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

Optimal. Leaf size=207 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4} \]

[Out]

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

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Rubi [A]  time = 0.276969, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {470, 527, 522, 205} \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, 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{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\int \frac{a c+(-2 b c-3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{2 b (b c-a d)}\\ &=\frac{(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\int \frac{6 a b c^2-6 b c (b c+2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 b c (b c-a d)^2}\\ &=\frac{(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{3 (b c+3 a d) x}{8 (b c-a d)^3 \left (c+d x^2\right )}-\frac{\int \frac{6 a b c^2 (3 b c+a d)-6 b^2 c^2 (b c+3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 b c^2 (b c-a d)^3}\\ &=\frac{(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{3 (b c+3 a d) x}{8 (b c-a d)^3 \left (c+d x^2\right )}-\frac{(3 a b (b c+a d)) \int \frac{1}{a+b x^2} \, dx}{2 (b c-a d)^4}+\frac{\left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{c+d x^2} \, dx}{8 (b c-a d)^4}\\ &=\frac{(b c+2 a d) x}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{3 (b c+3 a d) x}{8 (b c-a d)^3 \left (c+d x^2\right )}-\frac{3 \sqrt{a} \sqrt{b} (b c+a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4}+\frac{3 \left (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 \sqrt{c} \sqrt{d} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.353174, size = 166, normalized size = 0.8 \[ \frac{\frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d}}+\frac{2 c x (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{4 a b x (b c-a d)}{a+b x^2}+\frac{x (5 a d+3 b c) (b c-a d)}{c+d x^2}-12 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a*b*(b*c - a*d)*x)/(a + b*x^2) + (2*c*(b*c - a*d)^2*x)/(c + d*x^2)^2 + ((b*c - a*d)*(3*b*c + 5*a*d)*x)/(c
+ d*x^2) - 12*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*Arc
Tan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]))/(8*(b*c - a*d)^4)

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Maple [B]  time = 0.013, size = 388, normalized size = 1.9 \begin{align*} -{\frac{5\,{x}^{3}{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{x}^{3}abc{d}^{2}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}{b}^{2}{c}^{2}d}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{a}^{2}c{d}^{2}x}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab{c}^{2}dx}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,{b}^{2}{c}^{3}x}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{9\,cabd}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}bxd}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}axc}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}bd}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,{b}^{2}ac}{2\, \left ( ad-bc \right ) ^{4}}\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(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

