3.130 \(\int \frac {c+d x^2+e x^4+f x^6}{x^6 (a+b x^2)^2} \, dx\)

Optimal. Leaf size=152 \[ \frac {2 b c-a d}{3 a^3 x^3}-\frac {c}{5 a^2 x^5}-\frac {a^2 e-2 a b d+3 b^2 c}{a^4 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+3 a^2 b e-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt {b}}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )} \]

[Out]

-1/5*c/a^2/x^5+1/3*(-a*d+2*b*c)/a^3/x^3+(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x-1/2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^
4/(b*x^2+a)-1/2*(-a^3*f+3*a^2*b*e-5*a*b^2*d+7*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)/b^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1805, 1802, 205} \[ -\frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^2 b e+a^3 (-f)-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt {b}}-\frac {a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac {2 b c-a d}{3 a^3 x^3}-\frac {c}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^2),x]

[Out]

-c/(5*a^2*x^5) + (2*b*c - a*d)/(3*a^3*x^3) - (3*b^2*c - 2*a*b*d + a^2*e)/(a^4*x) - ((b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*x)/(2*a^4*(a + b*x^2)) - ((7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*
a^(9/2)*Sqrt[b])

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 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]

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 151, normalized size = 0.99 \[ \frac {2 b c-a d}{3 a^3 x^3}-\frac {c}{5 a^2 x^5}+\frac {a^2 (-e)+2 a b d-3 b^2 c}{a^4 x}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-3 a^2 b e+5 a b^2 d-7 b^3 c\right )}{2 a^{9/2} \sqrt {b}}+\frac {x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^4 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^2),x]

[Out]

-1/5*c/(a^2*x^5) + (2*b*c - a*d)/(3*a^3*x^3) + (-3*b^2*c + 2*a*b*d - a^2*e)/(a^4*x) + ((-(b^3*c) + a*b^2*d - a
^2*b*e + a^3*f)*x)/(2*a^4*(a + b*x^2)) + ((-7*b^3*c + 5*a*b^2*d - 3*a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a
]])/(2*a^(9/2)*Sqrt[b])

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fricas [A]  time = 0.71, size = 438, normalized size = 2.88 \[ \left [-\frac {30 \, {\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 12 \, a^{4} b c + 20 \, {\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 4 \, {\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} - 15 \, {\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{60 \, {\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}, -\frac {15 \, {\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 6 \, a^{4} b c + 10 \, {\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 2 \, {\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{30 \, {\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^6/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[-1/60*(30*(7*a*b^4*c - 5*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^6 + 12*a^4*b*c + 20*(7*a^2*b^3*c - 5*a^3*b^2*d
+ 3*a^4*b*e)*x^4 - 4*(7*a^3*b^2*c - 5*a^4*b*d)*x^2 - 15*((7*b^4*c - 5*a*b^3*d + 3*a^2*b^2*e - a^3*b*f)*x^7 + (
7*a*b^3*c - 5*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^5)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a
^5*b^2*x^7 + a^6*b*x^5), -1/30*(15*(7*a*b^4*c - 5*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^6 + 6*a^4*b*c + 10*(7*a
^2*b^3*c - 5*a^3*b^2*d + 3*a^4*b*e)*x^4 - 2*(7*a^3*b^2*c - 5*a^4*b*d)*x^2 + 15*((7*b^4*c - 5*a*b^3*d + 3*a^2*b
^2*e - a^3*b*f)*x^7 + (7*a*b^3*c - 5*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^5)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5
*b^2*x^7 + a^6*b*x^5)]

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giac [A]  time = 0.49, size = 151, normalized size = 0.99 \[ -\frac {{\left (7 \, b^{3} c - 5 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} - \frac {b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {45 \, b^{2} c x^{4} - 30 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^6/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(7*b^3*c - 5*a*b^2*d - a^3*f + 3*a^2*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/2*(b^3*c*x - a*b^2*d*
x - a^3*f*x + a^2*b*x*e)/((b*x^2 + a)*a^4) - 1/15*(45*b^2*c*x^4 - 30*a*b*d*x^4 + 15*a^2*x^4*e - 10*a*b*c*x^2 +
5*a^2*d*x^2 + 3*a^2*c)/(a^4*x^5)

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maple [A]  time = 0.02, size = 219, normalized size = 1.44 \[ \frac {f x}{2 \left (b \,x^{2}+a \right ) a}+\frac {f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {b e x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}+\frac {b^{2} d x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {b^{3} c x}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {7 b^{3} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{4}}-\frac {e}{a^{2} x}+\frac {2 b d}{a^{3} x}-\frac {3 b^{2} c}{a^{4} x}-\frac {d}{3 a^{2} x^{3}}+\frac {2 b c}{3 a^{3} x^{3}}-\frac {c}{5 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^6/(b*x^2+a)^2,x)

