3.76 \(\int \frac {x^4 (A+B x^2)}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=87 \[ -\frac {\sqrt {a} (3 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {a x (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {x (A b-2 a B)}{b^3}+\frac {B x^3}{3 b^2} \]

[Out]

(A*b-2*B*a)*x/b^3+1/3*B*x^3/b^2+1/2*a*(A*b-B*a)*x/b^3/(b*x^2+a)-1/2*(3*A*b-5*B*a)*arctan(x*b^(1/2)/a^(1/2))*a^
(1/2)/b^(7/2)

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Rubi [A]  time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 1153, 205} \[ \frac {a x (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {x (A b-2 a B)}{b^3}-\frac {\sqrt {a} (3 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {B x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + (a*(A*b - a*B)*x)/(2*b^3*(a + b*x^2)) - (Sqrt[a]*(3*A*b - 5*a*B)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

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 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {a (A b-a B) x}{2 b^3 \left (a+b x^2\right )}-\frac {\int \frac {a (A b-a B)-2 b (A b-a B) x^2-2 b^2 B x^4}{a+b x^2} \, dx}{2 b^3}\\ &=\frac {a (A b-a B) x}{2 b^3 \left (a+b x^2\right )}-\frac {\int \left (-2 (A b-2 a B)-2 b B x^2+\frac {3 a A b-5 a^2 B}{a+b x^2}\right ) \, dx}{2 b^3}\\ &=\frac {(A b-2 a B) x}{b^3}+\frac {B x^3}{3 b^2}+\frac {a (A b-a B) x}{2 b^3 \left (a+b x^2\right )}-\frac {(a (3 A b-5 a B)) \int \frac {1}{a+b x^2} \, dx}{2 b^3}\\ &=\frac {(A b-2 a B) x}{b^3}+\frac {B x^3}{3 b^2}+\frac {a (A b-a B) x}{2 b^3 \left (a+b x^2\right )}-\frac {\sqrt {a} (3 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 89, normalized size = 1.02 \[ \frac {x \left (a A b-a^2 B\right )}{2 b^3 \left (a+b x^2\right )}+\frac {\sqrt {a} (5 a B-3 A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{7/2}}+\frac {x (A b-2 a B)}{b^3}+\frac {B x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x^2))/(a + b*x^2)^2,x]

[Out]

((A*b - 2*a*B)*x)/b^3 + (B*x^3)/(3*b^2) + ((a*A*b - a^2*B)*x)/(2*b^3*(a + b*x^2)) + (Sqrt[a]*(-3*A*b + 5*a*B)*
ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2))

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fricas [A]  time = 0.49, size = 240, normalized size = 2.76 \[ \left [\frac {4 \, B b^{2} x^{5} - 4 \, {\left (5 \, B a b - 3 \, A b^{2}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} - 3 \, A a b + {\left (5 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 6 \, {\left (5 \, B a^{2} - 3 \, A a b\right )} x}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {2 \, B b^{2} x^{5} - 2 \, {\left (5 \, B a b - 3 \, A b^{2}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} - 3 \, A a b + {\left (5 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 3 \, {\left (5 \, B a^{2} - 3 \, A a b\right )} x}{6 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/12*(4*B*b^2*x^5 - 4*(5*B*a*b - 3*A*b^2)*x^3 - 3*(5*B*a^2 - 3*A*a*b + (5*B*a*b - 3*A*b^2)*x^2)*sqrt(-a/b)*lo
g((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 6*(5*B*a^2 - 3*A*a*b)*x)/(b^4*x^2 + a*b^3), 1/6*(2*B*b^2*x^5 -
2*(5*B*a*b - 3*A*b^2)*x^3 + 3*(5*B*a^2 - 3*A*a*b + (5*B*a*b - 3*A*b^2)*x^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a)
- 3*(5*B*a^2 - 3*A*a*b)*x)/(b^4*x^2 + a*b^3)]

