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

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

[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))

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Rubi [A]  time = 0.338634, antiderivative size = 223, normalized size of antiderivative = 1., 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 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]

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 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 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{\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*}

Mathematica [A]  time = 0.190922, size = 200, normalized size = 0.9 \[ \frac{\frac{\sqrt{d} \sqrt{e} x \left (a^2 e^4 \left (511 d^2 e x^2+279 d^3+385 d e^2 x^4+105 e^3 x^6\right )-6 a c d^2 e^2 \left (11 d^2 e x^2+3 d^3-11 d e^2 x^4-3 e^3 x^6\right )-c^2 d^4 \left (385 d^2 e x^2+105 d^3+511 d e^2 x^4+279 e^3 x^6\right )\right )}{\left (d+e x^2\right )^4}+3 \left (35 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{384 d^{9/2} e^{9/2}} \]

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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Maple [A]  time = 0.056, size = 231, normalized size = 1. \begin{align*}{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{4}} \left ({\frac{ \left ( 35\,{a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-93\,{c}^{2}{d}^{4} \right ){x}^{7}}{128\,{d}^{4}e}}+{\frac{ \left ( 385\,{a}^{2}{e}^{4}+66\,ac{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\,ac{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\,ac{d}^{2}{e}^{2}-35\,{c}^{2}{d}^{4} \right ) x}{128\,d{e}^{4}}} \right ) }+{\frac{35\,{a}^{2}}{128\,{d}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{64\,{d}^{2}{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}}{128\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

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(e*x/(d*e)^(1/2))*a^2+3/64/d^2/e^2/(d*e)^(1/2)*arctan(e*x/(d*e)
^(1/2))*a*c+35/128/e^4/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*c^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93905, size = 1697, normalized size = 7.61 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 5.93223, size = 335, normalized size = 1.5 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.12563, size = 267, normalized size = 1.2 \begin{align*} \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}} \end{align*}

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)