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

Optimal. Leaf size=173 \[ \frac {d x \left (17 c d^2-e (13 b d-9 a e)\right )}{8 e^5 \left (d+e x^2\right )}+\frac {x \left (6 c d^2-e (3 b d-a e)\right )}{e^5}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a e^2-35 b d e+63 c d^2\right )}{8 e^{11/2}}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{4 e^5 \left (d+e x^2\right )^2}-\frac {x^3 (3 c d-b e)}{3 e^4}+\frac {c x^5}{5 e^3} \]

[Out]

(6*c*d^2-e*(-a*e+3*b*d))*x/e^5-1/3*(-b*e+3*c*d)*x^3/e^4+1/5*c*x^5/e^3-1/4*d^2*(a*e^2-b*d*e+c*d^2)*x/e^5/(e*x^2
+d)^2+1/8*d*(17*c*d^2-e*(-9*a*e+13*b*d))*x/e^5/(e*x^2+d)-1/8*(15*a*e^2-35*b*d*e+63*c*d^2)*arctan(x*e^(1/2)/d^(
1/2))*d^(1/2)/e^(11/2)

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Rubi [A]  time = 0.32, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1257, 1814, 1810, 205} \[ \frac {d x \left (17 c d^2-e (13 b d-9 a e)\right )}{8 e^5 \left (d+e x^2\right )}-\frac {d^2 x \left (a e^2-b d e+c d^2\right )}{4 e^5 \left (d+e x^2\right )^2}+\frac {x \left (6 c d^2-e (3 b d-a e)\right )}{e^5}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a e^2-35 b d e+63 c d^2\right )}{8 e^{11/2}}-\frac {x^3 (3 c d-b e)}{3 e^4}+\frac {c x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*c*d^2 - e*(3*b*d - a*e))*x)/e^5 - ((3*c*d - b*e)*x^3)/(3*e^4) + (c*x^5)/(5*e^3) - (d^2*(c*d^2 - b*d*e + a*
e^2)*x)/(4*e^5*(d + e*x^2)^2) + (d*(17*c*d^2 - e*(13*b*d - 9*a*e))*x)/(8*e^5*(d + e*x^2)) - (Sqrt[d]*(63*c*d^2
- 35*b*d*e + 15*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*e^(11/2))

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 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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}-\frac {\int \frac {-d^2 \left (c d^2-b d e+a e^2\right )+4 d e \left (c d^2-b d e+a e^2\right ) x^2-4 e^2 \left (c d^2-b d e+a e^2\right ) x^4+4 e^3 (c d-b e) x^6-4 c e^4 x^8}{\left (d+e x^2\right )^2} \, dx}{4 e^5}\\ &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}+\frac {\int \frac {-d^2 \left (15 c d^2-e (11 b d-7 a e)\right )+8 d e \left (3 c d^2-e (2 b d-a e)\right ) x^2-8 d e^2 (2 c d-b e) x^4+8 c d e^3 x^6}{d+e x^2} \, dx}{8 d e^5}\\ &=-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}+\frac {\int \left (8 d \left (6 c d^2-e (3 b d-a e)\right )-8 d e (3 c d-b e) x^2+8 c d e^2 x^4+\frac {-63 c d^4+35 b d^3 e-15 a d^2 e^2}{d+e x^2}\right ) \, dx}{8 d e^5}\\ &=\frac {\left (6 c d^2-e (3 b d-a e)\right ) x}{e^5}-\frac {(3 c d-b e) x^3}{3 e^4}+\frac {c x^5}{5 e^3}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}+\frac {\left (-63 c d^4+35 b d^3 e-15 a d^2 e^2\right ) \int \frac {1}{d+e x^2} \, dx}{8 d e^5}\\ &=\frac {\left (6 c d^2-e (3 b d-a e)\right ) x}{e^5}-\frac {(3 c d-b e) x^3}{3 e^4}+\frac {c x^5}{5 e^3}-\frac {d^2 \left (c d^2-b d e+a e^2\right ) x}{4 e^5 \left (d+e x^2\right )^2}+\frac {d \left (17 c d^2-e (13 b d-9 a e)\right ) x}{8 e^5 \left (d+e x^2\right )}-\frac {\sqrt {d} \left (63 c d^2-5 e (7 b d-3 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 e^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 170, normalized size = 0.98 \[ \frac {x \left (d e (9 a e-13 b d)+17 c d^3\right )}{8 e^5 \left (d+e x^2\right )}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 e (3 a e-7 b d)+63 c d^2\right )}{8 e^{11/2}}+\frac {x \left (e (a e-3 b d)+6 c d^2\right )}{e^5}-\frac {x \left (d^2 e (a e-b d)+c d^4\right )}{4 e^5 \left (d+e x^2\right )^2}+\frac {x^3 (b e-3 c d)}{3 e^4}+\frac {c x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*c*d^2 + e*(-3*b*d + a*e))*x)/e^5 + ((-3*c*d + b*e)*x^3)/(3*e^4) + (c*x^5)/(5*e^3) - ((c*d^4 + d^2*e*(-(b*d
) + a*e))*x)/(4*e^5*(d + e*x^2)^2) + ((17*c*d^3 + d*e*(-13*b*d + 9*a*e))*x)/(8*e^5*(d + e*x^2)) - (Sqrt[d]*(63
*c*d^2 + 5*e*(-7*b*d + 3*a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*e^(11/2))

