3.133 \(\int \frac {x^8 (c+d x^2+e x^4+f x^6)}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=287 \[ \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {a^2 x \left (-17 a^3 f+13 a^2 b e-9 a b^2 d+5 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}-\frac {a x \left (-63 a^3 f+43 a^2 b e-27 a b^2 d+15 b^3 c\right )}{4 b^7}+\frac {x^3 \left (-23 a^3 f+15 a^2 b e-9 a b^2 d+5 b^3 c\right )}{6 b^6}-\frac {x^5 \left (-29 a^3 f+17 a^2 b e-9 a b^2 d+5 b^3 c\right )}{20 a b^5}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-143 a^3 f+99 a^2 b e-63 a b^2 d+35 b^3 c\right )}{8 b^{15/2}}+\frac {x^7 (b e-3 a f)}{7 b^4}+\frac {f x^9}{9 b^3} \]

[Out]

-1/4*a*(-63*a^3*f+43*a^2*b*e-27*a*b^2*d+15*b^3*c)*x/b^7+1/6*(-23*a^3*f+15*a^2*b*e-9*a*b^2*d+5*b^3*c)*x^3/b^6-1
/20*(-29*a^3*f+17*a^2*b*e-9*a*b^2*d+5*b^3*c)*x^5/a/b^5+1/7*(-3*a*f+b*e)*x^7/b^4+1/9*f*x^9/b^3+1/4*(c-a*(a^2*f-
a*b*e+b^2*d)/b^3)*x^9/a/(b*x^2+a)^2-1/8*a^2*(-17*a^3*f+13*a^2*b*e-9*a*b^2*d+5*b^3*c)*x/b^7/(b*x^2+a)+1/8*a^(3/
2)*(-143*a^3*f+99*a^2*b*e-63*a*b^2*d+35*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/b^(15/2)

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Rubi [A]  time = 0.49, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1804, 1585, 1257, 1810, 205} \[ \frac {x^9 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {x^5 \left (17 a^2 b e-29 a^3 f-9 a b^2 d+5 b^3 c\right )}{20 a b^5}+\frac {x^3 \left (15 a^2 b e-23 a^3 f-9 a b^2 d+5 b^3 c\right )}{6 b^6}-\frac {a^2 x \left (13 a^2 b e-17 a^3 f-9 a b^2 d+5 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}-\frac {a x \left (43 a^2 b e-63 a^3 f-27 a b^2 d+15 b^3 c\right )}{4 b^7}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (99 a^2 b e-143 a^3 f-63 a b^2 d+35 b^3 c\right )}{8 b^{15/2}}+\frac {x^7 (b e-3 a f)}{7 b^4}+\frac {f x^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]

[Out]

-(a*(15*b^3*c - 27*a*b^2*d + 43*a^2*b*e - 63*a^3*f)*x)/(4*b^7) + ((5*b^3*c - 9*a*b^2*d + 15*a^2*b*e - 23*a^3*f
)*x^3)/(6*b^6) - ((5*b^3*c - 9*a*b^2*d + 17*a^2*b*e - 29*a^3*f)*x^5)/(20*a*b^5) + ((b*e - 3*a*f)*x^7)/(7*b^4)
+ (f*x^9)/(9*b^3) + ((c - (a*(b^2*d - a*b*e + a^2*f))/b^3)*x^9)/(4*a*(a + b*x^2)^2) - (a^2*(5*b^3*c - 9*a*b^2*
d + 13*a^2*b*e - 17*a^3*f)*x)/(8*b^7*(a + b*x^2)) + (a^(3/2)*(35*b^3*c - 63*a*b^2*d + 99*a^2*b*e - 143*a^3*f)*
ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(15/2))

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 1257

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^
(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[1/(2*e^(2*p +
m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4
)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]

