3.1.84 \(\int \frac {A+B x^2}{x^4 (a+b x^2)^2} \, dx\)

Optimal. Leaf size=90 \[ \frac {\sqrt {b} (5 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}+\frac {b x (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{a^3 x}-\frac {A}{3 a^2 x^3} \]

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Rubi [A]  time = 0.11, antiderivative size = 90, 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, 1261, 205} \begin {gather*} \frac {b x (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{a^3 x}+\frac {\sqrt {b} (5 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {A}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]

[Out]

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

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 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 90, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b} (3 a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {b x (a B-A b)}{2 a^3 \left (a+b x^2\right )}+\frac {2 A b-a B}{a^3 x}-\frac {A}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]

[Out]

IntegrateAlgebraic[(A + B*x^2)/(x^4*(a + b*x^2)^2), x]

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fricas [A]  time = 0.46, size = 250, normalized size = 2.78 \begin {gather*} \left [-\frac {6 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2} + 3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (3 \, B a^{2} - 5 \, A a b\right )} 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 )}{12 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, -\frac {3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2} + 3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(6*(3*B*a*b - 5*A*b^2)*x^4 + 4*A*a^2 + 4*(3*B*a^2 - 5*A*a*b)*x^2 + 3*((3*B*a*b - 5*A*b^2)*x^5 + (3*B*a^
2 - 5*A*a*b)*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^3*b*x^5 + a^4*x^3), -1/6*(3*(
3*B*a*b - 5*A*b^2)*x^4 + 2*A*a^2 + 2*(3*B*a^2 - 5*A*a*b)*x^2 + 3*((3*B*a*b - 5*A*b^2)*x^5 + (3*B*a^2 - 5*A*a*b
)*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^3*b*x^5 + a^4*x^3)]

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giac [A]  time = 0.43, size = 85, normalized size = 0.94 \begin {gather*} -\frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} - \frac {B a b x - A b^{2} x}{2 \, {\left (b x^{2} + a\right )} a^{3}} - \frac {3 \, B a x^{2} - 6 \, A b x^{2} + A a}{3 \, a^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*B*a*b - 5*A*b^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) - 1/2*(B*a*b*x - A*b^2*x)/((b*x^2 + a)*a^3) - 1
/3*(3*B*a*x^2 - 6*A*b*x^2 + A*a)/(a^3*x^3)

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maple [A]  time = 0.02, size = 110, normalized size = 1.22 \begin {gather*} \frac {A \,b^{2} x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 A \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {B b x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 B b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}+\frac {2 A b}{a^{3} x}-\frac {B}{a^{2} x}-\frac {A}{3 a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.45, size = 93, normalized size = 1.03 \begin {gather*} -\frac {3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}}{6 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} - \frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.14, size = 83, normalized size = 0.92 \begin {gather*} \frac {\frac {x^2\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {A}{3\,a}+\frac {b\,x^4\,\left (5\,A\,b-3\,B\,a\right )}{2\,a^3}}{b\,x^5+a\,x^3}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (5\,A\,b-3\,B\,a\right )}{2\,a^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.59, size = 184, normalized size = 2.04 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right ) \log {\left (- \frac {a^{4} \sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right )}{- 5 A b^{2} + 3 B a b} + x \right )}}{4} - \frac {\sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right ) \log {\left (\frac {a^{4} \sqrt {- \frac {b}{a^{7}}} \left (- 5 A b + 3 B a\right )}{- 5 A b^{2} + 3 B a b} + x \right )}}{4} + \frac {- 2 A a^{2} + x^{4} \left (15 A b^{2} - 9 B a b\right ) + x^{2} \left (10 A a b - 6 B a^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-b/a**7)*(-5*A*b + 3*B*a)*log(-a**4*sqrt(-b/a**7)*(-5*A*b + 3*B*a)/(-5*A*b**2 + 3*B*a*b) + x)/4 - sqrt(-b
/a**7)*(-5*A*b + 3*B*a)*log(a**4*sqrt(-b/a**7)*(-5*A*b + 3*B*a)/(-5*A*b**2 + 3*B*a*b) + x)/4 + (-2*A*a**2 + x*
*4*(15*A*b**2 - 9*B*a*b) + x**2*(10*A*a*b - 6*B*a**2))/(6*a**4*x**3 + 6*a**3*b*x**5)

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