3.42 \(\int \frac{1}{(a+b x^2)^3 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=315 \[ \frac{3 d x (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{8 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}+\frac{d x \left (-2 a^2 d^2-13 a b c d+3 b^2 c^2\right )}{8 a^2 c \left (c+d x^2\right )^2 (b c-a d)^3}+\frac{3 b^{5/2} \left (21 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^5}-\frac{3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^5}+\frac{b x (3 b c-11 a d)}{8 a^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{b x}{4 a \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2) + (b*x)/(4*a*(b*c - a*d)*(a +
b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c
+ a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^2 - 6*a*b*
c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d
+ a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^5)

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Rubi [A]  time = 0.450865, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {414, 527, 522, 205} \[ \frac{3 d x (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{8 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}+\frac{d x \left (-2 a^2 d^2-13 a b c d+3 b^2 c^2\right )}{8 a^2 c \left (c+d x^2\right )^2 (b c-a d)^3}+\frac{3 b^{5/2} \left (21 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^5}-\frac{3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^5}+\frac{b x (3 b c-11 a d)}{8 a^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{b x}{4 a \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^3*(c + d*x^2)^3),x]

[Out]

(d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2) + (b*x)/(4*a*(b*c - a*d)*(a +
b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c
+ a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^2 - 6*a*b*
c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d
+ a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*(b*c - a*d)^5)

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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{1}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^3} \, dx &=\frac{b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}-\frac{\int \frac{-3 b c+4 a d-7 b d x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx}{4 a (b c-a d)}\\ &=\frac{b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\int \frac{3 b^2 c^2-3 a b c d+8 a^2 d^2+5 b d (3 b c-11 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac{d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac{b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\int \frac{12 \left (b^3 c^3-3 a b^2 c^2 d+8 a^2 b c d^2-2 a^3 d^3\right )+12 b d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{32 a^2 c (b c-a d)^3}\\ &=\frac{d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac{b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac{\int \frac{24 \left (b^4 c^4-5 a b^3 c^3 d+16 a^2 b^2 c^2 d^2-5 a^3 b c d^3+a^4 d^4\right )+24 b d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{64 a^2 c^2 (b c-a d)^4}\\ &=\frac{d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac{b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}-\frac{\left (3 d^3 \left (21 b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{c+d x^2} \, dx}{8 c^2 (b c-a d)^5}+\frac{\left (3 b^3 \left (b^2 c^2-6 a b c d+21 a^2 d^2\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^2 (b c-a d)^5}\\ &=\frac{d \left (3 b^2 c^2-13 a b c d-2 a^2 d^2\right ) x}{8 a^2 c (b c-a d)^3 \left (c+d x^2\right )^2}+\frac{b x}{4 a (b c-a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{b (3 b c-11 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{3 d (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) x}{8 a^2 c^2 (b c-a d)^4 \left (c+d x^2\right )}+\frac{3 b^{5/2} \left (b^2 c^2-6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^5}-\frac{3 d^{5/2} \left (21 b^2 c^2-6 a b c d+a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^5}\\ \end{align*}

Mathematica [A]  time = 0.914943, size = 233, normalized size = 0.74 \[ \frac{1}{8} \left (\frac{x (b c-a d) \left (\frac{3 b^4 c}{a^2 \left (a+b x^2\right )}+\frac{b^3 \left (-17 a d+2 b c-15 b d x^2\right )}{a \left (a+b x^2\right )^2}-\frac{d^3 \left (-2 a d+17 b c+15 b d x^2\right )}{c \left (c+d x^2\right )^2}+\frac{3 a d^4}{c^2 \left (c+d x^2\right )}\right )-\frac{3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2}}}{(b c-a d)^5}-\frac{3 b^{5/2} \left (21 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (a d-b c)^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)^3*(c + d*x^2)^3),x]

[Out]

((-3*b^(5/2)*(b^2*c^2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^5) + ((b*
c - a*d)*x*((3*b^4*c)/(a^2*(a + b*x^2)) + (3*a*d^4)/(c^2*(c + d*x^2)) + (b^3*(2*b*c - 17*a*d - 15*b*d*x^2))/(a
*(a + b*x^2)^2) - (d^3*(17*b*c - 2*a*d + 15*b*d*x^2))/(c*(c + d*x^2)^2)) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d
+ a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(5/2))/(b*c - a*d)^5)/8