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]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^7 \left (\left (5 b c-9 a d+\frac {9 a^2 e}{b}-\frac {9 a^3 f}{b^2}\right ) x-4 a \left (e-\frac {a f}{b}\right ) x^3-4 a f x^5\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^8 \left (5 b c-9 a d+\frac {9 a^2 e}{b}-\frac {9 a^3 f}{b^2}-4 a \left (e-\frac {a f}{b}\right ) x^2-4 a f x^4\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac {\int \frac {a^3 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right )-2 a^2 b \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x^2+2 a b^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x^4-2 b^3 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x^6+8 a b^4 (b e-2 a f) x^8+8 a b^5 f x^{10}}{a+b x^2} \, dx}{8 a b^7}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac {\int \left (-2 a^2 \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right )+4 a b \left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^2-2 b^2 \left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^4+8 a b^3 (b e-3 a f) x^6+8 a b^4 f x^8+\frac {35 a^3 b^3 c-63 a^4 b^2 d+99 a^5 b e-143 a^6 f}{a+b x^2}\right ) \, dx}{8 a b^7}\\ &=-\frac {a \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right ) x}{4 b^7}+\frac {\left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^3}{6 b^6}-\frac {\left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^5}{20 a b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^9}{9 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac {\left (a^2 \left (35 b^3 c-63 a b^2 d+99 a^2 b e-143 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^7}\\ &=-\frac {a \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right ) x}{4 b^7}+\frac {\left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^3}{6 b^6}-\frac {\left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^5}{20 a b^5}+\frac {(b e-3 a f) x^7}{7 b^4}+\frac {f x^9}{9 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac {a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac {a^{3/2} \left (35 b^3 c-63 a b^2 d+99 a^2 b e-143 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{15/2}}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 272, normalized size = 0.95 \[ \frac {x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {a^2 x \left (25 a^3 f-21 a^2 b e+17 a b^2 d-13 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}+\frac {a^3 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^7 \left (a+b x^2\right )^2}+\frac {a x \left (15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right )}{b^7}+\frac {x^3 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}-\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (143 a^3 f-99 a^2 b e+63 a b^2 d-35 b^3 c\right )}{8 b^{15/2}}+\frac {x^7 (b e-3 a f)}{7 b^4}+\frac {f x^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]

[Out]

(a*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x)/b^7 + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x^3)/(3
*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^5)/(5*b^5) + ((b*e - 3*a*f)*x^7)/(7*b^4) + (f*x^9)/(9*b^3) + (a^3*(b^3*
c - a*b^2*d + a^2*b*e - a^3*f)*x)/(4*b^7*(a + b*x^2)^2) + (a^2*(-13*b^3*c + 17*a*b^2*d - 21*a^2*b*e + 25*a^3*f
)*x)/(8*b^7*(a + b*x^2)) - (a^(3/2)*(-35*b^3*c + 63*a*b^2*d - 99*a^2*b*e + 143*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[
a]])/(8*b^(15/2))

