3.289 \(\int \frac{x^4}{(a+b x^2)^2 (c+d x^2)} \, dx\)
Optimal. Leaf size=109 \[ -\frac{\sqrt{a} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) (b c-a d)} \]
[Out]
(a*x)/(2*b*(b*c - a*d)*(a + b*x^2)) - (Sqrt[a]*(3*b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(3/2)*(b*c - a*
d)^2) + (c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d]*(b*c - a*d)^2)
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Rubi [A] time = 0.0854561, antiderivative size = 109, 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} \[ -\frac{\sqrt{a} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[x^4/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
(a*x)/(2*b*(b*c - a*d)*(a + b*x^2)) - (Sqrt[a]*(3*b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(3/2)*(b*c - a*
d)^2) + (c^(3/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d]*(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 )^2 \left (c+d x^2\right )} \, dx &=\frac{a x}{2 b (b c-a d) \left (a+b x^2\right )}-\frac{\int \frac{a c+(-2 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 b (b c-a d)}\\ &=\frac{a x}{2 b (b c-a d) \left (a+b x^2\right )}+\frac{c^2 \int \frac{1}{c+d x^2} \, dx}{(b c-a d)^2}-\frac{(a (3 b c-a d)) \int \frac{1}{a+b x^2} \, dx}{2 b (b c-a d)^2}\\ &=\frac{a x}{2 b (b c-a d) \left (a+b x^2\right )}-\frac{\sqrt{a} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{3/2} (b c-a d)^2}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d} (b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.148255, size = 95, normalized size = 0.87 \[ \frac{\frac{\sqrt{a} (a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}+\frac{a x (b c-a d)}{b \left (a+b x^2\right )}+\frac{2 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{d}}}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
((a*(b*c - a*d)*x)/(b*(a + b*x^2)) + (Sqrt[a]*(-3*b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + (2*c^(3/2)
*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d])/(2*(b*c - a*d)^2)
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Maple [A] time = 0.009, size = 144, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}dx}{2\, \left ( ad-bc \right ) ^{2}b \left ( b{x}^{2}+a \right ) }}+{\frac{axc}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{2}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,ac}{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)^2/(d*x^2+c),x)
[Out]
c^2/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))-1/2*a^2/(a*d-b*c)^2/b*x/(b*x^2+a)*d+1/2*a/(a*d-b*c)^2*x*c/
(b*x^2+a)+1/2*a^2/(a*d-b*c)^2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d-3/2*a/(a*d-b*c)^2/(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),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.40994, size = 1476, normalized size = 13.54 \begin{align*} \left [-\frac{{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 2 \,{\left (b^{2} c x^{2} + a b c\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 (a b c - a^{2} d\right )} x}{4 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac{{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) -{\left (b^{2} c x^{2} + a b c\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} + 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) -{\left (a b c - a^{2} d\right )} x}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, \frac{4 \,{\left (b^{2} c x^{2} + a b c\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) -{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 2 \,{\left (a b c - a^{2} d\right )} x}{4 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac{{\left (3 \, a b c - a^{2} d +{\left (3 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 2 \,{\left (b^{2} c x^{2} + a b c\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) -{\left (a b c - a^{2} d\right )} x}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\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)^2/(d*x^2+c),x, algorithm="fricas")
[Out]
[-1/4*((3*a*b*c - a^2*d + (3*b^2*c - a*b*d)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) -
2*(b^2*c*x^2 + a*b*c)*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) - 2*(a*b*c - a^2*d)*x)/(a*b^3
*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2), -1/2*((3*a*b*c - a^2*d + (3*b^2
*c - a*b*d)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - (b^2*c*x^2 + a*b*c)*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c
/d) - c)/(d*x^2 + c)) - (a*b*c - a^2*d)*x)/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a
^2*b^2*d^2)*x^2), 1/4*(4*(b^2*c*x^2 + a*b*c)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - (3*a*b*c - a^2*d + (3*b^2*c -
