3.2.28 \(\int \frac {c+d x^2+e x^4+f x^6}{x^{10} (a+b x^2)^2} \, dx\)

Optimal. Leaf size=230 \[ \frac {2 b c-a d}{7 a^3 x^7}-\frac {c}{9 a^2 x^9}-\frac {a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-5 a^3 f+7 a^2 b e-9 a b^2 d+11 b^3 c\right )}{2 a^{13/2}}-\frac {b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}-\frac {b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3} \]

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Rubi [A]  time = 0.38, antiderivative size = 230, 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} \begin {gather*} -\frac {b^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}+\frac {2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 x^3}-\frac {b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6 x}-\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (7 a^2 b e-5 a^3 f-9 a b^2 d+11 b^3 c\right )}{2 a^{13/2}}-\frac {a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {c}{9 a^2 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]

[Out]

-c/(9*a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) - (3*b^2*c - 2*a*b*d + a^2*e)/(5*a^4*x^5) + (4*b^3*c - 3*a*b^2*d +
2*a^2*b*e - a^3*f)/(3*a^5*x^3) - (b*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^6*x) - (b^2*(b^3*c - a*b^2
*d + a^2*b*e - a^3*f)*x)/(2*a^6*(a + b*x^2)) - (b^(3/2)*(11*b^3*c - 9*a*b^2*d + 7*a^2*b*e - 5*a^3*f)*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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 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^{10} \left (a+b x^2\right )^2} \, dx &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \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 {2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac {2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}+\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{10}}{a^5}}{x^{10} \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 c}{a x^{10}}-\frac {2 (-2 b c+a d)}{a^2 x^8}-\frac {2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^6}-\frac {2 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^4}+\frac {2 b \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^5 x^2}-\frac {b^2 \left (-11 b^3 c+9 a b^2 d-7 a^2 b e+5 a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac {c}{9 a^2 x^9}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac {\left (b^2 \left (11 b^3 c-9 a b^2 d+7 a^2 b e-5 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^6}\\ &=-\frac {c}{9 a^2 x^9}+\frac {2 b c-a d}{7 a^3 x^7}-\frac {3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac {b^{3/2} \left (11 b^3 c-9 a b^2 d+7 a^2 b e-5 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 230, normalized size = 1.00 \begin {gather*} \frac {2 b c-a d}{7 a^3 x^7}-\frac {c}{9 a^2 x^9}+\frac {a^2 (-e)+2 a b d-3 b^2 c}{5 a^4 x^5}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (5 a^3 f-7 a^2 b e+9 a b^2 d-11 b^3 c\right )}{2 a^{13/2}}+\frac {b^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^6 \left (a+b x^2\right )}+\frac {b \left (2 a^3 f-3 a^2 b e+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]

[Out]

-1/9*c/(a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) + (-3*b^2*c + 2*a*b*d - a^2*e)/(5*a^4*x^5) + (4*b^3*c - 3*a*b^2*d
+ 2*a^2*b*e - a^3*f)/(3*a^5*x^3) + (b*(-5*b^3*c + 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c)
+ a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^6*(a + b*x^2)) + (b^(3/2)*(-11*b^3*c + 9*a*b^2*d - 7*a^2*b*e + 5*a^3*f)*A
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]

[Out]

IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2), x]

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fricas [A]  time = 0.87, size = 582, normalized size = 2.53 \begin {gather*} \left [-\frac {630 \, {\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 420 \, {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 84 \, {\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 140 \, a^{5} c + 36 \, {\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 20 \, {\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2} + 315 \, {\left ({\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{11} + {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{9}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{1260 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}, -\frac {315 \, {\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 210 \, {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 42 \, {\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 70 \, a^{5} c + 18 \, {\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 10 \, {\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2} + 315 \, {\left ({\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{11} + {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{9}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{630 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[-1/1260*(630*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 + 420*(11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*
b^2*e - 5*a^4*b*f)*x^8 - 84*(11*a^2*b^3*c - 9*a^3*b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 140*a^5*c + 36*(11*a^3*b^
2*c - 9*a^4*b*d + 7*a^5*e)*x^4 - 20*(11*a^4*b*c - 9*a^5*d)*x^2 + 315*((11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*
a^3*b^2*f)*x^11 + (11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^9)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt
(-b/a) - a)/(b*x^2 + a)))/(a^6*b*x^11 + a^7*x^9), -1/630*(315*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*
f)*x^10 + 210*(11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^8 - 42*(11*a^2*b^3*c - 9*a^3*b^2*d + 7*a^
4*b*e - 5*a^5*f)*x^6 + 70*a^5*c + 18*(11*a^3*b^2*c - 9*a^4*b*d + 7*a^5*e)*x^4 - 10*(11*a^4*b*c - 9*a^5*d)*x^2
+ 315*((11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^11 + (11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a
^4*b*f)*x^9)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^6*b*x^11 + a^7*x^9)]