[Out]

1/2/a*x/(b*x^2+a)*f-1/2/a^2*x/(b*x^2+a)*b*e+1/2/a^3*x/(b*x^2+a)*b^2*d-1/2/a^4*x/(b*x^2+a)*b^3*c+1/2/a/(a*b)^(1
/2)*arctan(1/(a*b)^(1/2)*b*x)*f-3/2/a^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*b*e+5/2/a^3/(a*b)^(1/2)*arctan(1
/(a*b)^(1/2)*b*x)*b^2*d-7/2/a^4/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*b^3*c-1/5*c/a^2/x^5-1/3/a^2/x^3*d+2/3/a^
3/x^3*b*c-1/a^2/x*e+2/a^3/x*b*d-3/a^4/x*b^2*c

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maxima [A]  time = 2.99, size = 151, normalized size = 0.99 \[ -\frac {15 \, {\left (7 \, b^{3} c - 5 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} x^{6} + 10 \, {\left (7 \, a b^{2} c - 5 \, a^{2} b d + 3 \, a^{3} e\right )} x^{4} + 6 \, a^{3} c - 2 \, {\left (7 \, a^{2} b c - 5 \, a^{3} d\right )} x^{2}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} - \frac {{\left (7 \, b^{3} c - 5 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^6/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/30*(15*(7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*x^6 + 10*(7*a*b^2*c - 5*a^2*b*d + 3*a^3*e)*x^4 + 6*a^3*c -
2*(7*a^2*b*c - 5*a^3*d)*x^2)/(a^4*b*x^7 + a^5*x^5) - 1/2*(7*b^3*c - 5*a*b^2*d + 3*a^2*b*e - a^3*f)*arctan(b*x
/sqrt(a*b))/(sqrt(a*b)*a^4)

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mupad [B]  time = 1.00, size = 145, normalized size = 0.95 \[ -\frac {\frac {c}{5\,a}+\frac {x^6\,\left (-f\,a^3+3\,e\,a^2\,b-5\,d\,a\,b^2+7\,c\,b^3\right )}{2\,a^4}+\frac {x^2\,\left (5\,a\,d-7\,b\,c\right )}{15\,a^2}+\frac {x^4\,\left (3\,e\,a^2-5\,d\,a\,b+7\,c\,b^2\right )}{3\,a^3}}{b\,x^7+a\,x^5}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+3\,e\,a^2\,b-5\,d\,a\,b^2+7\,c\,b^3\right )}{2\,a^{9/2}\,\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2 + e*x^4 + f*x^6)/(x^6*(a + b*x^2)^2),x)

[Out]

- (c/(5*a) + (x^6*(7*b^3*c - a^3*f - 5*a*b^2*d + 3*a^2*b*e))/(2*a^4) + (x^2*(5*a*d - 7*b*c))/(15*a^2) + (x^4*(
7*b^2*c + 3*a^2*e - 5*a*b*d))/(3*a^3))/(a*x^5 + b*x^7) - (atan((b^(1/2)*x)/a^(1/2))*(7*b^3*c - a^3*f - 5*a*b^2
*d + 3*a^2*b*e))/(2*a^(9/2)*b^(1/2))

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sympy [A]  time = 32.02, size = 226, normalized size = 1.49 \[ - \frac {\sqrt {- \frac {1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log {\left (- a^{5} \sqrt {- \frac {1}{a^{9} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log {\left (a^{5} \sqrt {- \frac {1}{a^{9} b}} + x \right )}}{4} + \frac {- 6 a^{3} c + x^{6} \left (15 a^{3} f - 45 a^{2} b e + 75 a b^{2} d - 105 b^{3} c\right ) + x^{4} \left (- 30 a^{3} e + 50 a^{2} b d - 70 a b^{2} c\right ) + x^{2} \left (- 10 a^{3} d + 14 a^{2} b c\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**6/(b*x**2+a)**2,x)

[Out]

-sqrt(-1/(a**9*b))*(a**3*f - 3*a**2*b*e + 5*a*b**2*d - 7*b**3*c)*log(-a**5*sqrt(-1/(a**9*b)) + x)/4 + sqrt(-1/
(a**9*b))*(a**3*f - 3*a**2*b*e + 5*a*b**2*d - 7*b**3*c)*log(a**5*sqrt(-1/(a**9*b)) + x)/4 + (-6*a**3*c + x**6*
(15*a**3*f - 45*a**2*b*e + 75*a*b**2*d - 105*b**3*c) + x**4*(-30*a**3*e + 50*a**2*b*d - 70*a*b**2*c) + x**2*(-
10*a**3*d + 14*a**2*b*c))/(30*a**5*x**5 + 30*a**4*b*x**7)

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