-5/8/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a^2*d^3+1/4/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*b*c*d^2+3/8/(a*d-b*c)^4/(d*x^2+c)^2
*x^3*b^2*c^2*d-3/8/(a*d-b*c)^4/(d*x^2+c)^2*a^2*c*d^2*x-1/4/(a*d-b*c)^4/(d*x^2+c)^2*a*b*c^2*d*x+5/8/(a*d-b*c)^4
/(d*x^2+c)^2*b^2*c^3*x+3/8/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2*d^2+9/4/(a*d-b*c)^4/(c*d)^(1/2)
*arctan(x*d/(c*d)^(1/2))*c*a*b*d+3/8/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2*c^2-1/2*a^2*b/(a*d-b*
c)^4*x/(b*x^2+a)*d+1/2*a*b^2/(a*d-b*c)^4*x/(b*x^2+a)*c-3/2*a^2*b/(a*d-b*c)^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2
))*d-3/2*a*b^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.2699, size = 5736, normalized size = 27.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(6*(b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*x^5 + 2*(5*b^3*c^4*d + 9*a*b^2*c^3*d^2 - 9*a^2*b*c^2*
d^3 - 5*a^3*c*d^4)*x^3 + 12*(a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b^2*c^3*d^2 + 3*a*b*
c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*
b)*x - a)/(b*x^2 + a)) - 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4
)*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c
^2*d^2 + 2*a^3*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + 6*(3*a*b^2*c^4*d - 2*a^2
*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*c^7*d - 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^3 - 4*a^4*b*c^4*d^4 + a^5*c^3*
d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4 + 6*a^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a^4*b*c*d^7)*x^6 + (2*b^5*c^6*d
^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4*d^4 - 2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 + a^5*c*d^7)*x^4 + (b^5*c^7*d -
2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3 + 8*a^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a^5*c^2*d^6)*x^2), 1/8*(3*(b^3*
c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*x^5 + (5*b^3*c^4*d + 9*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3 - 5*a^3*c*d^
4)*x^3 + 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c
^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*a^3*c
*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + 6*(a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b
^2*c^3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2)*sqrt(-a*b)*log(
(b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 3*(3*a*b^2*c^4*d - 2*a^2*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*c^7*d
- 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^3 - 4*a^4*b*c^4*d^4 + a^5*c^3*d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4 + 6*a
^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a^4*b*c*d^7)*x^6 + (2*b^5*c^6*d^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4*d^4 -
2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 + a^5*c*d^7)*x^4 + (b^5*c^7*d - 2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3 + 8*a
^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a^5*c^2*d^6)*x^2), 1/16*(6*(b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4
)*x^5 + 2*(5*b^3*c^4*d + 9*a*b^2*c^3*d^2 - 9*a^2*b*c^2*d^3 - 5*a^3*c*d^4)*x^3 - 24*(a*b*c^4*d + a^2*c^3*d^2 +
(b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b^2*c^3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 +
2*a^2*c^2*d^3)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d
^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^
4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*a^3*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 +
c)) + 6*(3*a*b^2*c^4*d - 2*a^2*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*c^7*d - 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^
3 - 4*a^4*b*c^4*d^4 + a^5*c^3*d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4 + 6*a^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a
^4*b*c*d^7)*x^6 + (2*b^5*c^6*d^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4*d^4 - 2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 +
a^5*c*d^7)*x^4 + (b^5*c^7*d - 2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3 + 8*a^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a
^5*c^2*d^6)*x^2), 1/8*(3*(b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*x^5 + (5*b^3*c^4*d + 9*a*b^2*c^3*d^2
- 9*a^2*b*c^2*d^3 - 5*a^3*c*d^4)*x^3 - 12*(a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^2*d^3 + a*b*c*d^4)*x^6 + (2*b^2*c^
3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + (b^2*c^4*d + 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2)*sqrt(a*b)*arctan(sqr
t(a*b)*x/a) + 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*
b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*
a^3*c*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + 3*(3*a*b^2*c^4*d - 2*a^2*b*c^3*d^2 - a^3*c^2*d^3)*x)/(a*b^4*
c^7*d - 4*a^2*b^3*c^6*d^2 + 6*a^3*b^2*c^5*d^3 - 4*a^4*b*c^4*d^4 + a^5*c^3*d^5 + (b^5*c^5*d^3 - 4*a*b^4*c^4*d^4
+ 6*a^2*b^3*c^3*d^5 - 4*a^3*b^2*c^2*d^6 + a^4*b*c*d^7)*x^6 + (2*b^5*c^6*d^2 - 7*a*b^4*c^5*d^3 + 8*a^2*b^3*c^4
*d^4 - 2*a^3*b^2*c^3*d^5 - 2*a^4*b*c^2*d^6 + a^5*c*d^7)*x^4 + (b^5*c^7*d - 2*a*b^4*c^6*d^2 - 2*a^2*b^3*c^5*d^3
+ 8*a^3*b^2*c^4*d^4 - 7*a^4*b*c^3*d^5 + 2*a^5*c^2*d^6)*x^2)]

