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

Optimal. Leaf size=200 \[ -\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(-3*d*x)/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(11*b*c + a*d)*x)/
(8*c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^4
) - (Sqrt[d]*(15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*(b*c - a*d)^4)

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Rubi [A]  time = 0.252424, antiderivative size = 200, 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 = {471, 527, 522, 205} \[ -\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}+\frac{b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{d x (a d+11 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d x}{4 \left (c+d x^2\right )^2 (b c-a d)^2}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d*x)/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (d*(11*b*c + a*d)*x)/
(8*c*(b*c - a*d)^3*(c + d*x^2)) + (b^(3/2)*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*(b*c - a*d)^4
) - (Sqrt[d]*(15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*(b*c - a*d)^4)

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\int \frac{c-5 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{2 (b c-a d)}\\ &=-\frac{3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\int \frac{2 c (2 b c+a d)-18 b c d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{8 c (b c-a d)^2}\\ &=-\frac{3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{\int \frac{2 c \left (4 b^2 c^2+9 a b c d-a^2 d^2\right )-2 b c d (11 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{16 c^2 (b c-a d)^3}\\ &=-\frac{3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{\left (b^2 (b c+5 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 (b c-a d)^4}-\frac{\left (d \left (15 b^2 c^2+10 a b c d-a^2 d^2\right )\right ) \int \frac{1}{c+d x^2} \, dx}{8 c (b c-a d)^4}\\ &=-\frac{3 d x}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{d (11 b c+a d) x}{8 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{b^{3/2} (b c+5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} (b c-a d)^4}-\frac{\sqrt{d} \left (15 b^2 c^2+10 a b c d-a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.401621, size = 171, normalized size = 0.86 \[ \frac{\frac{\sqrt{d} \left (a^2 d^2-10 a b c d-15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}-\frac{4 b^2 x (b c-a d)}{a+b x^2}+\frac{4 b^{3/2} (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{d x (a d-b c) (a d+7 b c)}{c \left (c+d x^2\right )}-\frac{2 d x (b c-a d)^2}{\left (c+d x^2\right )^2}}{8 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*b^2*(b*c - a*d)*x)/(a + b*x^2) - (2*d*(b*c - a*d)^2*x)/(c + d*x^2)^2 + (d*(-(b*c) + a*d)*(7*b*c + a*d)*x)
/(c*(c + d*x^2)) + (4*b^(3/2)*(b*c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (Sqrt[d]*(-15*b^2*c^2 - 10*
a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(3/2))/(8*(b*c - a*d)^4)

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Maple [B]  time = 0.015, size = 391, normalized size = 2. \begin{align*}{\frac{{d}^{4}{x}^{3}{a}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{3\,{d}^{3}{x}^{3}ab}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{7\,{d}^{2}{x}^{3}{b}^{2}c}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,abc{d}^{2}x}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,{b}^{2}{c}^{2}dx}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}{d}^{3}x}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{4}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{5\,ab{d}^{2}}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{2}cd}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}xad}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xc}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}ad}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{3}c}{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^2/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

