3.260 \(\int \frac {(a+b x^2+c x^4)^2}{(d+e x^2)^5} \, dx\)

Optimal. Leaf size=317 \[ -\frac {x \left (-e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right )}{128 d^{9/2} e^{9/2}}+\frac {x \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (59 b d-3 a e)+163 c^2 d^4\right )}{192 d^3 e^4 \left (d+e x^2\right )^2}+\frac {x \left (a e^2-b d e+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4}-\frac {x \left (-7 a e^2-9 b d e+25 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{48 d^2 e^4 \left (d+e x^2\right )^3} \]

[Out]

1/8*(a*e^2-b*d*e+c*d^2)^2*x/d/e^4/(e*x^2+d)^4-1/48*(-7*a*e^2-9*b*d*e+25*c*d^2)*(a*e^2-b*d*e+c*d^2)*x/d^2/e^4/(
e*x^2+d)^3+1/192*(163*c^2*d^4-2*c*d^2*e*(-3*a*e+59*b*d)+e^2*(35*a^2*e^2+10*a*b*d*e+3*b^2*d^2))*x/d^3/e^4/(e*x^
2+d)^2-1/128*(93*c^2*d^4-2*c*d^2*e*(3*a*e+5*b*d)-e^2*(35*a^2*e^2+10*a*b*d*e+3*b^2*d^2))*x/d^4/e^4/(e*x^2+d)+1/
128*(35*c^2*d^4+2*c*d^2*e*(3*a*e+5*b*d)+e^2*(35*a^2*e^2+10*a*b*d*e+3*b^2*d^2))*arctan(x*e^(1/2)/d^(1/2))/d^(9/
2)/e^(9/2)

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Rubi [A]  time = 0.65, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1157, 1814, 385, 205} \[ -\frac {x \left (-e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {x \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (59 b d-3 a e)+163 c^2 d^4\right )}{192 d^3 e^4 \left (d+e x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right )}{128 d^{9/2} e^{9/2}}+\frac {x \left (a e^2-b d e+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4}-\frac {x \left (-7 a e^2-9 b d e+25 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{48 d^2 e^4 \left (d+e x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((c*d^2 - b*d*e + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) - ((25*c*d^2 - 9*b*d*e - 7*a*e^2)*(c*d^2 - b*d*e + a*e^2
)*x)/(48*d^2*e^4*(d + e*x^2)^3) + ((163*c^2*d^4 - 2*c*d^2*e*(59*b*d - 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 3
5*a^2*e^2))*x)/(192*d^3*e^4*(d + e*x^2)^2) - ((93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d^2 + 10*a*
b*d*e + 35*a^2*e^2))*x)/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 2*c*d^2*e*(5*b*d + 3*a*e) + e^2*(3*b^2*d^2
+ 10*a*b*d*e + 35*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(128*d^(9/2)*e^(9/2))