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fricas [A]  time = 0.58, size = 762, normalized size = 2.66 \[ \left [\frac {560 \, b^{6} f x^{13} + 80 \, {\left (9 \, b^{6} e - 13 \, a b^{5} f\right )} x^{11} + 16 \, {\left (63 \, b^{6} d - 99 \, a b^{5} e + 143 \, a^{2} b^{4} f\right )} x^{9} + 48 \, {\left (35 \, b^{6} c - 63 \, a b^{5} d + 99 \, a^{2} b^{4} e - 143 \, a^{3} b^{3} f\right )} x^{7} - 336 \, {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{5} - 1050 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{3} - 315 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f + {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} 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 (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f\right )} x}{5040 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}}, \frac {280 \, b^{6} f x^{13} + 40 \, {\left (9 \, b^{6} e - 13 \, a b^{5} f\right )} x^{11} + 8 \, {\left (63 \, b^{6} d - 99 \, a b^{5} e + 143 \, a^{2} b^{4} f\right )} x^{9} + 24 \, {\left (35 \, b^{6} c - 63 \, a b^{5} d + 99 \, a^{2} b^{4} e - 143 \, a^{3} b^{3} f\right )} x^{7} - 168 \, {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{5} - 525 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{3} + 315 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f + {\left (35 \, a b^{5} c - 63 \, a^{2} b^{4} d + 99 \, a^{3} b^{3} e - 143 \, a^{4} b^{2} f\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{4} c - 63 \, a^{3} b^{3} d + 99 \, a^{4} b^{2} e - 143 \, a^{5} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (35 \, a^{3} b^{3} c - 63 \, a^{4} b^{2} d + 99 \, a^{5} b e - 143 \, a^{6} f\right )} x}{2520 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/5040*(560*b^6*f*x^13 + 80*(9*b^6*e - 13*a*b^5*f)*x^11 + 16*(63*b^6*d - 99*a*b^5*e + 143*a^2*b^4*f)*x^9 + 48
*(35*b^6*c - 63*a*b^5*d + 99*a^2*b^4*e - 143*a^3*b^3*f)*x^7 - 336*(35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e -
143*a^4*b^2*f)*x^5 - 1050*(35*a^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)*x^3 - 315*(35*a^3*b^3*c -
63*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f + (35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 143*a^4*b^2*f)*x^4 + 2*(3
5*a^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b
*x^2 + a)) - 630*(35*a^3*b^3*c - 63*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f)*x)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7),
1/2520*(280*b^6*f*x^13 + 40*(9*b^6*e - 13*a*b^5*f)*x^11 + 8*(63*b^6*d - 99*a*b^5*e + 143*a^2*b^4*f)*x^9 + 24*(
35*b^6*c - 63*a*b^5*d + 99*a^2*b^4*e - 143*a^3*b^3*f)*x^7 - 168*(35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 14
3*a^4*b^2*f)*x^5 - 525*(35*a^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)*x^3 + 315*(35*a^3*b^3*c - 63
*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f + (35*a*b^5*c - 63*a^2*b^4*d + 99*a^3*b^3*e - 143*a^4*b^2*f)*x^4 + 2*(35*a
^2*b^4*c - 63*a^3*b^3*d + 99*a^4*b^2*e - 143*a^5*b*f)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(35*a^3*b^3
*c - 63*a^4*b^2*d + 99*a^5*b*e - 143*a^6*f)*x)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7)]

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giac [A]  time = 0.40, size = 301, normalized size = 1.05 \[ \frac {{\left (35 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d - 143 \, a^{5} f + 99 \, a^{4} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{7}} - \frac {13 \, a^{2} b^{4} c x^{3} - 17 \, a^{3} b^{3} d x^{3} - 25 \, a^{5} b f x^{3} + 21 \, a^{4} b^{2} x^{3} e + 11 \, a^{3} b^{3} c x - 15 \, a^{4} b^{2} d x - 23 \, a^{6} f x + 19 \, a^{5} b x e}{8 \, {\left (b x^{2} + a\right )}^{2} b^{7}} + \frac {35 \, b^{24} f x^{9} - 135 \, a b^{23} f x^{7} + 45 \, b^{24} x^{7} e + 63 \, b^{24} d x^{5} + 378 \, a^{2} b^{22} f x^{5} - 189 \, a b^{23} x^{5} e + 105 \, b^{24} c x^{3} - 315 \, a b^{23} d x^{3} - 1050 \, a^{3} b^{21} f x^{3} + 630 \, a^{2} b^{22} x^{3} e - 945 \, a b^{23} c x + 1890 \, a^{2} b^{22} d x + 4725 \, a^{4} b^{20} f x - 3150 \, a^{3} b^{21} x e}{315 \, b^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