a*b*d)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 2*(a*b*c - a^2*d)*x)/(a*b^3*c^2 - 2*
a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2), -1/2*((3*a*b*c - a^2*d + (3*b^2*c - a*b*
d)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 2*(b^2*c*x^2 + a*b*c)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - (a*b*c -
a^2*d)*x)/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2)]
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Sympy [B] time = 14.4067, size = 1850, normalized size = 16.97 \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),x)
[Out]
-a*x/(2*a**2*b*d - 2*a*b**2*c + x**2*(2*a*b**2*d - 2*b**3*c)) - sqrt(-a/b**3)*(a*d - 3*b*c)*log(x + (-a**5*b**
3*d**6*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) + 9*a**4*b**4*c*d**5*(-a/b**3)**(3/2)*(a*d - 3*b*c
)**3/(2*(a*d - b*c)**6) - a**4*d**4*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 13*a**3*b**5*c**2*d**4*(-
a/b**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 + 9*a**3*b*c*d**3*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2
) + 17*a**2*b**6*c**3*d**3*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-a/b
**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 21*a*b**7*c**4*d**2*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**
6) + 27*a*b**3*c**3*d*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) + 5*b**8*c**5*d*(-a/b**3)**(3/2)*(a*d - 3
*b*c)**3/(2*(a*d - b*c)**6) + 4*b**4*c**4*sqrt(-a/b**3)*(a*d - 3*b*c)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**
2*d + 12*b**2*c**3))/(4*(a*d - b*c)**2) + sqrt(-a/b**3)*(a*d - 3*b*c)*log(x + (a**5*b**3*d**6*(-a/b**3)**(3/2)
*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) - 9*a**4*b**4*c*d**5*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6)
+ a**4*d**4*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) + 13*a**3*b**5*c**2*d**4*(-a/b**3)**(3/2)*(a*d - 3
*b*c)**3/(a*d - b*c)**6 - 9*a**3*b*c*d**3*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 17*a**2*b**6*c**3*d
**3*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(a*d - b*c)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-a/b**3)*(a*d - 3*b*c)/(2*(
a*d - b*c)**2) + 21*a*b**7*c**4*d**2*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)**6) - 27*a*b**3*c**3*d*s
qrt(-a/b**3)*(a*d - 3*b*c)/(2*(a*d - b*c)**2) - 5*b**8*c**5*d*(-a/b**3)**(3/2)*(a*d - 3*b*c)**3/(2*(a*d - b*c)
**6) - 4*b**4*c**4*sqrt(-a/b**3)*(a*d - 3*b*c)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**2*d + 12*b**2*c**3))/(4
*(a*d - b*c)**2) - sqrt(-c**3/d)*log(x + (-4*a**5*b**3*d**6*(-c**3/d)**(3/2)/(a*d - b*c)**6 + 36*a**4*b**4*c*d
**5*(-c**3/d)**(3/2)/(a*d - b*c)**6 - a**4*d**4*sqrt(-c**3/d)/(a*d - b*c)**2 - 104*a**3*b**5*c**2*d**4*(-c**3/
d)**(3/2)/(a*d - b*c)**6 + 9*a**3*b*c*d**3*sqrt(-c**3/d)/(a*d - b*c)**2 + 136*a**2*b**6*c**3*d**3*(-c**3/d)**(
3/2)/(a*d - b*c)**6 - 27*a**2*b**2*c**2*d**2*sqrt(-c**3/d)/(a*d - b*c)**2 - 84*a*b**7*c**4*d**2*(-c**3/d)**(3/
2)/(a*d - b*c)**6 + 27*a*b**3*c**3*d*sqrt(-c**3/d)/(a*d - b*c)**2 + 20*b**8*c**5*d*(-c**3/d)**(3/2)/(a*d - b*c
)**6 + 8*b**4*c**4*sqrt(-c**3/d)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**2*d + 12*b**2*c**3))/(2*(a*d - b*c)**
2) + sqrt(-c**3/d)*log(x + (4*a**5*b**3*d**6*(-c**3/d)**(3/2)/(a*d - b*c)**6 - 36*a**4*b**4*c*d**5*(-c**3/d)**
(3/2)/(a*d - b*c)**6 + a**4*d**4*sqrt(-c**3/d)/(a*d - b*c)**2 + 104*a**3*b**5*c**2*d**4*(-c**3/d)**(3/2)/(a*d
- b*c)**6 - 9*a**3*b*c*d**3*sqrt(-c**3/d)/(a*d - b*c)**2 - 136*a**2*b**6*c**3*d**3*(-c**3/d)**(3/2)/(a*d - b*c
)**6 + 27*a**2*b**2*c**2*d**2*sqrt(-c**3/d)/(a*d - b*c)**2 + 84*a*b**7*c**4*d**2*(-c**3/d)**(3/2)/(a*d - b*c)*
*6 - 27*a*b**3*c**3*d*sqrt(-c**3/d)/(a*d - b*c)**2 - 20*b**8*c**5*d*(-c**3/d)**(3/2)/(a*d - b*c)**6 - 8*b**4*c
**4*sqrt(-c**3/d)/(a*d - b*c)**2)/(a**2*c*d**2 - 7*a*b*c**2*d + 12*b**2*c**3))/(2*(a*d - b*c)**2)
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Giac [A] time = 1.17898, size = 165, normalized size = 1.51 \begin{align*} \frac{c^{2} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d}} - \frac{{\left (3 \, a b c - a^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{a b}} + \frac{a x}{2 \,{\left (b^{2} c - a b d\right )}{\left (b x^{2} + a\right )}} \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),x, algorithm="giac")
[Out]
c^2*arctan(d*x/sqrt(c*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) - 1/2*(3*a*b*c - a^2*d)*arctan(b*x/sqrt(
a*b))/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt(a*b)) + 1/2*a*x/((b^2*c - a*b*d)*(b*x^2 + a))