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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 {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^5} \, dx &=\frac {\left (c d^2-b d e+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\int \frac {\frac {c^2 d^4-2 c d^2 e (b d-a e)+e^2 \left (b^2 d^2-2 a b d e-7 a^2 e^2\right )}{e^4}-\frac {8 d \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}+\frac {8 c d (c d-2 b e) x^4}{e^2}-\frac {8 c^2 d x^6}{e}}{\left (d+e x^2\right )^4} \, dx}{8 d}\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\left (25 c d^2-9 b d e-7 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{48 d^2 e^4 \left (d+e x^2\right )^3}+\frac {\int \frac {\frac {19 c^2 d^4-2 c d^2 e (11 b d-3 a e)+e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )}{e^4}-\frac {96 c d^2 (c d-b e) x^2}{e^3}+\frac {48 c^2 d^2 x^4}{e^2}}{\left (d+e x^2\right )^3} \, dx}{48 d^2}\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\left (25 c d^2-9 b d e-7 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{48 d^2 e^4 \left (d+e x^2\right )^3}+\frac {\left (163 c^2 d^4-2 c d^2 e (59 b d-3 a e)+e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) x}{192 d^3 e^4 \left (d+e x^2\right )^2}-\frac {\int \frac {\frac {3 \left (29 c^2 d^4-2 c d^2 e (5 b d+3 a e)-e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right )}{e^4}-\frac {192 c^2 d^3 x^2}{e^3}}{\left (d+e x^2\right )^2} \, dx}{192 d^3}\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\left (25 c d^2-9 b d e-7 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{48 d^2 e^4 \left (d+e x^2\right )^3}+\frac {\left (163 c^2 d^4-2 c d^2 e (59 b d-3 a e)+e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) x}{192 d^3 e^4 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-2 c d^2 e (5 b d+3 a e)-e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+2 c d^2 e (5 b d+3 a e)+e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) \int \frac {1}{d+e x^2} \, dx}{128 d^4 e^4}\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\left (25 c d^2-9 b d e-7 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{48 d^2 e^4 \left (d+e x^2\right )^3}+\frac {\left (163 c^2 d^4-2 c d^2 e (59 b d-3 a e)+e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) x}{192 d^3 e^4 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-2 c d^2 e (5 b d+3 a e)-e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+2 c d^2 e (5 b d+3 a e)+e^2 \left (3 b^2 d^2+10 a b d e+35 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{128 d^{9/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 345, normalized size = 1.09 \[ \frac {-\frac {3 \sqrt {d} \sqrt {e} x \left (-e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )-2 c d^2 e (3 a e+5 b d)+93 c^2 d^4\right )}{d+e x^2}+3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e+5 b d)+35 c^2 d^4\right )-\frac {8 d^{5/2} \sqrt {e} x \left (e^2 \left (-7 a^2 e^2-2 a b d e+9 b^2 d^2\right )+2 c d^2 e (9 a e-17 b d)+25 c^2 d^4\right )}{\left (d+e x^2\right )^3}+\frac {2 d^{3/2} \sqrt {e} x \left (e^2 \left (35 a^2 e^2+10 a b d e+3 b^2 d^2\right )+2 c d^2 e (3 a e-59 b d)+163 c^2 d^4\right )}{\left (d+e x^2\right )^2}+\frac {48 d^{7/2} \sqrt {e} x \left (e (a e-b d)+c d^2\right )^2}{\left (d+e x^2\right )^4}}{384 d^{9/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((48*d^(7/2)*Sqrt[e]*(c*d^2 + e*(-(b*d) + a*e))^2*x)/(d + e*x^2)^4 - (8*d^(5/2)*Sqrt[e]*(25*c^2*d^4 + 2*c*d^2*
e*(-17*b*d + 9*a*e) + e^2*(9*b^2*d^2 - 2*a*b*d*e - 7*a^2*e^2))*x)/(d + e*x^2)^3 + (2*d^(3/2)*Sqrt[e]*(163*c^2*
d^4 + 2*c*d^2*e*(-59*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(d + e*x^2)^2 - (3*Sqrt[d]*S
qrt[e]*(93*c^2*d^4 - 2*c*d^2*e*(5*b*d + 3*a*e) - e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*x)/(d + e*x^2) + 3
*(35*c^2*d^4 + 2*c*d^2*e*(5*b*d + 3*a*e) + e^2*(3*b^2*d^2 + 10*a*b*d*e + 35*a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[
d]])/(384*d^(9/2)*e^(9/2))

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fricas [B]  time = 0.74, size = 1266, normalized size = 3.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/(e*x^2+d)^5,x, algorithm="fricas")

[Out]

