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

[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))

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Rubi [A]  time = 0.492374, antiderivative size = 287, normalized size of antiderivative = 1., 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 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 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 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 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]

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]

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*}

Mathematica [A]  time = 0.154838, size = 272, normalized size = 0.95 \[ \frac{x^3 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{a^2 x \left (-21 a^2 b e+25 a^3 f+17 a b^2 d-13 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}+\frac{a^3 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^7 \left (a+b x^2\right )^2}+\frac{a x \left (-10 a^2 b e+15 a^3 f+6 a b^2 d-3 b^3 c\right )}{b^7}-\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-99 a^2 b e+143 a^3 f+63 a b^2 d-35 b^3 c\right )}{8 b^{15/2}}+\frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^9}{9 b^3} \]

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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Maple [A]  time = 0.014, size = 394, normalized size = 1.4 \begin{align*}{\frac{{x}^{7}e}{7\,{b}^{3}}}+{\frac{{x}^{5}d}{5\,{b}^{3}}}+{\frac{{x}^{3}c}{3\,{b}^{3}}}+{\frac{15\,{a}^{4}dx}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{25\,{a}^{5}{x}^{3}f}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{f{x}^{9}}{9\,{b}^{3}}}-{\frac{11\,{a}^{3}cx}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{143\,{a}^{5}f}{8\,{b}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{99\,{a}^{4}e}{8\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{a}^{3}d}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}c}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{21\,{a}^{4}{x}^{3}e}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{17\,{a}^{3}{x}^{3}d}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{x}^{3}{a}^{2}c}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{23\,{a}^{6}fx}{8\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{19\,{a}^{5}ex}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{10\,{a}^{3}{x}^{3}f}{3\,{b}^{6}}}+2\,{\frac{{x}^{3}{a}^{2}e}{{b}^{5}}}-{\frac{a{x}^{3}d}{{b}^{4}}}+15\,{\frac{{a}^{4}fx}{{b}^{7}}}-10\,{\frac{{a}^{3}ex}{{b}^{6}}}+6\,{\frac{{a}^{2}dx}{{b}^{5}}}-3\,{\frac{acx}{{b}^{4}}}-{\frac{3\,{x}^{7}af}{7\,{b}^{4}}}+{\frac{6\,{x}^{5}{a}^{2}f}{5\,{b}^{5}}}-{\frac{3\,{x}^{5}ae}{5\,{b}^{4}}} \end{align*}

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]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.25273, size = 1748, normalized size = 6.09 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 16.1898, size = 491, normalized size = 1.71 \begin{align*} \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}} - \frac{x^{7} \left (3 a f - b e\right )}{7 b^{4}} + \frac{x^{5} \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{5 b^{5}} - \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{x \left (15 a^{4} f - 10 a^{3} b e + 6 a^{2} b^{2} d - 3 a b^{3} c\right )}{b^{7}} \end{align*}

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]

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

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Giac [A]  time = 1.17982, size = 406, normalized size = 1.41 \begin{align*} \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}} \end{align*}

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