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giac [A]  time = 0.38, size = 252, normalized size = 1.10 \begin {gather*} -\frac {{\left (11 \, b^{5} c - 9 \, a b^{4} d - 5 \, a^{3} b^{2} f + 7 \, a^{2} b^{3} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{6}} - \frac {b^{5} c x - a b^{4} d x - a^{3} b^{2} f x + a^{2} b^{3} x e}{2 \, {\left (b x^{2} + a\right )} a^{6}} - \frac {1575 \, b^{4} c x^{8} - 1260 \, a b^{3} d x^{8} - 630 \, a^{3} b f x^{8} + 945 \, a^{2} b^{2} x^{8} e - 420 \, a b^{3} c x^{6} + 315 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 210 \, a^{3} b x^{6} e + 189 \, a^{2} b^{2} c x^{4} - 126 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 90 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{6} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(11*b^5*c - 9*a*b^4*d - 5*a^3*b^2*f + 7*a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6) - 1/2*(b^5*c*x -
a*b^4*d*x - a^3*b^2*f*x + a^2*b^3*x*e)/((b*x^2 + a)*a^6) - 1/315*(1575*b^4*c*x^8 - 1260*a*b^3*d*x^8 - 630*a^3
*b*f*x^8 + 945*a^2*b^2*x^8*e - 420*a*b^3*c*x^6 + 315*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 210*a^3*b*x^6*e + 189*a^2
*b^2*c*x^4 - 126*a^3*b*d*x^4 + 63*a^4*x^4*e - 90*a^3*b*c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^6*x^9)

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maple [A]  time = 0.02, size = 318, normalized size = 1.38 \begin {gather*} \frac {b^{2} f x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{2} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {b^{3} e x}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {7 b^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{4}}+\frac {b^{4} d x}{2 \left (b \,x^{2}+a \right ) a^{5}}+\frac {9 b^{4} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{5}}-\frac {b^{5} c x}{2 \left (b \,x^{2}+a \right ) a^{6}}-\frac {11 b^{5} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{6}}+\frac {2 b f}{a^{3} x}-\frac {3 b^{2} e}{a^{4} x}+\frac {4 b^{3} d}{a^{5} x}-\frac {5 b^{4} c}{a^{6} x}-\frac {f}{3 a^{2} x^{3}}+\frac {2 b e}{3 a^{3} x^{3}}-\frac {b^{2} d}{a^{4} x^{3}}+\frac {4 b^{3} c}{3 a^{5} x^{3}}-\frac {e}{5 a^{2} x^{5}}+\frac {2 b d}{5 a^{3} x^{5}}-\frac {3 b^{2} c}{5 a^{4} x^{5}}-\frac {d}{7 a^{2} x^{7}}+\frac {2 b c}{7 a^{3} x^{7}}-\frac {c}{9 a^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^2,x)

[Out]

1/2/a^3*b^2*x/(b*x^2+a)*f-1/2/a^4*b^3*x/(b*x^2+a)*e+1/2/a^5*b^4*x/(b*x^2+a)*d-1/2/a^6*b^5*x/(b*x^2+a)*c+5/2/a^
3*b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*f-7/2/a^4*b^3/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*e+9/2/a^5*b^4/
(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*d-11/2/a^6*b^5/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c-1/9*c/a^2/x^9-1/7
/a^2/x^7*d+2/7/a^3/x^7*b*c-1/5/a^2/x^5*e+2/5/a^3/x^5*b*d-3/5/a^4/x^5*b^2*c-1/3/a^2/x^3*f+2/3/a^3/x^3*b*e-1/a^4
/x^3*b^2*d+4/3/a^5/x^3*b^3*c+2*b/a^3/x*f-3*b^2/a^4/x*e+4*b^3/a^5/x*d-5*b^4/a^6/x*c

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maxima [A]  time = 2.99, size = 238, normalized size = 1.03 \begin {gather*} -\frac {315 \, {\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 210 \, {\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 42 \, {\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 70 \, a^{5} c + 18 \, {\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 10 \, {\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2}}{630 \, {\left (a^{6} b x^{11} + a^{7} x^{9}\right )}} - \frac {{\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/630*(315*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 + 210*(11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^
2*e - 5*a^4*b*f)*x^8 - 42*(11*a^2*b^3*c - 9*a^3*b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 70*a^5*c + 18*(11*a^3*b^2*c
- 9*a^4*b*d + 7*a^5*e)*x^4 - 10*(11*a^4*b*c - 9*a^5*d)*x^2)/(a^6*b*x^11 + a^7*x^9) - 1/2*(11*b^5*c - 9*a*b^4*
d + 7*a^2*b^3*e - 5*a^3*b^2*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6)

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mupad [B]  time = 1.01, size = 219, normalized size = 0.95 \begin {gather*} -\frac {\frac {c}{9\,a}-\frac {x^6\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{15\,a^4}+\frac {x^2\,\left (9\,a\,d-11\,b\,c\right )}{63\,a^2}+\frac {x^4\,\left (7\,e\,a^2-9\,d\,a\,b+11\,c\,b^2\right )}{35\,a^3}+\frac {b\,x^8\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{3\,a^5}+\frac {b^2\,x^{10}\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{2\,a^6}}{b\,x^{11}+a\,x^9}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-5\,f\,a^3+7\,e\,a^2\,b-9\,d\,a\,b^2+11\,c\,b^3\right )}{2\,a^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x)

[Out]

- (c/(9*a) - (x^6*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(15*a^4) + (x^2*(9*a*d - 11*b*c))/(63*a^2) + (
x^4*(11*b^2*c + 7*a^2*e - 9*a*b*d))/(35*a^3) + (b*x^8*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(3*a^5) +
(b^2*x^10*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(2*a^6))/(a*x^9 + b*x^11) - (b^(3/2)*atan((b^(1/2)*x)/
a^(1/2))*(11*b^3*c - 5*a^3*f - 9*a*b^2*d + 7*a^2*b*e))/(2*a^(13/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**10/(b*x**2+a)**2,x)

[Out]

Timed out

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