3.1.94 \(\int \frac {A+B x+C x^2+D x^3}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=93 \[ \frac {(a C+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {x (A b-a C)-a \left (B-\frac {a D}{b}\right )}{2 a b \left (a+b x^2\right )}+\frac {D \log \left (a+b x^2\right )}{2 b^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1814, 635, 205, 260} \begin {gather*} \frac {(a C+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}+\frac {D \log \left (a+b x^2\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]

[Out]

-(a*(B - (a*D)/b) - (A*b - a*C)*x)/(2*a*b*(a + b*x^2)) + ((A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*
b^(3/2)) + (D*Log[a + b*x^2])/(2*b^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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 83, normalized size = 0.89 \begin {gather*} \frac {\frac {\sqrt {b} (a C+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {a^2 D-a b (B+C x)+A b^2 x}{a \left (a+b x^2\right )}+D \log \left (a+b x^2\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]

[Out]

((a^2*D + A*b^2*x - a*b*(B + C*x))/(a*(a + b*x^2)) + (Sqrt[b]*(A*b + a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2)
+ D*Log[a + b*x^2])/(2*b^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]

[Out]

IntegrateAlgebraic[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2, x]

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fricas [A]  time = 0.85, size = 257, normalized size = 2.76 \begin {gather*} \left [\frac {2 \, D a^{3} - 2 \, B a^{2} b - {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (C a^{2} b - A a b^{2}\right )} x + 2 \, {\left (D a^{2} b x^{2} + D a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac {D a^{3} - B a^{2} b + {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (C a^{2} b - A a b^{2}\right )} x + {\left (D a^{2} b x^{2} + D a^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*D*a^3 - 2*B*a^2*b - (C*a^2 + A*a*b + (C*a*b + A*b^2)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/
(b*x^2 + a)) - 2*(C*a^2*b - A*a*b^2)*x + 2*(D*a^2*b*x^2 + D*a^3)*log(b*x^2 + a))/(a^2*b^3*x^2 + a^3*b^2), 1/2*
(D*a^3 - B*a^2*b + (C*a^2 + A*a*b + (C*a*b + A*b^2)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - (C*a^2*b - A*a*b^2)
*x + (D*a^2*b*x^2 + D*a^3)*log(b*x^2 + a))/(a^2*b^3*x^2 + a^3*b^2)]

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giac [A]  time = 0.38, size = 88, normalized size = 0.95 \begin {gather*} \frac {D \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} - \frac {{\left (C a - A b\right )} x - \frac {D a^{2} - B a b}{b}}{2 \, {\left (b x^{2} + a\right )} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*D*log(b*x^2 + a)/b^2 + 1/2*(C*a + A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b) - 1/2*((C*a - A*b)*x - (D*a^2
- B*a*b)/b)/((b*x^2 + a)*a*b)

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maple [A]  time = 0.01, size = 97, normalized size = 1.04 \begin {gather*} \frac {A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}+\frac {C \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {D \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {\frac {\left (A b -a C \right ) x}{2 a b}-\frac {b B -a D}{2 b^{2}}}{b \,x^{2}+a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x)

[Out]

(1/2*(A*b-C*a)/a/b*x-1/2*(B*b-D*a)/b^2)/(b*x^2+a)+1/2*D*ln(b*x^2+a)/b^2+1/2/a/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)
*b*x)*A+1/2/b/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*C

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maxima [A]  time = 2.94, size = 89, normalized size = 0.96 \begin {gather*} \frac {D a^{2} - B a b - {\left (C a b - A b^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {D \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*(D*a^2 - B*a*b - (C*a*b - A*b^2)*x)/(a*b^3*x^2 + a^2*b^2) + 1/2*D*log(b*x^2 + a)/b^2 + 1/2*(C*a + A*b)*arc
tan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b)

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mupad [B]  time = 1.32, size = 110, normalized size = 1.18 \begin {gather*} \frac {\left (\ln \left (b\,x^2+a\right )+\frac {a}{b\,x^2+a}\right )\,D}{2\,b^2}-\frac {B}{2\,b\,\left (b\,x^2+a\right )}+\frac {A\,x}{2\,a\,\left (b\,x^2+a\right )}-\frac {C\,x}{2\,b\,\left (b\,x^2+a\right )}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}}+\frac {C\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2 + x^3*D)/(a + b*x^2)^2,x)

[Out]

((log(a + b*x^2) + a/(a + b*x^2))*D)/(2*b^2) - B/(2*b*(a + b*x^2)) + (A*x)/(2*a*(a + b*x^2)) - (C*x)/(2*b*(a +
b*x^2)) + (A*atan((b^(1/2)*x)/a^(1/2)))/(2*a^(3/2)*b^(1/2)) + (C*atan((b^(1/2)*x)/a^(1/2)))/(2*a^(1/2)*b^(3/2
))

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sympy [B]  time = 3.09, size = 233, normalized size = 2.51 \begin {gather*} \left (\frac {D}{2 b^{2}} - \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log {\left (x + \frac {- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac {D}{2 b^{2}} - \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \left (\frac {D}{2 b^{2}} + \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log {\left (x + \frac {- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac {D}{2 b^{2}} + \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \frac {- B a b + D a^{2} + x \left (A b^{2} - C a b\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

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

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