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fricas [A]  time = 0.97, size = 504, normalized size = 2.91 \[ \left [\frac {48 \, c e^{4} x^{9} - 16 \, {\left (9 \, c d e^{3} - 5 \, b e^{4}\right )} x^{7} + 16 \, {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{5} + 50 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{3} + 15 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2} + {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{4} + 2 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{2}\right )} \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} - 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) + 30 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2}\right )} x}{240 \, {\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}}, \frac {24 \, c e^{4} x^{9} - 8 \, {\left (9 \, c d e^{3} - 5 \, b e^{4}\right )} x^{7} + 8 \, {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{5} + 25 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{3} - 15 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2} + {\left (63 \, c d^{2} e^{2} - 35 \, b d e^{3} + 15 \, a e^{4}\right )} x^{4} + 2 \, {\left (63 \, c d^{3} e - 35 \, b d^{2} e^{2} + 15 \, a d e^{3}\right )} x^{2}\right )} \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) + 15 \, {\left (63 \, c d^{4} - 35 \, b d^{3} e + 15 \, a d^{2} e^{2}\right )} x}{120 \, {\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/240*(48*c*e^4*x^9 - 16*(9*c*d*e^3 - 5*b*e^4)*x^7 + 16*(63*c*d^2*e^2 - 35*b*d*e^3 + 15*a*e^4)*x^5 + 50*(63*c
*d^3*e - 35*b*d^2*e^2 + 15*a*d*e^3)*x^3 + 15*(63*c*d^4 - 35*b*d^3*e + 15*a*d^2*e^2 + (63*c*d^2*e^2 - 35*b*d*e^
3 + 15*a*e^4)*x^4 + 2*(63*c*d^3*e - 35*b*d^2*e^2 + 15*a*d*e^3)*x^2)*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) -
d)/(e*x^2 + d)) + 30*(63*c*d^4 - 35*b*d^3*e + 15*a*d^2*e^2)*x)/(e^7*x^4 + 2*d*e^6*x^2 + d^2*e^5), 1/120*(24*c
*e^4*x^9 - 8*(9*c*d*e^3 - 5*b*e^4)*x^7 + 8*(63*c*d^2*e^2 - 35*b*d*e^3 + 15*a*e^4)*x^5 + 25*(63*c*d^3*e - 35*b*
d^2*e^2 + 15*a*d*e^3)*x^3 - 15*(63*c*d^4 - 35*b*d^3*e + 15*a*d^2*e^2 + (63*c*d^2*e^2 - 35*b*d*e^3 + 15*a*e^4)*
x^4 + 2*(63*c*d^3*e - 35*b*d^2*e^2 + 15*a*d*e^3)*x^2)*sqrt(d/e)*arctan(e*x*sqrt(d/e)/d) + 15*(63*c*d^4 - 35*b*
d^3*e + 15*a*d^2*e^2)*x)/(e^7*x^4 + 2*d*e^6*x^2 + d^2*e^5)]

