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

Optimal. Leaf size=240 \[ \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {a x \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{2 b^6}+\frac {x^3 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{6 b^5}-\frac {x^5 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{10 a b^4}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{2 b^{13/2}}+\frac {x^7 (b e-2 a f)}{7 b^3}+\frac {f x^9}{9 b^2} \]

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Rubi [A]  time = 0.29, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1804, 1585, 1261, 205} \begin {gather*} \frac {x^7 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {x^5 \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{10 a b^4}+\frac {x^3 \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{6 b^5}-\frac {a x \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{2 b^6}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{2 b^{13/2}}+\frac {x^7 (b e-2 a f)}{7 b^3}+\frac {f x^9}{9 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x)/(2*b^6) + ((5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x^
3)/(6*b^5) - ((5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x^5)/(10*a*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*
x^9)/(9*b^2) + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^7)/(2*a*(a + b*x^2)) + (a^(3/2)*(5*b^3*c - 7*a*b^2*d +
9*a^2*b*e - 11*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(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 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]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1804

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 227, normalized size = 0.95 \begin {gather*} \frac {x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac {a x \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )}{b^6}+\frac {x^3 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (11 a^3 f-9 a^2 b e+7 a b^2 d-5 b^3 c\right )}{2 b^{13/2}}-\frac {x \left (a^5 (-f)+a^4 b e-a^3 b^2 d+a^2 b^3 c\right )}{2 b^6 \left (a+b x^2\right )}+\frac {x^7 (b e-2 a f)}{7 b^3}+\frac {f x^9}{9 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.87, size = 572, normalized size = 2.38 \begin {gather*} \left [\frac {140 \, b^{5} f x^{11} + 20 \, {\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 36 \, {\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 84 \, {\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 420 \, {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} - 315 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f + {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{1260 \, {\left (b^{7} x^{2} + a b^{6}\right )}}, \frac {70 \, b^{5} f x^{11} + 10 \, {\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 18 \, {\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 42 \, {\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 210 \, {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} + 315 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f + {\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{630 \, {\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.39, size = 252, normalized size = 1.05 \begin {gather*} \frac {{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d - 11 \, a^{5} f + 9 \, a^{4} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{6}} - \frac {a^{2} b^{3} c x - a^{3} b^{2} d x - a^{5} f x + a^{4} b x e}{2 \, {\left (b x^{2} + a\right )} b^{6}} + \frac {35 \, b^{16} f x^{9} - 90 \, a b^{15} f x^{7} + 45 \, b^{16} x^{7} e + 63 \, b^{16} d x^{5} + 189 \, a^{2} b^{14} f x^{5} - 126 \, a b^{15} x^{5} e + 105 \, b^{16} c x^{3} - 210 \, a b^{15} d x^{3} - 420 \, a^{3} b^{13} f x^{3} + 315 \, a^{2} b^{14} x^{3} e - 630 \, a b^{15} c x + 945 \, a^{2} b^{14} d x + 1575 \, a^{4} b^{12} f x - 1260 \, a^{3} b^{13} x e}{315 \, b^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(5*a^2*b^3*c - 7*a^3*b^2*d - 11*a^5*f + 9*a^4*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) - 1/2*(a^2*b^3*c*
x - a^3*b^2*d*x - a^5*f*x + a^4*b*x*e)/((b*x^2 + a)*b^6) + 1/315*(35*b^16*f*x^9 - 90*a*b^15*f*x^7 + 45*b^16*x^
7*e + 63*b^16*d*x^5 + 189*a^2*b^14*f*x^5 - 126*a*b^15*x^5*e + 105*b^16*c*x^3 - 210*a*b^15*d*x^3 - 420*a^3*b^13
*f*x^3 + 315*a^2*b^14*x^3*e - 630*a*b^15*c*x + 945*a^2*b^14*d*x + 1575*a^4*b^12*f*x - 1260*a^3*b^13*x*e)/b^18

