\(\int \frac {A+B x^2}{x^6 (a+b x^2)} \, dx\) [68]

Optimal result

Integrand size = 20, antiderivative size = 80 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=-\frac {A}{5 a x^5}+\frac {A b-a B}{3 a^2 x^3}-\frac {b (A b-a B)}{a^3 x}-\frac {b^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {464, 331, 211} \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=-\frac {b^{3/2} (A b-a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}}-\frac {b (A b-a B)}{a^3 x}+\frac {A b-a B}{3 a^2 x^3}-\frac {A}{5 a x^5} \]

[In]

Int[(A + B*x^2)/(x^6*(a + b*x^2)),x]

[Out]

-1/5*A/(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)/S
qrt[a]])/a^(7/2)

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}{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] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=-\frac {A}{5 a x^5}+\frac {A b-a B}{3 a^2 x^3}+\frac {b (-A b+a B)}{a^3 x}+\frac {b^{3/2} (-A b+a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2}} \]

[In]

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

[Out]

-1/5*A/(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)

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {A}{5 a \,x^{5}}-\frac {-A b +B a}{3 x^{3} a^{2}}-\frac {b \left (A b -B a \right )}{a^{3} x}-\frac {b^{2} \left (A b -B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\)	\(74\)
risch	\(\frac {-\frac {b \left (A b -B a \right ) x^{4}}{a^{3}}+\frac {\left (A b -B a \right ) x^{2}}{3 a^{2}}-\frac {A}{5 a}}{x^{5}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} \textit {\_Z}^{2}+A^{2} b^{5}-2 A B a \,b^{4}+B^{2} a^{2} b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{7}+2 A^{2} b^{5}-4 A B a \,b^{4}+2 B^{2} a^{2} b^{3}\right ) x +\left (A \,a^{4} b^{2}-B \,a^{5} b \right ) \textit {\_R} \right )\right )}{2}\)	\(145\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.30 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=\left [-\frac {15 \, {\left (B a b - A b^{2}\right )} x^{5} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 30 \, {\left (B a b - A b^{2}\right )} x^{4} + 6 \, A a^{2} + 10 \, {\left (B a^{2} - A a b\right )} x^{2}}{30 \, a^{3} x^{5}}, \frac {15 \, {\left (B a b - A b^{2}\right )} x^{5} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 15 \, {\left (B a b - A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \, {\left (B a^{2} - A a b\right )} x^{2}}{15 \, a^{3} x^{5}}\right ] \]

[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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (68) = 136\).

Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.04 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=- \frac {\sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right ) \log {\left (- \frac {a^{4} \sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right )}{- A b^{3} + B a b^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right ) \log {\left (\frac {a^{4} \sqrt {- \frac {b^{3}}{a^{7}}} \left (- A b + B a\right )}{- A b^{3} + B a b^{2}} + x \right )}}{2} + \frac {- 3 A a^{2} + x^{4} \left (- 15 A b^{2} + 15 B a b\right ) + x^{2} \cdot \left (5 A a b - 5 B a^{2}\right )}{15 a^{3} x^{5}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=\frac {{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {15 \, {\left (B a b - A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \, {\left (B a^{2} - A a b\right )} x^{2}}{15 \, a^{3} x^{5}} \]

[In]

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

[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 - A*b^2)*x^4 - 3*A*a^2 - 5*(B*a^2 -
A*a*b)*x^2)/(a^3*x^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=\frac {{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {15 \, B a b x^{4} - 15 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 5 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{3} x^{5}} \]

[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)

Mupad [B] (verification not implemented)

Time = 4.93 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )} \, dx=-\frac {\frac {A}{5\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{3\,a^2}+\frac {b\,x^4\,\left (A\,b-B\,a\right )}{a^3}}{x^5}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{7/2}} \]

[In]

int((A + B*x^2)/(x^6*(a + b*x^2)),x)

[Out]

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