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giac [A]  time = 0.36, size = 160, normalized size = 0.92 \[ -\frac {{\left (63 \, c d^{3} - 35 \, b d^{2} e + 15 \, a d e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {11}{2}\right )}}{8 \, \sqrt {d}} + \frac {1}{15} \, {\left (3 \, c x^{5} e^{12} - 15 \, c d x^{3} e^{11} + 5 \, b x^{3} e^{12} + 90 \, c d^{2} x e^{10} - 45 \, b d x e^{11} + 15 \, a x e^{12}\right )} e^{\left (-15\right )} + \frac {{\left (17 \, c d^{3} x^{3} e - 13 \, b d^{2} x^{3} e^{2} + 15 \, c d^{4} x + 9 \, a d x^{3} e^{3} - 11 \, b d^{3} x e + 7 \, a d^{2} x e^{2}\right )} e^{\left (-5\right )}}{8 \, {\left (x^{2} e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(63*c*d^3 - 35*b*d^2*e + 15*a*d*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-11/2)/sqrt(d) + 1/15*(3*c*x^5*e^12 - 1
5*c*d*x^3*e^11 + 5*b*x^3*e^12 + 90*c*d^2*x*e^10 - 45*b*d*x*e^11 + 15*a*x*e^12)*e^(-15) + 1/8*(17*c*d^3*x^3*e -
13*b*d^2*x^3*e^2 + 15*c*d^4*x + 9*a*d*x^3*e^3 - 11*b*d^3*x*e + 7*a*d^2*x*e^2)*e^(-5)/(x^2*e + d)^2

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maple [A]  time = 0.02, size = 239, normalized size = 1.38 \[ \frac {9 a d \,x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{2}}-\frac {13 b \,d^{2} x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{3}}+\frac {17 c \,d^{3} x^{3}}{8 \left (e \,x^{2}+d \right )^{2} e^{4}}+\frac {c \,x^{5}}{5 e^{3}}+\frac {7 a \,d^{2} x}{8 \left (e \,x^{2}+d \right )^{2} e^{3}}-\frac {11 b \,d^{3} x}{8 \left (e \,x^{2}+d \right )^{2} e^{4}}+\frac {b \,x^{3}}{3 e^{3}}+\frac {15 c \,d^{4} x}{8 \left (e \,x^{2}+d \right )^{2} e^{5}}-\frac {c d \,x^{3}}{e^{4}}-\frac {15 a d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{3}}+\frac {35 b \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{4}}-\frac {63 c \,d^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}\, e^{5}}+\frac {a x}{e^{3}}-\frac {3 b d x}{e^{4}}+\frac {6 c \,d^{2} x}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5/e^3+1/3/e^3*x^3*b-1/e^4*x^3*c*d+1/e^3*a*x-3/e^4*d*b*x+6/e^5*c*d^2*x+9/8*d/e^2/(e*x^2+d)^2*x^3*a-13/8
*d^2/e^3/(e*x^2+d)^2*x^3*b+17/8*d^3/e^4/(e*x^2+d)^2*x^3*c+7/8*d^2/e^3/(e*x^2+d)^2*a*x-11/8*d^3/e^4/(e*x^2+d)^2
*b*x+15/8*d^4/e^5/(e*x^2+d)^2*c*x-15/8*d/e^3/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*a+35/8*d^2/e^4/(d*e)^(1/2)*
arctan(1/(d*e)^(1/2)*e*x)*b-63/8*d^3/e^5/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*c

