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

Optimal. Leaf size=223 \[ -\frac {x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {x \left (7 a^2-\frac {18 a c d^2}{e^2}-\frac {25 c^2 d^4}{e^4}\right )}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{128 d^{9/2} e^{9/2}}+\frac {x \left (35 a^2+\frac {6 a c d^2}{e^2}+\frac {163 c^2 d^4}{e^4}\right )}{192 d^3 \left (d+e x^2\right )^2}+\frac {x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4} \]

[Out]

1/8*(a*e^2+c*d^2)^2*x/d/e^4/(e*x^2+d)^4+1/48*(7*a^2-25*c^2*d^4/e^4-18*a*c*d^2/e^2)*x/d^2/(e*x^2+d)^3+1/192*(35
*a^2+163*c^2*d^4/e^4+6*a*c*d^2/e^2)*x/d^3/(e*x^2+d)^2-1/128*(-35*a^2*e^4-6*a*c*d^2*e^2+93*c^2*d^4)*x/d^4/e^4/(
e*x^2+d)+1/128*(35*a^2*e^4+6*a*c*d^2*e^2+35*c^2*d^4)*arctan(x*e^(1/2)/d^(1/2))/d^(9/2)/e^(9/2)

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Rubi [A]  time = 0.34, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1158, 1814, 1157, 385, 205} \[ -\frac {x \left (-35 a^2 e^4-6 a c d^2 e^2+93 c^2 d^4\right )}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {x \left (35 a^2+\frac {6 a c d^2}{e^2}+\frac {163 c^2 d^4}{e^4}\right )}{192 d^3 \left (d+e x^2\right )^2}+\frac {x \left (7 a^2-\frac {18 a c d^2}{e^2}-\frac {25 c^2 d^4}{e^4}\right )}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{128 d^{9/2} e^{9/2}}+\frac {x \left (a e^2+c d^2\right )^2}{8 d e^4 \left (d+e x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/(d + e*x^2)^5,x]

[Out]

((c*d^2 + a*e^2)^2*x)/(8*d*e^4*(d + e*x^2)^4) + ((7*a^2 - (25*c^2*d^4)/e^4 - (18*a*c*d^2)/e^2)*x)/(48*d^2*(d +
e*x^2)^3) + ((35*a^2 + (163*c^2*d^4)/e^4 + (6*a*c*d^2)/e^2)*x)/(192*d^3*(d + e*x^2)^2) - ((93*c^2*d^4 - 6*a*c
*d^2*e^2 - 35*a^2*e^4)*x)/(128*d^4*e^4*(d + e*x^2)) + ((35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*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 1158

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + c*
x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + 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, c, d, e}, x] && NeQ[c*d^2 + 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+c x^4\right )^2}{\left (d+e x^2\right )^5} \, dx &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}-\frac {\int \frac {-7 a^2+\frac {c^2 d^4}{e^4}+\frac {2 a c d^2}{e^2}-\frac {8 c d \left (c d^2+2 a e^2\right ) x^2}{e^3}+\frac {8 c^2 d^2 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+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\int \frac {35 a^2+\frac {19 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}-\frac {96 c^2 d^3 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+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\int \frac {-3 \left (35 a^2-\frac {29 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right )-\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+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\right ) \int \frac {1}{d+e x^2} \, dx}{128 d^4 e^4}\\ &=\frac {\left (c d^2+a e^2\right )^2 x}{8 d e^4 \left (d+e x^2\right )^4}+\frac {\left (7 a^2-\frac {25 c^2 d^4}{e^4}-\frac {18 a c d^2}{e^2}\right ) x}{48 d^2 \left (d+e x^2\right )^3}+\frac {\left (35 a^2+\frac {163 c^2 d^4}{e^4}+\frac {6 a c d^2}{e^2}\right ) x}{192 d^3 \left (d+e x^2\right )^2}-\frac {\left (93 c^2 d^4-6 a c d^2 e^2-35 a^2 e^4\right ) x}{128 d^4 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 a c d^2 e^2+35 a^2 e^4\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 = 0.19, size = 200, normalized size = 0.90 \[ \frac {3 \left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+\frac {\sqrt {d} \sqrt {e} x \left (a^2 e^4 \left (279 d^3+511 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )-6 a c d^2 e^2 \left (3 d^3+11 d^2 e x^2-11 d e^2 x^4-3 e^3 x^6\right )-c^2 d^4 \left (105 d^3+385 d^2 e x^2+511 d e^2 x^4+279 e^3 x^6\right )\right )}{\left (d+e x^2\right )^4}}{384 d^{9/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[d]*Sqrt[e]*x*(-6*a*c*d^2*e^2*(3*d^3 + 11*d^2*e*x^2 - 11*d*e^2*x^4 - 3*e^3*x^6) + a^2*e^4*(279*d^3 + 511
*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) - c^2*d^4*(105*d^3 + 385*d^2*e*x^2 + 511*d*e^2*x^4 + 279*e^3*x^6)))/
(d + e*x^2)^4 + 3*(35*c^2*d^4 + 6*a*c*d^2*e^2 + 35*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(384*d^(9/2)*e^(9/2))