[-1/768*(6*(93*c^2*d^5*e^4 - 10*b*c*d^4*e^5 - 10*a*b*d^2*e^7 - 35*a^2*d*e^8 - 3*(b^2 + 2*a*c)*d^3*e^6)*x^7 + 2
*(511*c^2*d^6*e^3 + 146*b*c*d^5*e^4 - 110*a*b*d^3*e^6 - 385*a^2*d^2*e^7 - 33*(b^2 + 2*a*c)*d^4*e^5)*x^5 + 2*(3
85*c^2*d^7*e^2 + 110*b*c*d^6*e^3 - 146*a*b*d^4*e^5 - 511*a^2*d^3*e^6 + 33*(b^2 + 2*a*c)*d^5*e^4)*x^3 + 3*(35*c
^2*d^8 + 10*b*c*d^7*e + 10*a*b*d^5*e^3 + 35*a^2*d^4*e^4 + 3*(b^2 + 2*a*c)*d^6*e^2 + (35*c^2*d^4*e^4 + 10*b*c*d
^3*e^5 + 10*a*b*d*e^7 + 35*a^2*e^8 + 3*(b^2 + 2*a*c)*d^2*e^6)*x^8 + 4*(35*c^2*d^5*e^3 + 10*b*c*d^4*e^4 + 10*a*
b*d^2*e^6 + 35*a^2*d*e^7 + 3*(b^2 + 2*a*c)*d^3*e^5)*x^6 + 6*(35*c^2*d^6*e^2 + 10*b*c*d^5*e^3 + 10*a*b*d^3*e^5
+ 35*a^2*d^2*e^6 + 3*(b^2 + 2*a*c)*d^4*e^4)*x^4 + 4*(35*c^2*d^7*e + 10*b*c*d^6*e^2 + 10*a*b*d^4*e^4 + 35*a^2*d
^3*e^5 + 3*(b^2 + 2*a*c)*d^5*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) + 6*(35*c^2*d^
8*e + 10*b*c*d^7*e^2 + 10*a*b*d^5*e^4 - 93*a^2*d^4*e^5 + 3*(b^2 + 2*a*c)*d^6*e^3)*x)/(d^5*e^9*x^8 + 4*d^6*e^8*
x^6 + 6*d^7*e^7*x^4 + 4*d^8*e^6*x^2 + d^9*e^5), -1/384*(3*(93*c^2*d^5*e^4 - 10*b*c*d^4*e^5 - 10*a*b*d^2*e^7 -
35*a^2*d*e^8 - 3*(b^2 + 2*a*c)*d^3*e^6)*x^7 + (511*c^2*d^6*e^3 + 146*b*c*d^5*e^4 - 110*a*b*d^3*e^6 - 385*a^2*d
^2*e^7 - 33*(b^2 + 2*a*c)*d^4*e^5)*x^5 + (385*c^2*d^7*e^2 + 110*b*c*d^6*e^3 - 146*a*b*d^4*e^5 - 511*a^2*d^3*e^
6 + 33*(b^2 + 2*a*c)*d^5*e^4)*x^3 - 3*(35*c^2*d^8 + 10*b*c*d^7*e + 10*a*b*d^5*e^3 + 35*a^2*d^4*e^4 + 3*(b^2 +
2*a*c)*d^6*e^2 + (35*c^2*d^4*e^4 + 10*b*c*d^3*e^5 + 10*a*b*d*e^7 + 35*a^2*e^8 + 3*(b^2 + 2*a*c)*d^2*e^6)*x^8 +
4*(35*c^2*d^5*e^3 + 10*b*c*d^4*e^4 + 10*a*b*d^2*e^6 + 35*a^2*d*e^7 + 3*(b^2 + 2*a*c)*d^3*e^5)*x^6 + 6*(35*c^2
*d^6*e^2 + 10*b*c*d^5*e^3 + 10*a*b*d^3*e^5 + 35*a^2*d^2*e^6 + 3*(b^2 + 2*a*c)*d^4*e^4)*x^4 + 4*(35*c^2*d^7*e +
10*b*c*d^6*e^2 + 10*a*b*d^4*e^4 + 35*a^2*d^3*e^5 + 3*(b^2 + 2*a*c)*d^5*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x
/d) + 3*(35*c^2*d^8*e + 10*b*c*d^7*e^2 + 10*a*b*d^5*e^4 - 93*a^2*d^4*e^5 + 3*(b^2 + 2*a*c)*d^6*e^3)*x)/(d^5*e^
9*x^8 + 4*d^6*e^8*x^6 + 6*d^7*e^7*x^4 + 4*d^8*e^6*x^2 + d^9*e^5)]