1/8*(35*a^2*b^3*c - 63*a^3*b^2*d - 143*a^5*f + 99*a^4*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^7) - 1/8*(13*a^2
*b^4*c*x^3 - 17*a^3*b^3*d*x^3 - 25*a^5*b*f*x^3 + 21*a^4*b^2*x^3*e + 11*a^3*b^3*c*x - 15*a^4*b^2*d*x - 23*a^6*f
*x + 19*a^5*b*x*e)/((b*x^2 + a)^2*b^7) + 1/315*(35*b^24*f*x^9 - 135*a*b^23*f*x^7 + 45*b^24*x^7*e + 63*b^24*d*x
^5 + 378*a^2*b^22*f*x^5 - 189*a*b^23*x^5*e + 105*b^24*c*x^3 - 315*a*b^23*d*x^3 - 1050*a^3*b^21*f*x^3 + 630*a^2
*b^22*x^3*e - 945*a*b^23*c*x + 1890*a^2*b^22*d*x + 4725*a^4*b^20*f*x - 3150*a^3*b^21*x*e)/b^27

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maple [A]  time = 0.02, size = 394, normalized size = 1.37 \[ \frac {f \,x^{9}}{9 b^{3}}-\frac {3 a f \,x^{7}}{7 b^{4}}+\frac {e \,x^{7}}{7 b^{3}}+\frac {25 a^{5} f \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{6}}-\frac {21 a^{4} e \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}+\frac {17 a^{3} d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}-\frac {13 a^{2} c \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {6 a^{2} f \,x^{5}}{5 b^{5}}-\frac {3 a e \,x^{5}}{5 b^{4}}+\frac {d \,x^{5}}{5 b^{3}}+\frac {23 a^{6} f x}{8 \left (b \,x^{2}+a \right )^{2} b^{7}}-\frac {19 a^{5} e x}{8 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {15 a^{4} d x}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {11 a^{3} c x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}-\frac {10 a^{3} f \,x^{3}}{3 b^{6}}+\frac {2 a^{2} e \,x^{3}}{b^{5}}-\frac {a d \,x^{3}}{b^{4}}+\frac {c \,x^{3}}{3 b^{3}}-\frac {143 a^{5} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{7}}+\frac {99 a^{4} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{6}}-\frac {63 a^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{5}}+\frac {35 a^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}+\frac {15 a^{4} f x}{b^{7}}-\frac {10 a^{3} e x}{b^{6}}+\frac {6 a^{2} d x}{b^{5}}-\frac {3 a c x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^3,x)

[Out]

1/3/b^3*x^3*c-3/7/b^4*x^7*a*f+6/5/b^5*x^5*a^2*f-3/5/b^4*x^5*a*e-10/3/b^6*x^3*a^3*f+2/b^5*x^3*a^2*e-1/b^4*x^3*a
*d+15/b^7*a^4*f*x-10/b^6*a^3*e*x+6/b^5*a^2*d*x-3/b^4*a*c*x+1/9*f*x^9/b^3+25/8*a^5/b^6/(b*x^2+a)^2*x^3*f+23/8*a
^6/b^7/(b*x^2+a)^2*f*x-19/8*a^5/b^6/(b*x^2+a)^2*e*x+15/8*a^4/b^5/(b*x^2+a)^2*d*x-11/8*a^3/b^4/(b*x^2+a)^2*c*x-
143/8*a^5/b^7/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*f+99/8*a^4/b^6/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*e-63/
8*a^3/b^5/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*d+35/8*a^2/b^4/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c+1/7/b^3
*x^7*e+1/5/b^3*x^5*d-21/8*a^4/b^5/(b*x^2+a)^2*x^3*e+17/8*a^3/b^4/(b*x^2+a)^2*x^3*d-13/8*a^2/b^3/(b*x^2+a)^2*x^
3*c

