)*(7*A*b - 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))

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]

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^2} \, dx &=-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {1}{2} b^2 \int \frac {-\frac {2 A}{a b^2}+\frac {2 (A b-a B) x^2}{a^2 b^2}-\frac {2 (A b-a B) x^4}{a^3 b}+\frac {(A b-a B) x^6}{a^4}}{x^6 \left (a+b x^2\right )} \, dx\\ &=-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {1}{2} b^2 \int \left (-\frac {2 A}{a^2 b^2 x^6}-\frac {2 (-2 A b+a B)}{a^3 b^2 x^4}+\frac {2 (-3 A b+2 a B)}{a^4 b x^2}+\frac {7 A b-5 a B}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {A}{5 a^2 x^5}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {\left (b^2 (7 A b-5 a B)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^4}\\ &=-\frac {A}{5 a^2 x^5}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 112, normalized size = 0.99 \[ \frac {b^{3/2} (5 a B-7 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2}}+\frac {b^2 x (a B-A b)}{2 a^4 \left (a+b x^2\right )}+\frac {b (2 a B-3 A b)}{a^4 x}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*A/(a^2*x^5) + (2*A*b - a*B)/(3*a^3*x^3) + (b*(-3*A*b + 2*a*B))/(a^4*x) + (b^2*(-(A*b) + a*B)*x)/(2*a^4*(a
+ b*x^2)) + (b^(3/2)*(-7*A*b + 5*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2))

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fricas [A]  time = 0.46, size = 308, normalized size = 2.73 \[ \left [\frac {30 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 12 \, A a^{3} - 4 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} - 15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{60 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, \frac {15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(30*(5*B*a*b^2 - 7*A*b^3)*x^6 + 20*(5*B*a^2*b - 7*A*a*b^2)*x^4 - 12*A*a^3 - 4*(5*B*a^3 - 7*A*a^2*b)*x^2
- 15*((5*B*a*b^2 - 7*A*b^3)*x^7 + (5*B*a^2*b - 7*A*a*b^2)*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(
b*x^2 + a)))/(a^4*b*x^7 + a^5*x^5), 1/30*(15*(5*B*a*b^2 - 7*A*b^3)*x^6 + 10*(5*B*a^2*b - 7*A*a*b^2)*x^4 - 6*A*
a^3 - 2*(5*B*a^3 - 7*A*a^2*b)*x^2 + 15*((5*B*a*b^2 - 7*A*b^3)*x^7 + (5*B*a^2*b - 7*A*a*b^2)*x^5)*sqrt(b/a)*arc
tan(x*sqrt(b/a)))/(a^4*b*x^7 + a^5*x^5)]

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giac [A]  time = 0.30, size = 112, normalized size = 0.99 \[ \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} + \frac {B a b^{2} x - A b^{3} x}{2 \, {\left (b x^{2} + a\right )} a^{4}} + \frac {30 \, B a b x^{4} - 45 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(5*B*a*b^2 - 7*A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/2*(B*a*b^2*x - A*b^3*x)/((b*x^2 + a)*a^4)
+ 1/15*(30*B*a*b*x^4 - 45*A*b^2*x^4 - 5*B*a^2*x^2 + 10*A*a*b*x^2 - 3*A*a^2)/(a^4*x^5)

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maple [A]  time = 0.02, size = 136, normalized size = 1.20 \[ -\frac {A \,b^{3} x}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {7 A \,b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{4}}+\frac {B \,b^{2} x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 B \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {3 A \,b^{2}}{a^{4} x}+\frac {2 B b}{a^{3} x}+\frac {2 A b}{3 a^{3} x^{3}}-\frac {B}{3 a^{2} x^{3}}-\frac {A}{5 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^4*b^3*x/(b*x^2+a)*A+1/2/a^3*b^2*x/(b*x^2+a)*B-7/2/a^4*b^3/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*A+5/2/a
^3*b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*B-1/5*A/a^2/x^5+2/3/a^3/x^3*A*b-1/3/a^2/x^3*B-3*b^2/a^4/x*A+2*b/a
^3/x*B

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maxima [A]  time = 2.31, size = 119, normalized size = 1.05 \[ \frac {15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} + \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(15*(5*B*a*b^2 - 7*A*b^3)*x^6 + 10*(5*B*a^2*b - 7*A*a*b^2)*x^4 - 6*A*a^3 - 2*(5*B*a^3 - 7*A*a^2*b)*x^2)/(
a^4*b*x^7 + a^5*x^5) + 1/2*(5*B*a*b^2 - 7*A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4)

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mupad [B]  time = 0.15, size = 104, normalized size = 0.92 \[ -\frac {\frac {A}{5\,a}-\frac {x^2\,\left (7\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {b^2\,x^6\,\left (7\,A\,b-5\,B\,a\right )}{2\,a^4}+\frac {b\,x^4\,\left (7\,A\,b-5\,B\,a\right )}{3\,a^3}}{b\,x^7+a\,x^5}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (7\,A\,b-5\,B\,a\right )}{2\,a^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (A/(5*a) - (x^2*(7*A*b - 5*B*a))/(15*a^2) + (b^2*x^6*(7*A*b - 5*B*a))/(2*a^4) + (b*x^4*(7*A*b - 5*B*a))/(3*a
^3))/(a*x^5 + b*x^7) - (b^(3/2)*atan((b^(1/2)*x)/a^(1/2))*(7*A*b - 5*B*a))/(2*a^(9/2))

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sympy [B]  time = 0.68, size = 218, normalized size = 1.93 \[ - \frac {\sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac {- 6 A a^{3} + x^{6} \left (- 105 A b^{3} + 75 B a b^{2}\right ) + x^{4} \left (- 70 A a b^{2} + 50 B a^{2} b\right ) + x^{2} \left (14 A a^{2} b - 10 B a^{3}\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b**3/a**9)*(-7*A*b + 5*B*a)*log(-a**5*sqrt(-b**3/a**9)*(-7*A*b + 5*B*a)/(-7*A*b**3 + 5*B*a*b**2) + x)/4
+ sqrt(-b**3/a**9)*(-7*A*b + 5*B*a)*log(a**5*sqrt(-b**3/a**9)*(-7*A*b + 5*B*a)/(-7*A*b**3 + 5*B*a*b**2) + x)/
4 + (-6*A*a**3 + x**6*(-105*A*b**3 + 75*B*a*b**2) + x**4*(-70*A*a*b**2 + 50*B*a**2*b) + x**2*(14*A*a**2*b - 10
*B*a**3))/(30*a**5*x**5 + 30*a**4*b*x**7)

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