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giac [A]  time = 0.30, size = 88, normalized size = 1.01 \[ \frac {{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} - \frac {B a^{2} x - A a b x}{2 \, {\left (b x^{2} + a\right )} b^{3}} + \frac {B b^{4} x^{3} - 6 \, B a b^{3} x + 3 \, A b^{4} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(5*B*a^2 - 3*A*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/2*(B*a^2*x - A*a*b*x)/((b*x^2 + a)*b^3) + 1/
3*(B*b^4*x^3 - 6*B*a*b^3*x + 3*A*b^4*x)/b^6

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maple [A]  time = 0.01, size = 105, normalized size = 1.21 \[ \frac {B \,x^{3}}{3 b^{2}}+\frac {A a x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 A a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {B \,a^{2} x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 B \,a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {A x}{b^{2}}-\frac {2 B a x}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

1/3*B*x^3/b^2+1/b^2*A*x-2/b^3*B*a*x+1/2*a/b^2*x/(b*x^2+a)*A-1/2*a^2/b^3*x/(b*x^2+a)*B-3/2*a/b^2/(a*b)^(1/2)*ar
ctan(1/(a*b)^(1/2)*b*x)*A+5/2*a^2/b^3/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*B

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maxima [A]  time = 2.42, size = 85, normalized size = 0.98 \[ -\frac {{\left (B a^{2} - A a b\right )} x}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} + \frac {{\left (5 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {B b x^{3} - 3 \, {\left (2 \, B a - A b\right )} x}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x^2+A)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*(B*a^2 - A*a*b)*x/(b^4*x^2 + a*b^3) + 1/2*(5*B*a^2 - 3*A*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/3
*(B*b*x^3 - 3*(2*B*a - A*b)*x)/b^3

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mupad [B]  time = 0.07, size = 104, normalized size = 1.20 \[ x\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )-\frac {x\,\left (\frac {B\,a^2}{2}-\frac {A\,a\,b}{2}\right )}{b^4\,x^2+a\,b^3}+\frac {B\,x^3}{3\,b^2}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (3\,A\,b-5\,B\,a\right )}{5\,B\,a^2-3\,A\,a\,b}\right )\,\left (3\,A\,b-5\,B\,a\right )}{2\,b^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(A + B*x^2))/(a + b*x^2)^2,x)

[Out]

x*(A/b^2 - (2*B*a)/b^3) - (x*((B*a^2)/2 - (A*a*b)/2))/(a*b^3 + b^4*x^2) + (B*x^3)/(3*b^2) + (a^(1/2)*atan((a^(
1/2)*b^(1/2)*x*(3*A*b - 5*B*a))/(5*B*a^2 - 3*A*a*b))*(3*A*b - 5*B*a))/(2*b^(7/2))

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sympy [A]  time = 0.64, size = 129, normalized size = 1.48 \[ \frac {B x^{3}}{3 b^{2}} + x \left (\frac {A}{b^{2}} - \frac {2 B a}{b^{3}}\right ) + \frac {x \left (A a b - B a^{2}\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {\sqrt {- \frac {a}{b^{7}}} \left (- 3 A b + 5 B a\right ) \log {\left (- b^{3} \sqrt {- \frac {a}{b^{7}}} + x \right )}}{4} + \frac {\sqrt {- \frac {a}{b^{7}}} \left (- 3 A b + 5 B a\right ) \log {\left (b^{3} \sqrt {- \frac {a}{b^{7}}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

B*x**3/(3*b**2) + x*(A/b**2 - 2*B*a/b**3) + x*(A*a*b - B*a**2)/(2*a*b**3 + 2*b**4*x**2) - sqrt(-a/b**7)*(-3*A*
b + 5*B*a)*log(-b**3*sqrt(-a/b**7) + x)/4 + sqrt(-a/b**7)*(-3*A*b + 5*B*a)*log(b**3*sqrt(-a/b**7) + x)/4

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