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maple [A]  time = 0.01, size = 309, normalized size = 1.29 \begin {gather*} \frac {f \,x^{9}}{9 b^{2}}-\frac {2 a f \,x^{7}}{7 b^{3}}+\frac {e \,x^{7}}{7 b^{2}}+\frac {3 a^{2} f \,x^{5}}{5 b^{4}}-\frac {2 a e \,x^{5}}{5 b^{3}}+\frac {d \,x^{5}}{5 b^{2}}-\frac {4 a^{3} f \,x^{3}}{3 b^{5}}+\frac {a^{2} e \,x^{3}}{b^{4}}-\frac {2 a d \,x^{3}}{3 b^{3}}+\frac {c \,x^{3}}{3 b^{2}}+\frac {a^{5} f x}{2 \left (b \,x^{2}+a \right ) b^{6}}-\frac {11 a^{5} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{6}}-\frac {a^{4} e x}{2 \left (b \,x^{2}+a \right ) b^{5}}+\frac {9 a^{4} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{5}}+\frac {a^{3} d x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {a^{2} c x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {5 a^{4} f x}{b^{6}}-\frac {4 a^{3} e x}{b^{5}}+\frac {3 a^{2} d x}{b^{4}}-\frac {2 a c x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, size = 227, normalized size = 0.95 \begin {gather*} -\frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{2 \, {\left (b^{7} x^{2} + a b^{6}\right )}} + \frac {{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{6}} + \frac {35 \, b^{4} f x^{9} + 45 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{7} + 63 \, {\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3} - 315 \, {\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x}{315 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x/(b^7*x^2 + a*b^6) + 1/2*(5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^4*b*
e - 11*a^5*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/315*(35*b^4*f*x^9 + 45*(b^4*e - 2*a*b^3*f)*x^7 + 63*(b
^4*d - 2*a*b^3*e + 3*a^2*b^2*f)*x^5 + 105*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^3 - 315*(2*a*b^3*c -
3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*x)/b^6

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mupad [B]  time = 0.10, size = 413, normalized size = 1.72 \begin {gather*} x^7\,\left (\frac {e}{7\,b^2}-\frac {2\,a\,f}{7\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )}{b}-\frac {a^2\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b^2}\right )-x^5\,\left (\frac {a^2\,f}{5\,b^4}-\frac {d}{5\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{5\,b}\right )+x^3\,\left (\frac {c}{3\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{3\,b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{3\,b}\right )+\frac {f\,x^9}{9\,b^2}+\frac {x\,\left (\frac {f\,a^5}{2}-\frac {e\,a^4\,b}{2}+\frac {d\,a^3\,b^2}{2}-\frac {c\,a^2\,b^3}{2}\right )}{b^7\,x^2+a\,b^6}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (-11\,f\,a^3+9\,e\,a^2\,b-7\,d\,a\,b^2+5\,c\,b^3\right )}{11\,f\,a^5-9\,e\,a^4\,b+7\,d\,a^3\,b^2-5\,c\,a^2\,b^3}\right )\,\left (-11\,f\,a^3+9\,e\,a^2\,b-7\,d\,a\,b^2+5\,c\,b^3\right )}{2\,b^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.07, size = 444, normalized size = 1.85 \begin {gather*} x^{7} \left (- \frac {2 a f}{7 b^{3}} + \frac {e}{7 b^{2}}\right ) + x^{5} \left (\frac {3 a^{2} f}{5 b^{4}} - \frac {2 a e}{5 b^{3}} + \frac {d}{5 b^{2}}\right ) + x^{3} \left (- \frac {4 a^{3} f}{3 b^{5}} + \frac {a^{2} e}{b^{4}} - \frac {2 a d}{3 b^{3}} + \frac {c}{3 b^{2}}\right ) + x \left (\frac {5 a^{4} f}{b^{6}} - \frac {4 a^{3} e}{b^{5}} + \frac {3 a^{2} d}{b^{4}} - \frac {2 a c}{b^{3}}\right ) + \frac {x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{2 a b^{6} + 2 b^{7} x^{2}} + \frac {\sqrt {- \frac {a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log {\left (- \frac {b^{6} \sqrt {- \frac {a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} - \frac {\sqrt {- \frac {a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log {\left (\frac {b^{6} \sqrt {- \frac {a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} + \frac {f x^{9}}{9 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**7*(-2*a*f/(7*b**3) + e/(7*b**2)) + x**5*(3*a**2*f/(5*b**4) - 2*a*e/(5*b**3) + d/(5*b**2)) + x**3*(-4*a**3*f
/(3*b**5) + a**2*e/b**4 - 2*a*d/(3*b**3) + c/(3*b**2)) + x*(5*a**4*f/b**6 - 4*a**3*e/b**5 + 3*a**2*d/b**4 - 2*
a*c/b**3) + x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/(2*a*b**6 + 2*b**7*x**2) + sqrt(-a**3/b**13)*(11
*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b**3*c)*log(-b**6*sqrt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d
- 5*b**3*c)/(11*a**4*f - 9*a**3*b*e + 7*a**2*b**2*d - 5*a*b**3*c) + x)/4 - sqrt(-a**3/b**13)*(11*a**3*f - 9*a
**2*b*e + 7*a*b**2*d - 5*b**3*c)*log(b**6*sqrt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b**3*c)/(
11*a**4*f - 9*a**3*b*e + 7*a**2*b**2*d - 5*a*b**3*c) + x)/4 + f*x**9/(9*b**2)

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