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giac [A]  time = 0.19, size = 364, normalized size = 1.15 \[ \frac {{\left (35 \, c^{2} d^{4} + 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \, a b d e^{3} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{128 \, d^{\frac {9}{2}}} - \frac {{\left (279 \, c^{2} d^{4} x^{7} e^{3} - 30 \, b c d^{3} x^{7} e^{4} + 511 \, c^{2} d^{5} x^{5} e^{2} - 9 \, b^{2} d^{2} x^{7} e^{5} - 18 \, a c d^{2} x^{7} e^{5} + 146 \, b c d^{4} x^{5} e^{3} + 385 \, c^{2} d^{6} x^{3} e - 30 \, a b d x^{7} e^{6} - 33 \, b^{2} d^{3} x^{5} e^{4} - 66 \, a c d^{3} x^{5} e^{4} + 110 \, b c d^{5} x^{3} e^{2} + 105 \, c^{2} d^{7} x - 105 \, a^{2} x^{7} e^{7} - 110 \, a b d^{2} x^{5} e^{5} + 33 \, b^{2} d^{4} x^{3} e^{3} + 66 \, a c d^{4} x^{3} e^{3} + 30 \, b c d^{6} x e - 385 \, a^{2} d x^{5} e^{6} - 146 \, a b d^{3} x^{3} e^{4} + 9 \, b^{2} d^{5} x e^{2} + 18 \, a c d^{5} x e^{2} - 511 \, a^{2} d^{2} x^{3} e^{5} + 30 \, a b d^{4} x e^{3} - 279 \, a^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )}}{384 \, {\left (x^{2} e + d\right )}^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/(e*x^2+d)^5,x, algorithm="giac")

[Out]

1/128*(35*c^2*d^4 + 10*b*c*d^3*e + 3*b^2*d^2*e^2 + 6*a*c*d^2*e^2 + 10*a*b*d*e^3 + 35*a^2*e^4)*arctan(x*e^(1/2)
/sqrt(d))*e^(-9/2)/d^(9/2) - 1/384*(279*c^2*d^4*x^7*e^3 - 30*b*c*d^3*x^7*e^4 + 511*c^2*d^5*x^5*e^2 - 9*b^2*d^2
*x^7*e^5 - 18*a*c*d^2*x^7*e^5 + 146*b*c*d^4*x^5*e^3 + 385*c^2*d^6*x^3*e - 30*a*b*d*x^7*e^6 - 33*b^2*d^3*x^5*e^
4 - 66*a*c*d^3*x^5*e^4 + 110*b*c*d^5*x^3*e^2 + 105*c^2*d^7*x - 105*a^2*x^7*e^7 - 110*a*b*d^2*x^5*e^5 + 33*b^2*
d^4*x^3*e^3 + 66*a*c*d^4*x^3*e^3 + 30*b*c*d^6*x*e - 385*a^2*d*x^5*e^6 - 146*a*b*d^3*x^3*e^4 + 9*b^2*d^5*x*e^2
+ 18*a*c*d^5*x*e^2 - 511*a^2*d^2*x^3*e^5 + 30*a*b*d^4*x*e^3 - 279*a^2*d^3*x*e^4)*e^(-4)/((x^2*e + d)^4*d^4)

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maple [A]  time = 0.01, size = 412, normalized size = 1.30 \[ \frac {35 a^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \sqrt {d e}\, d^{4}}+\frac {5 a b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{64 \sqrt {d e}\, d^{3} e}+\frac {3 a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{64 \sqrt {d e}\, d^{2} e^{2}}+\frac {3 b^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \sqrt {d e}\, d^{2} e^{2}}+\frac {5 b c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{64 \sqrt {d e}\, d \,e^{3}}+\frac {35 c^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \sqrt {d e}\, e^{4}}+\frac {\frac {\left (35 a^{2} e^{4}+10 d a b \,e^{3}+6 a c \,d^{2} e^{2}+3 b^{2} d^{2} e^{2}+10 b c \,d^{3} e -93 c^{2} d^{4}\right ) x^{7}}{128 d^{4} e}+\frac {\left (385 a^{2} e^{4}+110 d a b \,e^{3}+66 a c \,d^{2} e^{2}+33 b^{2} d^{2} e^{2}-146 b c \,d^{3} e -511 c^{2} d^{4}\right ) x^{5}}{384 d^{3} e^{2}}+\frac {\left (511 a^{2} e^{4}+146 d a b \,e^{3}-66 a c \,d^{2} e^{2}-33 b^{2} d^{2} e^{2}-110 b c \,d^{3} e -385 c^{2} d^{4}\right ) x^{3}}{384 d^{2} e^{3}}+\frac {\left (93 a^{2} e^{4}-10 d a b \,e^{3}-6 a c \,d^{2} e^{2}-3 b^{2} d^{2} e^{2}-10 b c \,d^{3} e -35 c^{2} d^{4}\right ) x}{128 d \,e^{4}}}{\left (e \,x^{2}+d \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/(e*x^2+d)^5,x)