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maxima [A]  time = 2.47, size = 175, normalized size = 1.01 \[ \frac {{\left (17 \, c d^{3} e - 13 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{3} + {\left (15 \, c d^{4} - 11 \, b d^{3} e + 7 \, a d^{2} e^{2}\right )} x}{8 \, {\left (e^{7} x^{4} + 2 \, d e^{6} x^{2} + d^{2} e^{5}\right )}} - \frac {{\left (63 \, c d^{3} - 35 \, b d^{2} e + 15 \, a d e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {d e} e^{5}} + \frac {3 \, c e^{2} x^{5} - 5 \, {\left (3 \, c d e - b e^{2}\right )} x^{3} + 15 \, {\left (6 \, c d^{2} - 3 \, b d e + a e^{2}\right )} x}{15 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((17*c*d^3*e - 13*b*d^2*e^2 + 9*a*d*e^3)*x^3 + (15*c*d^4 - 11*b*d^3*e + 7*a*d^2*e^2)*x)/(e^7*x^4 + 2*d*e^6
*x^2 + d^2*e^5) - 1/8*(63*c*d^3 - 35*b*d^2*e + 15*a*d*e^2)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*e^5) + 1/15*(3*c*e
^2*x^5 - 5*(3*c*d*e - b*e^2)*x^3 + 15*(6*c*d^2 - 3*b*d*e + a*e^2)*x)/e^5

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mupad [B]  time = 0.35, size = 223, normalized size = 1.29 \[ x^3\,\left (\frac {b}{3\,e^3}-\frac {c\,d}{e^4}\right )-x\,\left (\frac {3\,c\,d^2}{e^5}-\frac {a}{e^3}+\frac {3\,d\,\left (\frac {b}{e^3}-\frac {3\,c\,d}{e^4}\right )}{e}\right )+\frac {\left (\frac {17\,c\,d^3\,e}{8}-\frac {13\,b\,d^2\,e^2}{8}+\frac {9\,a\,d\,e^3}{8}\right )\,x^3+\left (\frac {15\,c\,d^4}{8}-\frac {11\,b\,d^3\,e}{8}+\frac {7\,a\,d^2\,e^2}{8}\right )\,x}{d^2\,e^5+2\,d\,e^6\,x^2+e^7\,x^4}+\frac {c\,x^5}{5\,e^3}-\frac {\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e}\,x\,\left (63\,c\,d^2-35\,b\,d\,e+15\,a\,e^2\right )}{63\,c\,d^3-35\,b\,d^2\,e+15\,a\,d\,e^2}\right )\,\left (63\,c\,d^2-35\,b\,d\,e+15\,a\,e^2\right )}{8\,e^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.58, size = 235, normalized size = 1.36 \[ \frac {c x^{5}}{5 e^{3}} + x^{3} \left (\frac {b}{3 e^{3}} - \frac {c d}{e^{4}}\right ) + x \left (\frac {a}{e^{3}} - \frac {3 b d}{e^{4}} + \frac {6 c d^{2}}{e^{5}}\right ) + \frac {\sqrt {- \frac {d}{e^{11}}} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \log {\left (- e^{5} \sqrt {- \frac {d}{e^{11}}} + x \right )}}{16} - \frac {\sqrt {- \frac {d}{e^{11}}} \left (15 a e^{2} - 35 b d e + 63 c d^{2}\right ) \log {\left (e^{5} \sqrt {- \frac {d}{e^{11}}} + x \right )}}{16} + \frac {x^{3} \left (9 a d e^{3} - 13 b d^{2} e^{2} + 17 c d^{3} e\right ) + x \left (7 a d^{2} e^{2} - 11 b d^{3} e + 15 c d^{4}\right )}{8 d^{2} e^{5} + 16 d e^{6} x^{2} + 8 e^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x**5/(5*e**3) + x**3*(b/(3*e**3) - c*d/e**4) + x*(a/e**3 - 3*b*d/e**4 + 6*c*d**2/e**5) + sqrt(-d/e**11)*(15*
a*e**2 - 35*b*d*e + 63*c*d**2)*log(-e**5*sqrt(-d/e**11) + x)/16 - sqrt(-d/e**11)*(15*a*e**2 - 35*b*d*e + 63*c*
d**2)*log(e**5*sqrt(-d/e**11) + x)/16 + (x**3*(9*a*d*e**3 - 13*b*d**2*e**2 + 17*c*d**3*e) + x*(7*a*d**2*e**2 -
11*b*d**3*e + 15*c*d**4))/(8*d**2*e**5 + 16*d*e**6*x**2 + 8*e**7*x**4)

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