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maxima [A]  time = 3.10, size = 281, normalized size = 0.98 \[ -\frac {{\left (13 \, a^{2} b^{4} c - 17 \, a^{3} b^{3} d + 21 \, a^{4} b^{2} e - 25 \, a^{5} b f\right )} x^{3} + {\left (11 \, a^{3} b^{3} c - 15 \, a^{4} b^{2} d + 19 \, a^{5} b e - 23 \, a^{6} f\right )} x}{8 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac {{\left (35 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 99 \, a^{4} b e - 143 \, a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{7}} + \frac {35 \, b^{4} f x^{9} + 45 \, {\left (b^{4} e - 3 \, a b^{3} f\right )} x^{7} + 63 \, {\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3} - 315 \, {\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x}{315 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/8*((13*a^2*b^4*c - 17*a^3*b^3*d + 21*a^4*b^2*e - 25*a^5*b*f)*x^3 + (11*a^3*b^3*c - 15*a^4*b^2*d + 19*a^5*b*
e - 23*a^6*f)*x)/(b^9*x^4 + 2*a*b^8*x^2 + a^2*b^7) + 1/8*(35*a^2*b^3*c - 63*a^3*b^2*d + 99*a^4*b*e - 143*a^5*f
)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^7) + 1/315*(35*b^4*f*x^9 + 45*(b^4*e - 3*a*b^3*f)*x^7 + 63*(b^4*d - 3*a*b
^3*e + 6*a^2*b^2*f)*x^5 + 105*(b^4*c - 3*a*b^3*d + 6*a^2*b^2*e - 10*a^3*b*f)*x^3 - 315*(3*a*b^3*c - 6*a^2*b^2*
d + 10*a^3*b*e - 15*a^4*f)*x)/b^7

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mupad [B]  time = 1.00, size = 506, normalized size = 1.76 \[ x^7\,\left (\frac {e}{7\,b^3}-\frac {3\,a\,f}{7\,b^4}\right )+x^3\,\left (\frac {c}{3\,b^3}-\frac {a^3\,f}{3\,b^6}-\frac {a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )+\frac {x\,\left (\frac {23\,f\,a^6}{8}-\frac {19\,e\,a^5\,b}{8}+\frac {15\,d\,a^4\,b^2}{8}-\frac {11\,c\,a^3\,b^3}{8}\right )-x^3\,\left (-\frac {25\,f\,a^5\,b}{8}+\frac {21\,e\,a^4\,b^2}{8}-\frac {17\,d\,a^3\,b^3}{8}+\frac {13\,c\,a^2\,b^4}{8}\right )}{a^2\,b^7+2\,a\,b^8\,x^2+b^9\,x^4}-x\,\left (\frac {3\,a\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )}{b}-\frac {3\,a^2\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b^2}+\frac {a^3\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^3}\right )-x^5\,\left (\frac {3\,a^2\,f}{5\,b^5}-\frac {d}{5\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{5\,b}\right )+\frac {f\,x^9}{9\,b^3}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (-143\,f\,a^3+99\,e\,a^2\,b-63\,d\,a\,b^2+35\,c\,b^3\right )}{143\,f\,a^5-99\,e\,a^4\,b+63\,d\,a^3\,b^2-35\,c\,a^2\,b^3}\right )\,\left (-143\,f\,a^3+99\,e\,a^2\,b-63\,d\,a\,b^2+35\,c\,b^3\right )}{8\,b^{15/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x)

[Out]

x^7*(e/(7*b^3) - (3*a*f)/(7*b^4)) + x^3*(c/(3*b^3) - (a^3*f)/(3*b^6) - (a^2*(e/b^3 - (3*a*f)/b^4))/b^2 + (a*((
3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b) + (x*((23*a^6*f)/8 - (11*a^3*b^3*c)/8 + (15*a^4*b^2*
d)/8 - (19*a^5*b*e)/8) - x^3*((13*a^2*b^4*c)/8 - (17*a^3*b^3*d)/8 + (21*a^4*b^2*e)/8 - (25*a^5*b*f)/8))/(a^2*b
^7 + b^9*x^4 + 2*a*b^8*x^2) - x*((3*a*(c/b^3 - (a^3*f)/b^6 - (3*a^2*(e/b^3 - (3*a*f)/b^4))/b^2 + (3*a*((3*a^2*
f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)/b^4))/b))/b))/b - (3*a^2*((3*a^2*f)/b^5 - d/b^3 + (3*a*(e/b^3 - (3*a*f)
/b^4))/b))/b^2 + (a^3*(e/b^3 - (3*a*f)/b^4))/b^3) - x^5*((3*a^2*f)/(5*b^5) - d/(5*b^3) + (3*a*(e/b^3 - (3*a*f)
/b^4))/(5*b)) + (f*x^9)/(9*b^3) - (a^(3/2)*atan((a^(3/2)*b^(1/2)*x*(35*b^3*c - 143*a^3*f - 63*a*b^2*d + 99*a^2
*b*e))/(143*a^5*f - 35*a^2*b^3*c + 63*a^3*b^2*d - 99*a^4*b*e))*(35*b^3*c - 143*a^3*f - 63*a*b^2*d + 99*a^2*b*e
))/(8*b^(15/2))