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fricas [A]  time = 0.45, size = 806, normalized size = 3.61 \[ \left [-\frac {6 \, {\left (93 \, c^{2} d^{5} e^{4} - 6 \, a c d^{3} e^{6} - 35 \, a^{2} d e^{8}\right )} x^{7} + 2 \, {\left (511 \, c^{2} d^{6} e^{3} - 66 \, a c d^{4} e^{5} - 385 \, a^{2} d^{2} e^{7}\right )} x^{5} + 2 \, {\left (385 \, c^{2} d^{7} e^{2} + 66 \, a c d^{5} e^{4} - 511 \, a^{2} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (35 \, c^{2} d^{8} + 6 \, a c d^{6} e^{2} + 35 \, a^{2} d^{4} e^{4} + {\left (35 \, c^{2} d^{4} e^{4} + 6 \, a c d^{2} e^{6} + 35 \, a^{2} e^{8}\right )} x^{8} + 4 \, {\left (35 \, c^{2} d^{5} e^{3} + 6 \, a c d^{3} e^{5} + 35 \, a^{2} d e^{7}\right )} x^{6} + 6 \, {\left (35 \, c^{2} d^{6} e^{2} + 6 \, a c d^{4} e^{4} + 35 \, a^{2} d^{2} e^{6}\right )} x^{4} + 4 \, {\left (35 \, c^{2} d^{7} e + 6 \, a c d^{5} e^{3} + 35 \, a^{2} d^{3} e^{5}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 6 \, {\left (35 \, c^{2} d^{8} e + 6 \, a c d^{6} e^{3} - 93 \, a^{2} d^{4} e^{5}\right )} x}{768 \, {\left (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}\right )}}, -\frac {3 \, {\left (93 \, c^{2} d^{5} e^{4} - 6 \, a c d^{3} e^{6} - 35 \, a^{2} d e^{8}\right )} x^{7} + {\left (511 \, c^{2} d^{6} e^{3} - 66 \, a c d^{4} e^{5} - 385 \, a^{2} d^{2} e^{7}\right )} x^{5} + {\left (385 \, c^{2} d^{7} e^{2} + 66 \, a c d^{5} e^{4} - 511 \, a^{2} d^{3} e^{6}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{8} + 6 \, a c d^{6} e^{2} + 35 \, a^{2} d^{4} e^{4} + {\left (35 \, c^{2} d^{4} e^{4} + 6 \, a c d^{2} e^{6} + 35 \, a^{2} e^{8}\right )} x^{8} + 4 \, {\left (35 \, c^{2} d^{5} e^{3} + 6 \, a c d^{3} e^{5} + 35 \, a^{2} d e^{7}\right )} x^{6} + 6 \, {\left (35 \, c^{2} d^{6} e^{2} + 6 \, a c d^{4} e^{4} + 35 \, a^{2} d^{2} e^{6}\right )} x^{4} + 4 \, {\left (35 \, c^{2} d^{7} e + 6 \, a c d^{5} e^{3} + 35 \, a^{2} d^{3} e^{5}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + 3 \, {\left (35 \, c^{2} d^{8} e + 6 \, a c d^{6} e^{3} - 93 \, a^{2} d^{4} e^{5}\right )} x}{384 \, {\left (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}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(6*(93*c^2*d^5*e^4 - 6*a*c*d^3*e^6 - 35*a^2*d*e^8)*x^7 + 2*(511*c^2*d^6*e^3 - 66*a*c*d^4*e^5 - 385*a^2
*d^2*e^7)*x^5 + 2*(385*c^2*d^7*e^2 + 66*a*c*d^5*e^4 - 511*a^2*d^3*e^6)*x^3 + 3*(35*c^2*d^8 + 6*a*c*d^6*e^2 + 3
5*a^2*d^4*e^4 + (35*c^2*d^4*e^4 + 6*a*c*d^2*e^6 + 35*a^2*e^8)*x^8 + 4*(35*c^2*d^5*e^3 + 6*a*c*d^3*e^5 + 35*a^2
*d*e^7)*x^6 + 6*(35*c^2*d^6*e^2 + 6*a*c*d^4*e^4 + 35*a^2*d^2*e^6)*x^4 + 4*(35*c^2*d^7*e + 6*a*c*d^5*e^3 + 35*a
^2*d^3*e^5)*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 + 6*a*c*d^6*e^3 -
93*a^2*d^4*e^5)*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 - 6*a*c*d^3*e^6 - 35*a^2*d*e^8)*x^7 + (511*c^2*d^6*e^3 - 66*a*c*d^4*e^5 - 385*a^2*d^2*e^7)*x^5 + (385*
c^2*d^7*e^2 + 66*a*c*d^5*e^4 - 511*a^2*d^3*e^6)*x^3 - 3*(35*c^2*d^8 + 6*a*c*d^6*e^2 + 35*a^2*d^4*e^4 + (35*c^2
*d^4*e^4 + 6*a*c*d^2*e^6 + 35*a^2*e^8)*x^8 + 4*(35*c^2*d^5*e^3 + 6*a*c*d^3*e^5 + 35*a^2*d*e^7)*x^6 + 6*(35*c^2
