3.242 \(\int \frac{x^4}{(a+b x^2) (c+d x^2)^2} \, dx\)
Optimal. Leaf size=108 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left (c+d x^2\right ) (b c-a d)} \]
[Out]
-(c*x)/(2*d*(b*c - a*d)*(c + d*x^2)) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d)^2) + (Sqrt[c
]*(b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(3/2)*(b*c - a*d)^2)
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Rubi [A] time = 0.0846266, antiderivative size = 108, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used =
{470, 522, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[x^4/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
-(c*x)/(2*d*(b*c - a*d)*(c + d*x^2)) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d)^2) + (Sqrt[c
]*(b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(3/2)*(b*c - a*d)^2)
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 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 ) \left (c+d x^2\right )^2} \, dx &=-\frac{c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac{\int \frac{a c+(b c-2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 d (b c-a d)}\\ &=-\frac{c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac{a^2 \int \frac{1}{a+b x^2} \, dx}{(b c-a d)^2}+\frac{(c (b c-3 a d)) \int \frac{1}{c+d x^2} \, dx}{2 d (b c-a d)^2}\\ &=-\frac{c x}{2 d (b c-a d) \left (c+d x^2\right )}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.128526, size = 108, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (a d-b c)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}+\frac{c x}{2 d \left (c+d x^2\right ) (a d-b c)} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
(c*x)/(2*d*(-(b*c) + a*d)*(c + d*x^2)) + (a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(-(b*c) + a*d)^2) + (S
qrt[c]*(b*c - 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*d^(3/2)*(b*c - a*d)^2)
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Maple [A] time = 0.009, size = 144, normalized size = 1.3 \begin{align*}{\frac{cxa}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}xb}{2\,d \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,ac}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{c}^{2}b}{2\,d \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{a}^{2}}{ \left ( ad-bc \right ) ^{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(x^4/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
1/2*c/(a*d-b*c)^2*x/(d*x^2+c)*a-1/2*c^2/(a*d-b*c)^2/d*x/(d*x^2+c)*b-3/2*c/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(
c*d)^(1/2))*a+1/2*c^2/(a*d-b*c)^2/d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b+a^2/(a*d-b*c)^2/(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^4/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.04322, size = 1472, normalized size = 13.63 \begin{align*} \left [\frac{2 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) -{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{2} - a c d\right )} x}{4 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{4 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) -{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{2} - a c d\right )} x}{4 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) +{\left (a d^{2} x^{2} + a c d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) -{\left (b c^{2} - a c d\right )} x}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{2 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) +{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) -{\left (b c^{2} - a c d\right )} x}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(a*d^2*x^2 + a*c*d)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - (b*c^2 - 3*a*c*d + (b
*c*d - 3*a*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) - 2*(b*c^2 - a*c*d)*x)/(b^2*c^
3*d - 2*a*b*c^2*d^2 + a^2*c*d^3 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2), 1/4*(4*(a*d^2*x^2 + a*c*d)*sqrt(
a/b)*arctan(b*x*sqrt(a/b)/a) - (b*c^2 - 3*a*c*d + (b*c*d - 3*a*d^2)*x^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c
/d) - c)/(d*x^2 + c)) - 2*(b*c^2 - a*c*d)*x)/(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3 + (b^2*c^2*d^2 - 2*a*b*c*d
^3 + a^2*d^4)*x^2), 1/2*((b*c^2 - 3*a*c*d + (b*c*d - 3*a*d^2)*x^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) + (a*d^2*
x^2 + a*c*d)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - (b*c^2 - a*c*d)*x)/(b^2*c^3*d - 2*a*