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sympy [A]  time = 16.43, size = 503, normalized size = 1.75 \[ x^{7} \left (- \frac {3 a f}{7 b^{4}} + \frac {e}{7 b^{3}}\right ) + x^{5} \left (\frac {6 a^{2} f}{5 b^{5}} - \frac {3 a e}{5 b^{4}} + \frac {d}{5 b^{3}}\right ) + x^{3} \left (- \frac {10 a^{3} f}{3 b^{6}} + \frac {2 a^{2} e}{b^{5}} - \frac {a d}{b^{4}} + \frac {c}{3 b^{3}}\right ) + x \left (\frac {15 a^{4} f}{b^{7}} - \frac {10 a^{3} e}{b^{6}} + \frac {6 a^{2} d}{b^{5}} - \frac {3 a c}{b^{4}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right ) \log {\left (- \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right )}{143 a^{4} f - 99 a^{3} b e + 63 a^{2} b^{2} d - 35 a b^{3} c} + x \right )}}{16} - \frac {\sqrt {- \frac {a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right ) \log {\left (\frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right )}{143 a^{4} f - 99 a^{3} b e + 63 a^{2} b^{2} d - 35 a b^{3} c} + x \right )}}{16} + \frac {x^{3} \left (25 a^{5} b f - 21 a^{4} b^{2} e + 17 a^{3} b^{3} d - 13 a^{2} b^{4} c\right ) + x \left (23 a^{6} f - 19 a^{5} b e + 15 a^{4} b^{2} d - 11 a^{3} b^{3} c\right )}{8 a^{2} b^{7} + 16 a b^{8} x^{2} + 8 b^{9} x^{4}} + \frac {f x^{9}}{9 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**3,x)

[Out]

x**7*(-3*a*f/(7*b**4) + e/(7*b**3)) + x**5*(6*a**2*f/(5*b**5) - 3*a*e/(5*b**4) + d/(5*b**3)) + x**3*(-10*a**3*
f/(3*b**6) + 2*a**2*e/b**5 - a*d/b**4 + c/(3*b**3)) + x*(15*a**4*f/b**7 - 10*a**3*e/b**6 + 6*a**2*d/b**5 - 3*a
*c/b**4) + sqrt(-a**3/b**15)*(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b**3*c)*log(-b**7*sqrt(-a**3/b**15)*
(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b**3*c)/(143*a**4*f - 99*a**3*b*e + 63*a**2*b**2*d - 35*a*b**3*c)
+ x)/16 - sqrt(-a**3/b**15)*(143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b**3*c)*log(b**7*sqrt(-a**3/b**15)*(
143*a**3*f - 99*a**2*b*e + 63*a*b**2*d - 35*b**3*c)/(143*a**4*f - 99*a**3*b*e + 63*a**2*b**2*d - 35*a*b**3*c)
+ x)/16 + (x**3*(25*a**5*b*f - 21*a**4*b**2*e + 17*a**3*b**3*d - 13*a**2*b**4*c) + x*(23*a**6*f - 19*a**5*b*e
+ 15*a**4*b**2*d - 11*a**3*b**3*c))/(8*a**2*b**7 + 16*a*b**8*x**2 + 8*b**9*x**4) + f*x**9/(9*b**3)

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