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(x*b^(1/2)/a^(1/2))/b^(3/2)/(-a*d+b*c)-c^(3/2)*arctan(x*d^(1/2)/c^(1/2))/d^(3/2)/(-a*d+b*c
)

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Rubi [A]  time = 0.08, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.09, 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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fricas [A]  time = 0.58, size = 391, normalized size = 5.01 \[ \left [-\frac {a d \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + b c \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 2 \, {\left (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, \frac {2 \, a d \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - b c \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) + 2 \, {\left (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, -\frac {2 \, b c \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + a d \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 2 \, {\left (b c - a d\right )} x}{2 \, {\left (b^{2} c d - a b d^{2}\right )}}, \frac {a d \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - b c \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + {\left (b c - a d\right )} x}{b^{2} c d - a b d^{2}}\right ] \]

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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giac [A]  time = 0.38, size = 72, normalized size = 0.92 \[ \frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {a b}} - \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d - a d^{2}\right )} \sqrt {c d}} + \frac {x}{b d} \]

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]

a^2*arctan(b*x/sqrt(a*b))/((b^2*c - a*b*d)*sqrt(a*b)) - c^2*arctan(d*x/sqrt(c*d))/((b*c*d - a*d^2)*sqrt(c*d))
+ x/(b*d)

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maple [A]  time = 0.01, size = 73, normalized size = 0.94 \[ -\frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \sqrt {a b}\, b}+\frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \sqrt {c d}\, d}+\frac {x}{b d} \]

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/b*a^2/(a*d-b*c)/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)+1/d*c^2/(a*d-b*c)/(c*d)^(1/2)*arctan(1/(c*d)^(1/
2)*d*x)

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maxima [A]  time = 2.35, size = 72, normalized size = 0.92 \[ \frac {a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {a b}} - \frac {c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c d - a d^{2}\right )} \sqrt {c d}} + \frac {x}{b d} \]

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]

a^2*arctan(b*x/sqrt(a*b))/((b^2*c - a*b*d)*sqrt(a*b)) - c^2*arctan(d*x/sqrt(c*d))/((b*c*d - a*d^2)*sqrt(c*d))
+ x/(b*d)

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mupad [B]  time = 0.61, size = 343, normalized size = 4.40 \[ \frac {\ln \left (a^5\,b^4\,d^3-a^2\,b^7\,c^3+d^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+b^6\,c^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,b^4\,c-2\,a\,b^3\,d}-\frac {\ln \left (a^2\,b^7\,c^3-a^5\,b^4\,d^3+d^3\,x\,{\left (-a^3\,b^3\right )}^{3/2}+b^6\,c^3\,x\,\sqrt {-a^3\,b^3}\right )\,\sqrt {-a^3\,b^3}}{2\,\left (b^4\,c-a\,b^3\,d\right )}+\frac {x}{b\,d}-\frac {\ln \left (a^3\,c^2\,d^7-b^3\,c^5\,d^4+b^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+a^3\,d^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,\left (a\,d^4-b\,c\,d^3\right )}+\frac {\ln \left (b^3\,c^5\,d^4-a^3\,c^2\,d^7+b^3\,x\,{\left (-c^3\,d^3\right )}^{3/2}+a^3\,d^6\,x\,\sqrt {-c^3\,d^3}\right )\,\sqrt {-c^3\,d^3}}{2\,a\,d^4-2\,b\,c\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x^2)*(c + d*x^2)),x)

[Out]

(log(a^5*b^4*d^3 - a^2*b^7*c^3 + d^3*x*(-a^3*b^3)^(3/2) + b^6*c^3*x*(-a^3*b^3)^(1/2))*(-a^3*b^3)^(1/2))/(2*b^4
*c - 2*a*b^3*d) - (log(a^2*b^7*c^3 - a^5*b^4*d^3 + d^3*x*(-a^3*b^3)^(3/2) + b^6*c^3*x*(-a^3*b^3)^(1/2))*(-a^3*
b^3)^(1/2))/(2*(b^4*c - a*b^3*d)) + x/(b*d) - (log(a^3*c^2*d^7 - b^3*c^5*d^4 + b^3*x*(-c^3*d^3)^(3/2) + a^3*d^
6*x*(-c^3*d^3)^(1/2))*(-c^3*d^3)^(1/2))/(2*(a*d^4 - b*c*d^3)) + (log(b^3*c^5*d^4 - a^3*c^2*d^7 + b^3*x*(-c^3*d
^3)^(3/2) + a^3*d^6*x*(-c^3*d^3)^(1/2))*(-c^3*d^3)^(1/2))/(2*a*d^4 - 2*b*c*d^3)

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sympy [B]  time = 12.39, size = 921, normalized size = 11.81 \[ - \frac {\sqrt {- \frac {a^{3}}{b^{3}}} \log {\left (x + \frac {- \frac {a^{4} d^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c} - \frac {a^{3} b^{3} d^{6} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{4} c d^{5} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a b^{5} c^{2} d^{4} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{3} d^{3} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {a^{3}}{b^{3}}} \log {\left (x + \frac {\frac {a^{4} d^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c} + \frac {a^{3} b^{3} d^{6} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{4} c d^{5} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a b^{5} c^{2} d^{4} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{3} d^{3} \left (- \frac {a^{3}}{b^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} \sqrt {- \frac {a^{3}}{b^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {c^{3}}{d^{3}}} \log {\left (x + \frac {- \frac {a^{4} d^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c} - \frac {a^{3} b^{3} d^{6} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{4} c d^{5} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a b^{5} c^{2} d^{4} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{3} d^{3} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {c^{3}}{d^{3}}} \log {\left (x + \frac {\frac {a^{4} d^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c} + \frac {a^{3} b^{3} d^{6} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{4} c d^{5} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a b^{5} c^{2} d^{4} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{3} d^{3} \left (- \frac {c^{3}}{d^{3}}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} \sqrt {- \frac {c^{3}}{d^{3}}}}{a d - b c}}{a^{3} c d^{2} + a^{2} b c^{2} d + a b^{2} c^{3}} \right )}}{2 \left (a d - b c\right )} + \frac {x}{b d} \]

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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