1/8*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^3*a^2+3/4*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*b-7/8*d^2/(a*d-b*c)^4/(d*x^2+c
)^2*x^3*b^2*c+5/4*d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*a*b*x-9/8*d/(a*d-b*c)^4/(d*x^2+c)^2*b^2*c^2*x-1/8*d^3/(a*d-b*c
)^4/(d*x^2+c)^2*a^2*x+1/8*d^3/(a*d-b*c)^4/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-5/4*d^2/(a*d-b*c)^4/(c*d)^
(1/2)*arctan(x*d/(c*d)^(1/2))*a*b-15/8*d/(a*d-b*c)^4*c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2+1/2*b^2/(a*d-b*
c)^4*x/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4*x/(b*x^2+a)*c+5/2*b^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
*a*d+1/2*b^3/(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^2/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 9.21907, size = 5824, normalized size = 29.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(2*(11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + 2*(17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d
^3 - a^3*d^4)*x^3 - 4*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2
*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*
x*sqrt(-b/a) - a)/(b*x^2 + a)) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d
^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2
*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(
4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2
- 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4
*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5
*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d
^5)*x^2), -1/8*((11*b^3*c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + (17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b
*c*d^3 - a^3*d^4)*x^3 + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*
b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d +
19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - 2*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d
^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10
*a^2*b*c^2*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + (4*b^3*c^4 + 5*a*b^2*c^3*d -
10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3
*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*
d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*
a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/16*(2*(11*b^3*
c^2*d^2 - 10*a*b^2*c*d^3 - a^2*b*d^4)*x^5 + 2*(17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3
- 8*(a*b^2*c^4 + 5*a^2*b*c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b
*c*d^3)*x^4 + (b^3*c^4 + 7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (15*a*b^2*c^4
+ 10*a^2*b*c^3*d - a^3*c^2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*
c^2*d^2 + 8*a^2*b*c*d^3 - a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*s
qrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 +
a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2
- 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2
+ 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*
b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^5*c^2*d^5)*x^2), -1/8*((11*b^3*c^2*d^2 - 10*a*b^2*c*d^
3 - a^2*b*d^4)*x^5 + (17*b^3*c^3*d - 11*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 - a^3*d^4)*x^3 - 4*(a*b^2*c^4 + 5*a^2*b*
c^3*d + (b^3*c^2*d^2 + 5*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d + 11*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3)*x^4 + (b^3*c^4 +
7*a*b^2*c^3*d + 10*a^2*b*c^2*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) + (15*a*b^2*c^4 + 10*a^2*b*c^3*d - a^3*c^
2*d^2 + (15*b^3*c^2*d^2 + 10*a*b^2*c*d^3 - a^2*b*d^4)*x^6 + (30*b^3*c^3*d + 35*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 -
a^3*d^4)*x^4 + (15*b^3*c^4 + 40*a*b^2*c^3*d + 19*a^2*b*c^2*d^2 - 2*a^3*c*d^3)*x^2)*sqrt(d/c)*arctan(x*sqrt(d/
c)) + (4*b^3*c^4 + 5*a*b^2*c^3*d - 10*a^2*b*c^2*d^2 + a^3*c*d^3)*x)/(a*b^4*c^7 - 4*a^2*b^3*c^6*d + 6*a^3*b^2*c
^5*d^2 - 4*a^4*b*c^4*d^3 + a^5*c^3*d^4 + (b^5*c^5*d^2 - 4*a*b^4*c^4*d^3 + 6*a^2*b^3*c^3*d^4 - 4*a^3*b^2*c^2*d^
5 + a^4*b*c*d^6)*x^6 + (2*b^5*c^6*d - 7*a*b^4*c^5*d^2 + 8*a^2*b^3*c^4*d^3 - 2*a^3*b^2*c^3*d^4 - 2*a^4*b*c^2*d^
5 + a^5*c*d^6)*x^4 + (b^5*c^7 - 2*a*b^4*c^6*d - 2*a^2*b^3*c^5*d^2 + 8*a^3*b^2*c^4*d^3 - 7*a^4*b*c^3*d^4 + 2*a^
5*c^2*d^5)*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(x**2/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.19951, size = 428, normalized size = 2.14 \begin{align*} -\frac{b^{2} 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{{\left (b^{3} c + 5 \, a b^{2} 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{{\left (15 \, b^{2} c^{2} d + 10 \, a b c d^{2} - a^{2} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d + 6 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{3} b c^{2} d^{3} + a^{4} c d^{4}\right )} \sqrt{c d}} - \frac{7 \, b c d^{2} x^{3} + a d^{3} x^{3} + 9 \, b c^{2} d x - a c d^{2} x}{8 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c 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^2/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-1/2*b^2*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)) + 1/2*(b^3*c + 5*a*b^2*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)) - 1/8*(15*b
^2*c^2*d + 10*a*b*c*d^2 - a^2*d^3)*arctan(d*x/sqrt(c*d))/((b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3
*b*c^2*d^3 + a^4*c*d^4)*sqrt(c*d)) - 1/8*(7*b*c*d^2*x^3 + a*d^3*x^3 + 9*b*c^2*d*x - a*c*d^2*x)/((b^3*c^4 - 3*a
*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*x^2 + c)^2)