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

Optimal. Leaf size=78 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} (b c-a d)}-\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{3/2} (b c-a d)}+\frac{x}{b d} \]

[Out]

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

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Rubi [A]  time = 0.0834722, antiderivative size = 78, 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 = {479, 522, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} (b c-a d)}-\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{3/2} (b c-a d)}+\frac{x}{b d} \]

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

Rule 479

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

Mathematica [A]  time = 0.0899106, size = 74, normalized size = 0.95 \[ \frac{\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}-\frac{a x}{b}-\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{3/2}}+\frac{c x}{d}}{b c-a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 73, normalized size = 0.9 \begin{align*}{\frac{x}{bd}}+{\frac{{c}^{2}}{ \left ( ad-bc \right ) d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{ \left ( ad-bc \right ) b}\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),x)

[Out]

x/b/d+1/d*c^2/(a*d-b*c)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))-1/b*a^2/(a*d-b*c)/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64524, size = 810, normalized size = 10.38 \begin{align*} \left [-\frac{a d \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + b c \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 - a d\right )} x}{2 \,{\left (b^{2} c d - a b d^{2}\right )}}, \frac{2 \, a d \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - b c \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 - a d\right )} x}{2 \,{\left (b^{2} c d - a b d^{2}\right )}}, -\frac{2 \, b c \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) + a d \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 c - a d\right )} x}{2 \,{\left (b^{2} c d - a b d^{2}\right )}}, \frac{a d \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - b c \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) +{\left (b c - a d\right )} x}{b^{2} c d - a b d^{2}}\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),x, algorithm="fricas")

[Out]

[-1/2*(a*d*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + b*c*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt
(-c/d) - c)/(d*x^2 + c)) - 2*(b*c - a*d)*x)/(b^2*c*d - a*b*d^2), 1/2*(2*a*d*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a)
- b*c*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 2*(b*c - a*d)*x)/(b^2*c*d - a*b*d^2), -1/2*
(2*b*c*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) + a*d*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 2*
(b*c - a*d)*x)/(b^2*c*d - a*b*d^2), (a*d*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - b*c*sqrt(c/d)*arctan(d*x*sqrt(c/d
)/c) + (b*c - a*d)*x)/(b^2*c*d - a*b*d^2)]

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Sympy [B]  time = 4.57926, size = 921, normalized size = 11.81 \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),x)

[Out]

-sqrt(-a**3/b**3)*log(x + (-a**4*d**4*sqrt(-a**3/b**3)/(a*d - b*c) - a**3*b**3*d**6*(-a**3/b**3)**(3/2)/(a*d -
b*c)**3 + a**2*b**4*c*d**5*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 + a*b**5*c**2*d**4*(-a**3/b**3)**(3/2)/(a*d - b
*c)**3 - b**6*c**3*d**3*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - b**4*c**4*sqrt(-a**3/b**3)/(a*d - b*c))/(a**3*c*d
**2 + a**2*b*c**2*d + a*b**2*c**3))/(2*(a*d - b*c)) + sqrt(-a**3/b**3)*log(x + (a**4*d**4*sqrt(-a**3/b**3)/(a*
d - b*c) + a**3*b**3*d**6*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 - a**2*b**4*c*d**5*(-a**3/b**3)**(3/2)/(a*d - b*c
)**3 - a*b**5*c**2*d**4*(-a**3/b**3)**(3/2)/(a*d - b*c)**3 + b**6*c**3*d**3*(-a**3/b**3)**(3/2)/(a*d - b*c)**3
+ b**4*c**4*sqrt(-a**3/b**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c**2*d + a*b**2*c**3))/(2*(a*d - b*c)) - sqrt
(-c**3/d**3)*log(x + (-a**4*d**4*sqrt(-c**3/d**3)/(a*d - b*c) - a**3*b**3*d**6*(-c**3/d**3)**(3/2)/(a*d - b*c)
**3 + a**2*b**4*c*d**5*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + a*b**5*c**2*d**4*(-c**3/d**3)**(3/2)/(a*d - b*c)**
3 - b**6*c**3*d**3*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - b**4*c**4*sqrt(-c**3/d**3)/(a*d - b*c))/(a**3*c*d**2 +
a**2*b*c**2*d + a*b**2*c**3))/(2*(a*d - b*c)) + sqrt(-c**3/d**3)*log(x + (a**4*d**4*sqrt(-c**3/d**3)/(a*d - b
*c) + a**3*b**3*d**6*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 - a**2*b**4*c*d**5*(-c**3/d**3)**(3/2)/(a*d - b*c)**3
- a*b**5*c**2*d**4*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + b**6*c**3*d**3*(-c**3/d**3)**(3/2)/(a*d - b*c)**3 + b*
*4*c**4*sqrt(-c**3/d**3)/(a*d - b*c))/(a**3*c*d**2 + a**2*b*c**2*d + a*b**2*c**3))/(2*(a*d - b*c)) + x/(b*d)