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Maple [A]  time = 0.015, size = 568, normalized size = 1.8 \begin{align*}{\frac{3\,{d}^{6}{x}^{3}{a}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}{c}^{2}}}-{\frac{9\,{d}^{5}{x}^{3}ab}{4\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{15\,{d}^{4}{x}^{3}{b}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,{d}^{5}x{a}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}c}}-{\frac{11\,{d}^{4}xab}{4\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{17\,{d}^{3}cx{b}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{d}^{5}{a}^{2}}{8\, \left ( ad-bc \right ) ^{5}{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{9\,a{d}^{4}b}{4\, \left ( ad-bc \right ) ^{5}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{63\,{d}^{3}{b}^{2}}{8\, \left ( ad-bc \right ) ^{5}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{4}{x}^{3}{d}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,{b}^{5}{x}^{3}cd}{4\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{3\,{b}^{6}{x}^{3}{c}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}-{\frac{17\,{b}^{3}xa{d}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{b}^{4}xcd}{4\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{b}^{5}x{c}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{63\,{d}^{2}{b}^{3}}{8\, \left ( ad-bc \right ) ^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{b}^{4}cd}{4\, \left ( ad-bc \right ) ^{5}a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,{b}^{5}{c}^{2}}{8\, \left ( ad-bc \right ) ^{5}{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^3/(d*x^2+c)^3,x)

[Out]