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Sympy [B]  time = 146.596, size = 4041, normalized size = 19.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*sqrt(-a*b)*(a*d + b*c)*log(x + (-432*a**10*c*d**11*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 864*a**9*b
*c**2*d**10*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 20304*a**8*b**2*c**3*d**9*(-a*b)**(3/2)*(a*d + b*c)
**3/(a*d - b*c)**12 - 79488*a**7*b**3*c**4*d**8*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 151200*a**6*b**
4*c**5*d**7*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 27*a**6*d**6*sqrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4
- 157248*a**5*b**5*c**6*d**6*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 486*a**5*b*c*d**5*sqrt(-a*b)*(a*d
+ b*c)/(a*d - b*c)**4 + 78624*a**4*b**6*c**7*d**5*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 4725*a**4*b**
2*c**2*d**4*sqrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4 + 3456*a**3*b**7*c**8*d**4*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d
- b*c)**12 - 11988*a**3*b**3*c**3*d**3*sqrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4 - 26352*a**2*b**8*c**9*d**3*(-a*
b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 8181*a**2*b**4*c**4*d**2*sqrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4 + 12
960*a*b**9*c**10*d**2*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 2214*a*b**5*c**5*d*sqrt(-a*b)*(a*d + b*c)
/(a*d - b*c)**4 - 2160*b**10*c**11*d*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 27*b**6*c**6*sqrt(-a*b)*(a
*d + b*c)/(a*d - b*c)**4)/(27*a**3*b*d**3 + 189*a**2*b**2*c*d**2 + 189*a*b**3*c**2*d + 27*b**4*c**3))/(4*(a*d
- b*c)**4) - 3*sqrt(-a*b)*(a*d + b*c)*log(x + (432*a**10*c*d**11*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12
+ 864*a**9*b*c**2*d**10*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 20304*a**8*b**2*c**3*d**9*(-a*b)**(3/2)
*(a*d + b*c)**3/(a*d - b*c)**12 + 79488*a**7*b**3*c**4*d**8*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 - 151
200*a**6*b**4*c**5*d**7*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 27*a**6*d**6*sqrt(-a*b)*(a*d + b*c)/(a*
d - b*c)**4 + 157248*a**5*b**5*c**6*d**6*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 486*a**5*b*c*d**5*sqrt
(-a*b)*(a*d + b*c)/(a*d - b*c)**4 - 78624*a**4*b**6*c**7*d**5*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 4
725*a**4*b**2*c**2*d**4*sqrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4 - 3456*a**3*b**7*c**8*d**4*(-a*b)**(3/2)*(a*d +
b*c)**3/(a*d - b*c)**12 + 11988*a**3*b**3*c**3*d**3*sqrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4 + 26352*a**2*b**8*c*
*9*d**3*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 8181*a**2*b**4*c**4*d**2*sqrt(-a*b)*(a*d + b*c)/(a*d -
b*c)**4 - 12960*a*b**9*c**10*d**2*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 2214*a*b**5*c**5*d*sqrt(-a*b)
*(a*d + b*c)/(a*d - b*c)**4 + 2160*b**10*c**11*d*(-a*b)**(3/2)*(a*d + b*c)**3/(a*d - b*c)**12 + 27*b**6*c**6*s
qrt(-a*b)*(a*d + b*c)/(a*d - b*c)**4)/(27*a**3*b*d**3 + 189*a**2*b**2*c*d**2 + 189*a*b**3*c**2*d + 27*b**4*c**
3))/(4*(a*d - b*c)**4) + 3*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)*log(x + (-27*a**10*c*d**11*(-1/(
c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) - 27*a**9*b*c**2*d**10*(-1/(c*d))**(3/
2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(2*(a*d - b*c)**12) + 1269*a**8*b**2*c**3*d**9*(-1/(c*d))**(3/2)*(a*
*2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) - 1242*a**7*b**3*c**4*d**8*(-1/(c*d))**(3/2)*(a**2*d**
2 + 6*a*b*c*d + b**2*c**2)**3/(a*d - b*c)**12 + 4725*a**6*b**4*c**5*d**7*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*
c*d + b**2*c**2)**3/(2*(a*d - b*c)**12) - 27*a**6*d**6*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(
a*d - b*c)**4) - 2457*a**5*b**5*c**6*d**6*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(a*d - b*c)
**12 - 243*a**5*b*c*d**5*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(2*(a*d - b*c)**4) + 2457*a**4*b**
6*c**7*d**5*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(2*(a*d - b*c)**12) - 4725*a**4*b**2*c**2
*d**4*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(a*d - b*c)**4) + 54*a**3*b**7*c**8*d**4*(-1/(c*d)