*d^6*e^2 + 6*a*c*d^4*e^4 + 35*a^2*d^2*e^6)*x^4 + 4*(35*c^2*d^7*e + 6*a*c*d^5*e^3 + 35*a^2*d^3*e^5)*x^2)*sqrt(d
*e)*arctan(sqrt(d*e)*x/d) + 3*(35*c^2*d^8*e + 6*a*c*d^6*e^3 - 93*a^2*d^4*e^5)*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 = 0.25, size = 198, normalized size = 0.89 \[ \frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{128 \, d^{\frac {9}{2}}} - \frac {{\left (279 \, c^{2} d^{4} x^{7} e^{3} + 511 \, c^{2} d^{5} x^{5} e^{2} - 18 \, a c d^{2} x^{7} e^{5} + 385 \, c^{2} d^{6} x^{3} e - 66 \, a c d^{3} x^{5} e^{4} + 105 \, c^{2} d^{7} x - 105 \, a^{2} x^{7} e^{7} + 66 \, a c d^{4} x^{3} e^{3} - 385 \, a^{2} d x^{5} e^{6} + 18 \, a c d^{5} x e^{2} - 511 \, a^{2} d^{2} x^{3} e^{5} - 279 \, a^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )}}{384 \, {\left (x^{2} e + d\right )}^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(35*c^2*d^4 + 6*a*c*d^2*e^2 + 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 + 511*c^2*d^5*x^5*e^2 - 18*a*c*d^2*x^7*e^5 + 385*c^2*d^6*x^3*e - 66*a*c*d^3*x^5*e^4 + 105*c^2*d^7*x
- 105*a^2*x^7*e^7 + 66*a*c*d^4*x^3*e^3 - 385*a^2*d*x^5*e^6 + 18*a*c*d^5*x*e^2 - 511*a^2*d^2*x^3*e^5 - 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 = 231, normalized size = 1.04 \[ \frac {35 a^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \sqrt {d e}\, d^{4}}+\frac {3 a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{64 \sqrt {d e}\, d^{2} e^{2}}+\frac {35 c^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \sqrt {d e}\, e^{4}}+\frac {\frac {\left (35 a^{2} e^{4}+6 a c \,d^{2} e^{2}-93 c^{2} d^{4}\right ) x^{7}}{128 d^{4} e}+\frac {\left (385 a^{2} e^{4}+66 a c \,d^{2} e^{2}-511 c^{2} d^{4}\right ) x^{5}}{384 d^{3} e^{2}}+\frac {\left (511 a^{2} e^{4}-66 a c \,d^{2} e^{2}-385 c^{2} d^{4}\right ) x^{3}}{384 d^{2} e^{3}}+\frac {\left (93 a^{2} e^{4}-6 a c \,d^{2} e^{2}-35 c^{2} d^{4}\right ) x}{128 d \,e^{4}}}{\left (e \,x^{2}+d \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.41, size = 244, normalized size = 1.09 \[ -\frac {3 \, {\left (93 \, c^{2} d^{4} e^{3} - 6 \, a c d^{2} e^{5} - 35 \, a^{2} e^{7}\right )} x^{7} + {\left (511 \, c^{2} d^{5} e^{2} - 66 \, a c d^{3} e^{4} - 385 \, a^{2} d e^{6}\right )} x^{5} + {\left (385 \, c^{2} d^{6} e + 66 \, a c d^{4} e^{3} - 511 \, a^{2} d^{2} e^{5}\right )} x^{3} + 3 \, {\left (35 \, c^{2} d^{7} + 6 \, a c d^{5} e^{2} - 93 \, a^{2} d^{3} e^{4}\right )} x}{384 \, {\left (d^{4} e^{8} x^{8} + 4 \, d^{5} e^{7} x^{6} + 6 \, d^{6} e^{6} x^{4} + 4 \, d^{7} e^{5} x^{2} + d^{8} e^{4}\right )}} + \frac {{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 35 \, a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{128 \, \sqrt {d e} d^{4} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(3*(93*c^2*d^4*e^3 - 6*a*c*d^2*e^5 - 35*a^2*e^7)*x^7 + (511*c^2*d^5*e^2 - 66*a*c*d^3*e^4 - 385*a^2*d*e^
6)*x^5 + (385*c^2*d^6*e + 66*a*c*d^4*e^3 - 511*a^2*d^2*e^5)*x^3 + 3*(35*c^2*d^7 + 6*a*c*d^5*e^2 - 93*a^2*d^3*e
^4)*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 + 6*a*c*d^2
*e^2 + 35*a^2*e^4)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d^4*e^4)