3/8*d^6/(a*d-b*c)^5/(d*x^2+c)^2/c^2*x^3*a^2-9/4*d^5/(a*d-b*c)^5/(d*x^2+c)^2/c*x^3*a*b+15/8*d^4/(a*d-b*c)^5/(d*
x^2+c)^2*x^3*b^2+5/8*d^5/(a*d-b*c)^5/(d*x^2+c)^2/c*x*a^2-11/4*d^4/(a*d-b*c)^5/(d*x^2+c)^2*x*a*b+17/8*d^3/(a*d-
b*c)^5/(d*x^2+c)^2*c*x*b^2+3/8*d^5/(a*d-b*c)^5/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-9/4*d^4/(a*d-b*c)^5
/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b+63/8*d^3/(a*d-b*c)^5/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2-15/8*b
^4/(a*d-b*c)^5/(b*x^2+a)^2*x^3*d^2+9/4*b^5/(a*d-b*c)^5/(b*x^2+a)^2/a*x^3*c*d-3/8*b^6/(a*d-b*c)^5/(b*x^2+a)^2/a
^2*x^3*c^2-17/8*b^3/(a*d-b*c)^5/(b*x^2+a)^2*x*a*d^2+11/4*b^4/(a*d-b*c)^5/(b*x^2+a)^2*x*c*d-5/8*b^5/(a*d-b*c)^5
/(b*x^2+a)^2*x/a*c^2-63/8*b^3/(a*d-b*c)^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^2+9/4*b^4/(a*d-b*c)^5/a/(a*b)^
(1/2)*arctan(b*x/(a*b)^(1/2))*c*d-3/8*b^5/(a*d-b*c)^5/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(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(1/(b*x^2+a)^3/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 72.7818, size = 9997, normalized size = 31.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[1/16*(6*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a*b^5*c^4*d
^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*a*b^5*c^5*
d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 - 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*
d + 21*a^4*b^2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4
*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^
3*c^3*d^3 + 21*a^4*b^2*c^2*d^4)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3
)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 +
a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^
2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^
5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt
(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^
2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^
2 - 10*a^7*b^2*c^6*d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5
*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 +
5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d
- 9*a^4*b^5*c^7*d^2 + 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^
7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*
d^6)*x^2), 1/16*(6*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a
*b^5*c^4*d^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*
a*b^5*c^5*d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 - 6*(21*a^4*b^2*c^4*d^2
- 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^
3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a
^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 +
a^6*c*d^5)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^4*b^2*c^4*d^2 + (b^6*
c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21
*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 + 21*a^4*b^2*c^2*d^4
)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x^2)*sqrt(-b/a)*log((b*x^2 -
2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*
d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*
d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c
^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 -
5*a^6*b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d - 9*a^4*b^5*c^7*d^2
+ 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6
*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2), 1/16*(6*
(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a*b^5*c^4*d^2 - 9*a^
2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*a*b^5*c^5*d - 29*a^
2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + 6*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^
4*b^2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15
*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3
+ 21*a^4*b^2*c^2*d^4)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x^2)*sq
rt(b/a)*arctan(x*sqrt(b/a)) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*
a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x
^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b
^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) -
c)/(d*x^2 + c)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*
d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*d^3 + 5*a^8*b*c^5*d
^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*
c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5
+ 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d - 9*a^4*b^5*c^7*d^2 + 25*a^5*b^4*c^6*d
^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^
8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2), 1/8*(3*(b^6*c^4*d^2 - 6*a*b
^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + (6*b^6*c^5*d - 31*a*b^5*c^4*d^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*
b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + (3*b^6*c^6 - 8*a*b^5*c^5*d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b
^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^4*b^2*c^4*d^2 + (b^6*c^4
*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^
3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 + 21*a^4*b^2*c^2*d^4)*x
^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/
a)) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*
d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2
+ 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*
c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (5*a*b^5*c^6 - 22*a^2*b^4*c^5*d +
17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d +
10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3
+ 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a
^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9
- a^3*b^6*c^8*d - 9*a^4*b^5*c^7*d^2 + 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3
*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c
^4*d^5 - a^9*c^3*d^6)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)**3/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.88356, size = 4545, normalized size = 14.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-3/8*(sqrt(c*d)*a^2*b^9*c^10*abs(d) - 10*sqrt(c*d)*a^3*b^8*c^9*d*abs(d) + 72*sqrt(c*d)*a^4*b^7*c^8*d^2*abs(d)
- 214*sqrt(c*d)*a^5*b^6*c^7*d^3*abs(d) + 302*sqrt(c*d)*a^6*b^5*c^6*d^4*abs(d) - 214*sqrt(c*d)*a^7*b^4*c^5*d^5*
abs(d) + 72*sqrt(c*d)*a^8*b^3*c^4*d^6*abs(d) - 10*sqrt(c*d)*a^9*b^2*c^3*d^7*abs(d) + sqrt(c*d)*a^10*b*c^2*d^8*
abs(d) - sqrt(c*d)*b^4*c^3*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6
*b*c^3*d^4 - a^7*c^2*d^5)*abs(d) + 5*sqrt(c*d)*a*b^3*c^2*d*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*
d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*abs(d) + 5*sqrt(c*d)*a^2*b^2*c*d^2*abs(a^2*b^5*c^7 -
5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*abs(d) - sqrt(c*d)
*a^3*b*d^3*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7
*c^2*d^5)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4
*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5 + sqrt((a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4
*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5)^2 - 4*(a^3*b^4*c^7 - 4*a^4*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d
^3 + a^7*c^3*d^4)*(a^2*b^5*c^6*d - 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5))
)/(a^2*b^5*c^6*d - 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5)))/(a^2*b^5*c^7*d
*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)
- 3*a^3*b^4*c^6*d^2*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*
d^4 - a^7*c^2*d^5) + 2*a^4*b^3*c^5*d^3*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4
*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) + 2*a^5*b^2*c^4*d^4*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d
^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) - 3*a^6*b*c^3*d^5*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d +
10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) + a^7*c^2*d^6*abs(a^2*b^5*c^7 - 5*a^
3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) + (a^2*b^5*c^7 - 5*a^3*
b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)^2*d) + 3/8*(sqrt(a*b)*a^2
*b^8*c^10*d*abs(b) - 10*sqrt(a*b)*a^3*b^7*c^9*d^2*abs(b) + 72*sqrt(a*b)*a^4*b^6*c^8*d^3*abs(b) - 214*sqrt(a*b)
*a^5*b^5*c^7*d^4*abs(b) + 302*sqrt(a*b)*a^6*b^4*c^6*d^5*abs(b) - 214*sqrt(a*b)*a^7*b^3*c^5*d^6*abs(b) + 72*sqr
t(a*b)*a^8*b^2*c^4*d^7*abs(b) - 10*sqrt(a*b)*a^9*b*c^3*d^8*abs(b) + sqrt(a*b)*a^10*c^2*d^9*abs(b) + sqrt(a*b)*
b^3*c^3*d*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*
c^2*d^5)*abs(b) - 5*sqrt(a*b)*a*b^2*c^2*d^2*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^
2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*abs(b) - 5*sqrt(a*b)*a^2*b*c*d^3*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d
+ 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*abs(b) + sqrt(a*b)*a^3*d^4*abs(a^2*
b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*abs(b))*a
rctan(2*sqrt(1/2)*x/sqrt((a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*
d^4 + a^7*c^2*d^5 - sqrt((a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*
d^4 + a^7*c^2*d^5)^2 - 4*(a^3*b^4*c^7 - 4*a^4*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d^3 + a^7*c^3*d^4)*(
a^2*b^5*c^6*d - 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5)))/(a^2*b^5*c^6*d -
4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5)))/(a^2*b^6*c^7*abs(a^2*b^5*c^7 - 5*
a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) - 3*a^3*b^5*c^6*d*abs
(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) + 2*
a^4*b^4*c^5*d^2*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4
- a^7*c^2*d^5) + 2*a^5*b^3*c^4*d^3*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3
+ 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) - 3*a^6*b^2*c^3*d^4*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 -
10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) + a^7*b*c^2*d^5*abs(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^
4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5) - (a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*
b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)^2*b) + 1/8*(3*b^5*c^3*d^2*x^7 - 15*a*b^4*c^2
*d^3*x^7 - 15*a^2*b^3*c*d^4*x^7 + 3*a^3*b^2*d^5*x^7 + 6*b^5*c^4*d*x^5 - 25*a*b^4*c^3*d^2*x^5 - 34*a^2*b^3*c^2*
d^3*x^5 - 25*a^3*b^2*c*d^4*x^5 + 6*a^4*b*d^5*x^5 + 3*b^5*c^5*x^3 - 5*a*b^4*c^4*d*x^3 - 34*a^2*b^3*c^3*d^2*x^3
- 34*a^3*b^2*c^2*d^3*x^3 - 5*a^4*b*c*d^4*x^3 + 3*a^5*d^5*x^3 + 5*a*b^4*c^5*x - 17*a^2*b^3*c^4*d*x - 17*a^4*b*c
^2*d^3*x + 5*a^5*c*d^4*x)/((a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4)
*(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c)^2)