b*c^2*d^2 + a^2*c*d^3 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2), 1/2*(2*(a*d^2*x^2 + a*c*d)*sqrt(a/b)*arcta
n(b*x*sqrt(a/b)/a) + (b*c^2 - 3*a*c*d + (b*c*d - 3*a*d^2)*x^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - (b*c^2 - a*
c*d)*x)/(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2)]
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Sympy [B] time = 13.8954, size = 1850, normalized size = 17.13 \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)/(d*x**2+c)**2,x)
[Out]
c*x/(2*a*c*d**2 - 2*b*c**2*d + x**2*(2*a*d**3 - 2*b*c*d**2)) + sqrt(-a**3/b)*log(x + (-20*a**5*b*d**8*(-a**3/b
)**(3/2)/(a*d - b*c)**6 + 84*a**4*b**2*c*d**7*(-a**3/b)**(3/2)/(a*d - b*c)**6 - 8*a**4*d**4*sqrt(-a**3/b)/(a*d
- b*c)**2 - 136*a**3*b**3*c**2*d**6*(-a**3/b)**(3/2)/(a*d - b*c)**6 - 27*a**3*b*c*d**3*sqrt(-a**3/b)/(a*d - b
*c)**2 + 104*a**2*b**4*c**3*d**5*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-a**3/b)/(a*d -
b*c)**2 - 36*a*b**5*c**4*d**4*(-a**3/b)**(3/2)/(a*d - b*c)**6 - 9*a*b**3*c**3*d*sqrt(-a**3/b)/(a*d - b*c)**2
+ 4*b**6*c**5*d**3*(-a**3/b)**(3/2)/(a*d - b*c)**6 + b**4*c**4*sqrt(-a**3/b)/(a*d - b*c)**2)/(12*a**3*d**2 - 7
*a**2*b*c*d + a*b**2*c**2))/(2*(a*d - b*c)**2) - sqrt(-a**3/b)*log(x + (20*a**5*b*d**8*(-a**3/b)**(3/2)/(a*d -
b*c)**6 - 84*a**4*b**2*c*d**7*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 8*a**4*d**4*sqrt(-a**3/b)/(a*d - b*c)**2 + 13
6*a**3*b**3*c**2*d**6*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 27*a**3*b*c*d**3*sqrt(-a**3/b)/(a*d - b*c)**2 - 104*a*
*2*b**4*c**3*d**5*(-a**3/b)**(3/2)/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-a**3/b)/(a*d - b*c)**2 + 36*a
*b**5*c**4*d**4*(-a**3/b)**(3/2)/(a*d - b*c)**6 + 9*a*b**3*c**3*d*sqrt(-a**3/b)/(a*d - b*c)**2 - 4*b**6*c**5*d
**3*(-a**3/b)**(3/2)/(a*d - b*c)**6 - b**4*c**4*sqrt(-a**3/b)/(a*d - b*c)**2)/(12*a**3*d**2 - 7*a**2*b*c*d + a
*b**2*c**2))/(2*(a*d - b*c)**2) + sqrt(-c/d**3)*(3*a*d - b*c)*log(x + (-5*a**5*b*d**8*(-c/d**3)**(3/2)*(3*a*d
- b*c)**3/(2*(a*d - b*c)**6) + 21*a**4*b**2*c*d**7*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - 4*a*
*4*d**4*sqrt(-c/d**3)*(3*a*d - b*c)/(a*d - b*c)**2 - 17*a**3*b**3*c**2*d**6*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/
(a*d - b*c)**6 - 27*a**3*b*c*d**3*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + 13*a**2*b**4*c**3*d**5*(-c/
d**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*
c)**2) - 9*a*b**5*c**4*d**4*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - 9*a*b**3*c**3*d*sqrt(-c/d**
3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + b**6*c**5*d**3*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + b*
*4*c**4*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2))/(12*a**3*d**2 - 7*a**2*b*c*d + a*b**2*c**2))/(4*(a*d -
b*c)**2) - sqrt(-c/d**3)*(3*a*d - b*c)*log(x + (5*a**5*b*d**8*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c
)**6) - 21*a**4*b**2*c*d**7*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + 4*a**4*d**4*sqrt(-c/d**3)*(
3*a*d - b*c)/(a*d - b*c)**2 + 17*a**3*b**3*c**2*d**6*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(a*d - b*c)**6 + 27*a**
3*b*c*d**3*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) - 13*a**2*b**4*c**3*d**5*(-c/d**3)**(3/2)*(3*a*d - b
*c)**3/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*d - b*c)**2) + 9*a*b**5*c**4*
d**4*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) + 9*a*b**3*c**3*d*sqrt(-c/d**3)*(3*a*d - b*c)/(2*(a*
d - b*c)**2) - b**6*c**5*d**3*(-c/d**3)**(3/2)*(3*a*d - b*c)**3/(2*(a*d - b*c)**6) - b**4*c**4*sqrt(-c/d**3)*(
3*a*d - b*c)/(2*(a*d - b*c)**2))/(12*a**3*d**2 - 7*a**2*b*c*d + a*b**2*c**2))/(4*(a*d - b*c)**2)
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Giac [A] time = 1.21317, size = 163, normalized size = 1.51 \begin{align*} \frac{a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} + \frac{{\left (b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt{c d}} - \frac{c x}{2 \,{\left (b c d - a d^{2}\right )}{\left (d x^{2} + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")
[Out]
a^2*arctan(b*x/sqrt(a*b))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*b)) + 1/2*(b*c^2 - 3*a*c*d)*arctan(d*x/sqrt(
c*d))/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*sqrt(c*d)) - 1/2*c*x/((b*c*d - a*d^2)*(d*x^2 + c))