[Out]

(1/128*(35*a^2*e^4+10*a*b*d*e^3+6*a*c*d^2*e^2+3*b^2*d^2*e^2+10*b*c*d^3*e-93*c^2*d^4)/d^4/e*x^7+1/384*(385*a^2*
e^4+110*a*b*d*e^3+66*a*c*d^2*e^2+33*b^2*d^2*e^2-146*b*c*d^3*e-511*c^2*d^4)/d^3/e^2*x^5+1/384*(511*a^2*e^4+146*
a*b*d*e^3-66*a*c*d^2*e^2-33*b^2*d^2*e^2-110*b*c*d^3*e-385*c^2*d^4)/d^2/e^3*x^3+1/128*(93*a^2*e^4-10*a*b*d*e^3-
6*a*c*d^2*e^2-3*b^2*d^2*e^2-10*b*c*d^3*e-35*c^2*d^4)/d/e^4*x)/(e*x^2+d)^4+35/128/(d*e)^(1/2)*a^2/d^4*arctan(1/
(d*e)^(1/2)*e*x)+5/64/d^3/e/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*a*b+3/64/(d*e)^(1/2)*a*c/d^2/e^2*arctan(1/(d
*e)^(1/2)*e*x)+3/128/d^2/e^2/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*b^2+5/64/d/e^3/(d*e)^(1/2)*arctan(1/(d*e)^(
1/2)*e*x)*b*c+35/128/(d*e)^(1/2)*c^2/e^4*arctan(1/(d*e)^(1/2)*e*x)

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maxima [A]  time = 2.52, size = 366, normalized size = 1.15 \[ -\frac {3 \, {\left (93 \, c^{2} d^{4} e^{3} - 10 \, b c d^{3} e^{4} - 10 \, a b d e^{6} - 35 \, a^{2} e^{7} - 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{7} + {\left (511 \, c^{2} d^{5} e^{2} + 146 \, b c d^{4} e^{3} - 110 \, a b d^{2} e^{5} - 385 \, a^{2} d e^{6} - 33 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{5} + {\left (385 \, c^{2} d^{6} e + 110 \, b c d^{5} e^{2} - 146 \, a b d^{3} e^{4} - 511 \, a^{2} d^{2} e^{5} + 33 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x^{3} + 3 \, {\left (35 \, c^{2} d^{7} + 10 \, b c d^{6} e + 10 \, a b d^{4} e^{3} - 93 \, a^{2} d^{3} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{5} e^{2}\right )} x}{384 \, {\left (d^{4} e^{8} x^{8} + 4 \, d^{5} e^{7} x^{6} + 6 \, d^{6} e^{6} x^{4} + 4 \, d^{7} e^{5} x^{2} + d^{8} e^{4}\right )}} + \frac {{\left (35 \, c^{2} d^{4} + 10 \, b c d^{3} e + 10 \, a b d e^{3} + 35 \, a^{2} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \, \sqrt {d e} d^{4} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/(e*x^2+d)^5,x, algorithm="maxima")

[Out]

