\(\int \frac {A+B x^2}{x^4 (a+b x^2)} \, dx\) [66]

Optimal result

Integrand size = 20, antiderivative size = 59 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {A}{3 a x^3}+\frac {A b-a B}{a^2 x}+\frac {\sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {464, 331, 211} \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {b} (A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A b-a B}{a^2 x}-\frac {A}{3 a x^3} \]

[In]

Int[(A + B*x^2)/(x^4*(a + b*x^2)),x]

[Out]

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

Rule 211

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

Rule 331

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 464

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]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {A}{3 a x^3}+\frac {A b-a B}{a^2 x}-\frac {\sqrt {b} (-A b+a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {A}{3 a \,x^{3}}-\frac {-A b +B a}{x \,a^{2}}+\frac {b \left (A b -B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\)	\(54\)
risch	\(\frac {\frac {\left (A b -B a \right ) x^{2}}{a^{2}}-\frac {A}{3 a}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{2}+A^{2} b^{3}-2 A B a \,b^{2}+B^{2} a^{2} b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{5}+2 A^{2} b^{3}-4 A B a \,b^{2}+2 B^{2} a^{2} b \right ) x +\left (-A \,a^{3} b +B \,a^{4}\right ) \textit {\_R} \right )\right )}{2}\)	\(120\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.29 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=\left [-\frac {3 \, {\left (B a - A b\right )} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, {\left (B a - A b\right )} x^{2} + 2 \, A a}{6 \, a^{2} x^{3}}, -\frac {3 \, {\left (B a - A b\right )} x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, {\left (B a - A b\right )} x^{2} + A a}{3 \, a^{2} x^{3}}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (49) = 98\).

Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.19 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right ) \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right ) \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} + \frac {- A a + x^{2} \cdot \left (3 A b - 3 B a\right )}{3 a^{2} x^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {{\left (B a b - A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {3 \, {\left (B a - A b\right )} x^{2} + A a}{3 \, a^{2} x^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {{\left (B a b - A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {3 \, B a x^{2} - 3 \, A b x^{2} + A a}{3 \, a^{2} x^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{5/2}}-\frac {\frac {A}{3\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{a^2}}{x^3} \]

[In]

int((A + B*x^2)/(x^4*(a + b*x^2)),x)

[Out]

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