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

Optimal. Leaf size=155 \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d} (b c-a d)^3}-\frac{\sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (a d+3 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{x}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.140216, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {471, 527, 522, 205} \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{3/2} \sqrt{d} (b c-a d)^3}-\frac{\sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (a d+3 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{x}{4 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 ) \left (c+d x^2\right )^3} \, dx &=\frac{x}{4 (b c-a d) \left (c+d x^2\right )^2}-\frac{\int \frac{a-3 b x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{4 (b c-a d)}\\ &=\frac{x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac{(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac{\int \frac{a (5 b c-a d)-b (3 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 c (b c-a d)^2}\\ &=\frac{x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac{(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac{\left (a b^2\right ) \int \frac{1}{a+b x^2} \, dx}{(b c-a d)^3}+\frac{\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \int \frac{1}{c+d x^2} \, dx}{8 c (b c-a d)^3}\\ &=\frac{x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac{(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac{\sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{\left (3 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^{3/2} \sqrt{d} (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.227899, size = 151, normalized size = 0.97 \[ \frac{1}{8} \left (\frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2} \sqrt{d} (b c-a d)^3}+\frac{8 \sqrt{a} b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(a d-b c)^3}+\frac{x (a d+3 b c)}{c \left (c+d x^2\right ) (b c-a d)^2}+\frac{2 x}{\left (c+d x^2\right )^2 (b c-a d)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.012, size = 298, normalized size = 1.9 \begin{align*}{\frac{{x}^{3}{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{{x}^{3}ab{d}^{2}}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{x}^{3}{b}^{2}cd}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,cabdx}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{5\,{b}^{2}{c}^{2}x}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{a}^{2}{d}^{2}x}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{3}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,adb}{4\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,{b}^{2}c}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}a}{ \left ( ad-bc \right ) ^{3}}\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)/(d*x^2+c)^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.83158, size = 3168, normalized size = 20.44 \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)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 58.7035, size = 3386, normalized size = 21.85 \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)/(d*x**2+c)**3,x)

[Out]