)**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(a*d - b*c)**12 - 2997*a**3*b**3*c**3*d**3*sqrt(-1/(c*d))*(a**
2*d**2 + 6*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 - 1647*a**2*b**8*c**9*d**3*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b
*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) - 8181*a**2*b**4*c**4*d**2*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b*
*2*c**2)/(4*(a*d - b*c)**4) + 405*a*b**9*c**10*d**2*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(
2*(a*d - b*c)**12) - 1107*a*b**5*c**5*d*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(2*(a*d - b*c)**4)
- 135*b**10*c**11*d*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) - 27*b**6*c**
6*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(a*d - b*c)**4))/(27*a**3*b*d**3 + 189*a**2*b**2*c*d**
2 + 189*a*b**3*c**2*d + 27*b**4*c**3))/(16*(a*d - b*c)**4) - 3*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c*
*2)*log(x + (27*a**10*c*d**11*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) + 2
7*a**9*b*c**2*d**10*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(2*(a*d - b*c)**12) - 1269*a**8*b
**2*c**3*d**9*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) + 1242*a**7*b**3*c*
*4*d**8*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(a*d - b*c)**12 - 4725*a**6*b**4*c**5*d**7*(-
1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(2*(a*d - b*c)**12) + 27*a**6*d**6*sqrt(-1/(c*d))*(a**2
*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(a*d - b*c)**4) + 2457*a**5*b**5*c**6*d**6*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*
a*b*c*d + b**2*c**2)**3/(a*d - b*c)**12 + 243*a**5*b*c*d**5*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)
/(2*(a*d - b*c)**4) - 2457*a**4*b**6*c**7*d**5*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(2*(a*
d - b*c)**12) + 4725*a**4*b**2*c**2*d**4*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(a*d - b*c)**4)
- 54*a**3*b**7*c**8*d**4*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(a*d - b*c)**12 + 2997*a**3
*b**3*c**3*d**3*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 + 1647*a**2*b**8*c**9*d**3*(
-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**3/(4*(a*d - b*c)**12) + 8181*a**2*b**4*c**4*d**2*sqrt(-1
/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(a*d - b*c)**4) - 405*a*b**9*c**10*d**2*(-1/(c*d))**(3/2)*(a**2
*d**2 + 6*a*b*c*d + b**2*c**2)**3/(2*(a*d - b*c)**12) + 1107*a*b**5*c**5*d*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c
*d + b**2*c**2)/(2*(a*d - b*c)**4) + 135*b**10*c**11*d*(-1/(c*d))**(3/2)*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)**
3/(4*(a*d - b*c)**12) + 27*b**6*c**6*sqrt(-1/(c*d))*(a**2*d**2 + 6*a*b*c*d + b**2*c**2)/(4*(a*d - b*c)**4))/(2
7*a**3*b*d**3 + 189*a**2*b**2*c*d**2 + 189*a*b**3*c**2*d + 27*b**4*c**3))/(16*(a*d - b*c)**4) - (x**5*(9*a*b*d
**2 + 3*b**2*c*d) + x**3*(5*a**2*d**2 + 14*a*b*c*d + 5*b**2*c**2) + x*(3*a**2*c*d + 9*a*b*c**2))/(8*a**4*c**2*
d**3 - 24*a**3*b*c**3*d**2 + 24*a**2*b**2*c**4*d - 8*a*b**3*c**5 + x**6*(8*a**3*b*d**5 - 24*a**2*b**2*c*d**4 +
24*a*b**3*c**2*d**3 - 8*b**4*c**3*d**2) + x**4*(8*a**4*d**5 - 8*a**3*b*c*d**4 - 24*a**2*b**2*c**2*d**3 + 40*a
*b**3*c**3*d**2 - 16*b**4*c**4*d) + x**2*(16*a**4*c*d**4 - 40*a**3*b*c**2*d**3 + 24*a**2*b**2*c**3*d**2 + 8*a*
b**3*c**4*d - 8*b**4*c**5))

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Giac [A]  time = 1.16043, size = 406, normalized size = 1.96 \begin{align*} \frac{a b x}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} - \frac{3 \,{\left (a b^{2} c + a^{2} b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b}} + \frac{3 \,{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{c d}} + \frac{3 \, b c d x^{3} + 5 \, a d^{2} x^{3} + 5 \, b c^{2} x + 3 \, a c d x}{8 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*b*x/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) - 3/2*(a*b^2*c + a^2*b*d)*arctan(b
*x/sqrt(a*b))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*b)) + 3/8*(b^2*c
^2 + 6*a*b*c*d + a^2*d^2)*arctan(d*x/sqrt(c*d))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3
+ a^4*d^4)*sqrt(c*d)) + 1/8*(3*b*c*d*x^3 + 5*a*d^2*x^3 + 5*b*c^2*x + 3*a*c*d*x)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*
a^2*b*c*d^2 - a^3*d^3)*(d*x^2 + c)^2)