-1/384*(3*(93*c^2*d^4*e^3 - 10*b*c*d^3*e^4 - 10*a*b*d*e^6 - 35*a^2*e^7 - 3*(b^2 + 2*a*c)*d^2*e^5)*x^7 + (511*c
^2*d^5*e^2 + 146*b*c*d^4*e^3 - 110*a*b*d^2*e^5 - 385*a^2*d*e^6 - 33*(b^2 + 2*a*c)*d^3*e^4)*x^5 + (385*c^2*d^6*
e + 110*b*c*d^5*e^2 - 146*a*b*d^3*e^4 - 511*a^2*d^2*e^5 + 33*(b^2 + 2*a*c)*d^4*e^3)*x^3 + 3*(35*c^2*d^7 + 10*b
*c*d^6*e + 10*a*b*d^4*e^3 - 93*a^2*d^3*e^4 + 3*(b^2 + 2*a*c)*d^5*e^2)*x)/(d^4*e^8*x^8 + 4*d^5*e^7*x^6 + 6*d^6*
e^6*x^4 + 4*d^7*e^5*x^2 + d^8*e^4) + 1/128*(35*c^2*d^4 + 10*b*c*d^3*e + 10*a*b*d*e^3 + 35*a^2*e^4 + 3*(b^2 + 2
*a*c)*d^2*e^2)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^4*e^4)

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mupad [B]  time = 4.57, size = 375, normalized size = 1.18 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (35\,a^2\,e^4+10\,a\,b\,d\,e^3+6\,a\,c\,d^2\,e^2+3\,b^2\,d^2\,e^2+10\,b\,c\,d^3\,e+35\,c^2\,d^4\right )}{128\,d^{9/2}\,e^{9/2}}-\frac {\frac {x\,\left (-93\,a^2\,e^4+10\,a\,b\,d\,e^3+6\,a\,c\,d^2\,e^2+3\,b^2\,d^2\,e^2+10\,b\,c\,d^3\,e+35\,c^2\,d^4\right )}{128\,d\,e^4}-\frac {x^7\,\left (35\,a^2\,e^4+10\,a\,b\,d\,e^3+6\,a\,c\,d^2\,e^2+3\,b^2\,d^2\,e^2+10\,b\,c\,d^3\,e-93\,c^2\,d^4\right )}{128\,d^4\,e}+\frac {x^3\,\left (-511\,a^2\,e^4-146\,a\,b\,d\,e^3+66\,a\,c\,d^2\,e^2+33\,b^2\,d^2\,e^2+110\,b\,c\,d^3\,e+385\,c^2\,d^4\right )}{384\,d^2\,e^3}-\frac {x^5\,\left (385\,a^2\,e^4+110\,a\,b\,d\,e^3+66\,a\,c\,d^2\,e^2+33\,b^2\,d^2\,e^2-146\,b\,c\,d^3\,e-511\,c^2\,d^4\right )}{384\,d^3\,e^2}}{d^4+4\,d^3\,e\,x^2+6\,d^2\,e^2\,x^4+4\,d\,e^3\,x^6+e^4\,x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/(d + e*x^2)^5,x)

[Out]

(atan((e^(1/2)*x)/d^(1/2))*(35*a^2*e^4 + 35*c^2*d^4 + 3*b^2*d^2*e^2 + 10*a*b*d*e^3 + 10*b*c*d^3*e + 6*a*c*d^2*
e^2))/(128*d^(9/2)*e^(9/2)) - ((x*(35*c^2*d^4 - 93*a^2*e^4 + 3*b^2*d^2*e^2 + 10*a*b*d*e^3 + 10*b*c*d^3*e + 6*a
*c*d^2*e^2))/(128*d*e^4) - (x^7*(35*a^2*e^4 - 93*c^2*d^4 + 3*b^2*d^2*e^2 + 10*a*b*d*e^3 + 10*b*c*d^3*e + 6*a*c
*d^2*e^2))/(128*d^4*e) + (x^3*(385*c^2*d^4 - 511*a^2*e^4 + 33*b^2*d^2*e^2 - 146*a*b*d*e^3 + 110*b*c*d^3*e + 66
*a*c*d^2*e^2))/(384*d^2*e^3) - (x^5*(385*a^2*e^4 - 511*c^2*d^4 + 33*b^2*d^2*e^2 + 110*a*b*d*e^3 - 146*b*c*d^3*
e + 66*a*c*d^2*e^2))/(384*d^3*e^2))/(d^4 + e^4*x^8 + 4*d^3*e*x^2 + 4*d*e^3*x^6 + 6*d^2*e^2*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/(e*x**2+d)**5,x)

[Out]

Timed out

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