-sqrt(-a*b**3)*log(x + (-64*a**8*c**3*d**9*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 768*a**7*b*c**4*d**8*(-a*b**3)**(
3/2)/(a*d - b*c)**9 - 2560*a**6*b**2*c**5*d**7*(-a*b**3)**(3/2)/(a*d - b*c)**9 - a**6*d**6*sqrt(-a*b**3)/(a*d
- b*c)**3 + 2816*a**5*b**3*c**6*d**6*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 18*a**5*b*c*d**5*sqrt(-a*b**3)/(a*d - b
*c)**3 + 1920*a**4*b**4*c**7*d**5*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 99*a**4*b**2*c**2*d**4*sqrt(-a*b**3)/(a*d
- b*c)**3 - 7936*a**3*b**5*c**8*d**4*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 108*a**3*b**3*c**3*d**3*sqrt(-a*b**3)/(
a*d - b*c)**3 + 8192*a**2*b**6*c**9*d**3*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 297*a**2*b**4*c**4*d**2*sqrt(-a*b**
3)/(a*d - b*c)**3 - 3840*a*b**7*c**10*d**2*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 674*a*b**5*c**5*d*sqrt(-a*b**3)/(
a*d - b*c)**3 + 704*b**8*c**11*d*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 27*b**6*c**6*sqrt(-a*b**3)/(a*d - b*c)**3)/
(a**3*b**2*d**3 - 15*a**2*b**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*c**3))/(2*(a*d - b*c)**3) + sqrt(-a*b**3)*l
og(x + (64*a**8*c**3*d**9*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 768*a**7*b*c**4*d**8*(-a*b**3)**(3/2)/(a*d - b*c)*
*9 + 2560*a**6*b**2*c**5*d**7*(-a*b**3)**(3/2)/(a*d - b*c)**9 + a**6*d**6*sqrt(-a*b**3)/(a*d - b*c)**3 - 2816*
a**5*b**3*c**6*d**6*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 18*a**5*b*c*d**5*sqrt(-a*b**3)/(a*d - b*c)**3 - 1920*a**
4*b**4*c**7*d**5*(-a*b**3)**(3/2)/(a*d - b*c)**9 + 99*a**4*b**2*c**2*d**4*sqrt(-a*b**3)/(a*d - b*c)**3 + 7936*
a**3*b**5*c**8*d**4*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 108*a**3*b**3*c**3*d**3*sqrt(-a*b**3)/(a*d - b*c)**3 - 8
192*a**2*b**6*c**9*d**3*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 297*a**2*b**4*c**4*d**2*sqrt(-a*b**3)/(a*d - b*c)**3
+ 3840*a*b**7*c**10*d**2*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 674*a*b**5*c**5*d*sqrt(-a*b**3)/(a*d - b*c)**3 - 7
04*b**8*c**11*d*(-a*b**3)**(3/2)/(a*d - b*c)**9 - 27*b**6*c**6*sqrt(-a*b**3)/(a*d - b*c)**3)/(a**3*b**2*d**3 -
15*a**2*b**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*c**3))/(2*(a*d - b*c)**3) - sqrt(-1/(c**3*d))*(a**2*d**2 - 6
*a*b*c*d - 3*b**2*c**2)*log(x + (-a**8*c**3*d**9*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3
/(8*(a*d - b*c)**9) + 3*a**7*b*c**4*d**8*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d
- b*c)**9) - 5*a**6*b**2*c**5*d**7*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(a*d - b*c)*
*9 - a**6*d**6*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3) + 11*a**5*b**3*c**6*
d**6*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) + 9*a**5*b*c*d**5*sqrt(-
1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**3) + 15*a**4*b**4*c**7*d**5*(-1/(c**3*d))**(
3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(4*(a*d - b*c)**9) - 99*a**4*b**2*c**2*d**4*sqrt(-1/(c**3*d))*(a
**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3) - 31*a**3*b**5*c**8*d**4*(-1/(c**3*d))**(3/2)*(a**2*d**
2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) + 27*a**3*b**3*c**3*d**3*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a
*b*c*d - 3*b**2*c**2)/(2*(a*d - b*c)**3) + 16*a**2*b**6*c**9*d**3*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d
- 3*b**2*c**2)**3/(a*d - b*c)**9 + 297*a**2*b**4*c**4*d**2*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c
**2)/(8*(a*d - b*c)**3) - 15*a*b**7*c**10*d**2*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(
2*(a*d - b*c)**9) + 337*a*b**5*c**5*d*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**
3) + 11*b**8*c**11*d*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(8*(a*d - b*c)**9) + 27*b**
6*c**6*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3))/(a**3*b**2*d**3 - 15*a**2*b
**3*c*d**2 + 51*a*b**4*c**2*d + 27*b**5*c**3))/(16*(a*d - b*c)**3) + sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d
- 3*b**2*c**2)*log(x + (a**8*c**3*d**9*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(8*(a*d -
b*c)**9) - 3*a**7*b*c**4*d**8*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9
) + 5*a**6*b**2*c**5*d**7*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(a*d - b*c)**9 + a**6*
d**6*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3) - 11*a**5*b**3*c**6*d**6*(-1/(
c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) - 9*a**5*b*c*d**5*sqrt(-1/(c**3*d)
)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**3) - 15*a**4*b**4*c**7*d**5*(-1/(c**3*d))**(3/2)*(a**2
*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(4*(a*d - b*c)**9) + 99*a**4*b**2*c**2*d**4*sqrt(-1/(c**3*d))*(a**2*d**2 -
6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3) + 31*a**3*b**5*c**8*d**4*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*
c*d - 3*b**2*c**2)**3/(2*(a*d - b*c)**9) - 27*a**3*b**3*c**3*d**3*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3
*b**2*c**2)/(2*(a*d - b*c)**3) - 16*a**2*b**6*c**9*d**3*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c
**2)**3/(a*d - b*c)**9 - 297*a**2*b**4*c**4*d**2*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a
*d - b*c)**3) + 15*a*b**7*c**10*d**2*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(2*(a*d - b
*c)**9) - 337*a*b**5*c**5*d*sqrt(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(4*(a*d - b*c)**3) - 11*b*
*8*c**11*d*(-1/(c**3*d))**(3/2)*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)**3/(8*(a*d - b*c)**9) - 27*b**6*c**6*sqr
t(-1/(c**3*d))*(a**2*d**2 - 6*a*b*c*d - 3*b**2*c**2)/(8*(a*d - b*c)**3))/(a**3*b**2*d**3 - 15*a**2*b**3*c*d**2
+ 51*a*b**4*c**2*d + 27*b**5*c**3))/(16*(a*d - b*c)**3) + (x**3*(a*d**2 + 3*b*c*d) + x*(-a*c*d + 5*b*c**2))/(
8*a**2*c**3*d**2 - 16*a*b*c**4*d + 8*b**2*c**5 + x**4*(8*a**2*c*d**4 - 16*a*b*c**2*d**3 + 8*b**2*c**3*d**2) +
x**2*(16*a**2*c**2*d**3 - 32*a*b*c**3*d**2 + 16*b**2*c**4*d))

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Giac [A]  time = 1.18821, size = 278, normalized size = 1.79 \begin{align*} -\frac{a b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b}} + \frac{{\left (3 \, 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^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt{c d}} + \frac{3 \, b c d x^{3} + a d^{2} x^{3} + 5 \, b c^{2} x - a c d x}{8 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\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)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

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