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mupad [B]  time = 4.49, size = 240, normalized size = 1.08 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (35\,a^2\,e^4+6\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{128\,d^{9/2}\,e^{9/2}}-\frac {\frac {x\,\left (-93\,a^2\,e^4+6\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{128\,d\,e^4}-\frac {x^7\,\left (35\,a^2\,e^4+6\,a\,c\,d^2\,e^2-93\,c^2\,d^4\right )}{128\,d^4\,e}+\frac {x^3\,\left (-511\,a^2\,e^4+66\,a\,c\,d^2\,e^2+385\,c^2\,d^4\right )}{384\,d^2\,e^3}-\frac {x^5\,\left (385\,a^2\,e^4+66\,a\,c\,d^2\,e^2-511\,c^2\,d^4\right )}{384\,d^3\,e^2}}{d^4+4\,d^3\,e\,x^2+6\,d^2\,e^2\,x^4+4\,d\,e^3\,x^6+e^4\,x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + 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 + 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 + 6*a*c*d^2*e^2))/(128*d*e^4) - (x^7*(35*a^2*e^4 - 93*c^2*d^4 + 6*a*c*d^2*e^2))/(128*d^4*e) + (x^
3*(385*c^2*d^4 - 511*a^2*e^4 + 66*a*c*d^2*e^2))/(384*d^2*e^3) - (x^5*(385*a^2*e^4 - 511*c^2*d^4 + 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 [A]  time = 4.11, size = 335, normalized size = 1.50 \[ - \frac {\sqrt {- \frac {1}{d^{9} e^{9}}} \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (- d^{5} e^{4} \sqrt {- \frac {1}{d^{9} e^{9}}} + x \right )}}{256} + \frac {\sqrt {- \frac {1}{d^{9} e^{9}}} \left (35 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log {\left (d^{5} e^{4} \sqrt {- \frac {1}{d^{9} e^{9}}} + x \right )}}{256} + \frac {x^{7} \left (105 a^{2} e^{7} + 18 a c d^{2} e^{5} - 279 c^{2} d^{4} e^{3}\right ) + x^{5} \left (385 a^{2} d e^{6} + 66 a c d^{3} e^{4} - 511 c^{2} d^{5} e^{2}\right ) + x^{3} \left (511 a^{2} d^{2} e^{5} - 66 a c d^{4} e^{3} - 385 c^{2} d^{6} e\right ) + x \left (279 a^{2} d^{3} e^{4} - 18 a c d^{5} e^{2} - 105 c^{2} d^{7}\right )}{384 d^{8} e^{4} + 1536 d^{7} e^{5} x^{2} + 2304 d^{6} e^{6} x^{4} + 1536 d^{5} e^{7} x^{6} + 384 d^{4} e^{8} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(d**9*e**9))*(35*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*d**4)*log(-d**5*e**4*sqrt(-1/(d**9*e**9)) + x)
/256 + sqrt(-1/(d**9*e**9))*(35*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*d**4)*log(d**5*e**4*sqrt(-1/(d**9*e**9))
+ x)/256 + (x**7*(105*a**2*e**7 + 18*a*c*d**2*e**5 - 279*c**2*d**4*e**3) + x**5*(385*a**2*d*e**6 + 66*a*c*d**
3*e**4 - 511*c**2*d**5*e**2) + x**3*(511*a**2*d**2*e**5 - 66*a*c*d**4*e**3 - 385*c**2*d**6*e) + x*(279*a**2*d*
*3*e**4 - 18*a*c*d**5*e**2 - 105*c**2*d**7))/(384*d**8*e**4 + 1536*d**7*e**5*x**2 + 2304*d**6*e**6*x**4 + 1536
*d**5*e**7*x**6 + 384*d**4*e**8*x**8)

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