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Giac [B]  time = 1.30069, size = 576, normalized size = 7.38 \begin{align*} -\frac{{\left (\sqrt{a b} b^{3} c^{2} d{\left | b \right |} + \sqrt{a b} a^{2} b d^{3}{\left | b \right |} + \sqrt{a b} b c{\left | -b^{2} c d + a b d^{2} \right |}{\left | b \right |} + \sqrt{a b} a d{\left | -b^{2} c d + a b d^{2} \right |}{\left | b \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b^{2} c d + a b d^{2} + \sqrt{-4 \, a b^{3} c d^{3} +{\left (b^{2} c d + a b d^{2}\right )}^{2}}}{b^{2} d^{2}}}}\right )}{b^{4} c d{\left | -b^{2} c d + a b d^{2} \right |} + a b^{3} d^{2}{\left | -b^{2} c d + a b d^{2} \right |} +{\left (b^{2} c d - a b d^{2}\right )}^{2} b^{2}} + \frac{{\left (\sqrt{c d} b^{3} c^{2} d{\left | d \right |} + \sqrt{c d} a^{2} b d^{3}{\left | d \right |} - \sqrt{c d} b c{\left | -b^{2} c d + a b d^{2} \right |}{\left | d \right |} - \sqrt{c d} a d{\left | -b^{2} c d + a b d^{2} \right |}{\left | d \right |}\right )} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b^{2} c d + a b d^{2} - \sqrt{-4 \, a b^{3} c d^{3} +{\left (b^{2} c d + a b d^{2}\right )}^{2}}}{b^{2} d^{2}}}}\right )}{b^{2} c d^{3}{\left | -b^{2} c d + a b d^{2} \right |} + a b d^{4}{\left | -b^{2} c d + a b d^{2} \right |} -{\left (b^{2} c d - a b d^{2}\right )}^{2} d^{2}} + \frac{x}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(a*b)*b^3*c^2*d*abs(b) + sqrt(a*b)*a^2*b*d^3*abs(b) + sqrt(a*b)*b*c*abs(-b^2*c*d + a*b*d^2)*abs(b) + sqr
t(a*b)*a*d*abs(-b^2*c*d + a*b*d^2)*abs(b))*arctan(2*sqrt(1/2)*x/sqrt((b^2*c*d + a*b*d^2 + sqrt(-4*a*b^3*c*d^3
+ (b^2*c*d + a*b*d^2)^2))/(b^2*d^2)))/(b^4*c*d*abs(-b^2*c*d + a*b*d^2) + a*b^3*d^2*abs(-b^2*c*d + a*b*d^2) + (
b^2*c*d - a*b*d^2)^2*b^2) + (sqrt(c*d)*b^3*c^2*d*abs(d) + sqrt(c*d)*a^2*b*d^3*abs(d) - sqrt(c*d)*b*c*abs(-b^2*
c*d + a*b*d^2)*abs(d) - sqrt(c*d)*a*d*abs(-b^2*c*d + a*b*d^2)*abs(d))*arctan(2*sqrt(1/2)*x/sqrt((b^2*c*d + a*b
*d^2 - sqrt(-4*a*b^3*c*d^3 + (b^2*c*d + a*b*d^2)^2))/(b^2*d^2)))/(b^2*c*d^3*abs(-b^2*c*d + a*b*d^2) + a*b*d^4*
abs(-b^2*c*d + a*b*d^2) - (b^2*c*d - a*b*d^2)^2*d^2) + x/(b*d)