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

Optimal. Leaf size=73 \[ \frac {(A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {(b B-a D) \log \left (a+b x^2\right )}{2 b^2}+\frac {C x}{b}+\frac {D x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1810, 635, 205, 260} \[ \frac {(A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {(b B-a D) \log \left (a+b x^2\right )}{2 b^2}+\frac {C x}{b}+\frac {D x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*x)/b + (D*x^2)/(2*b) + ((A*b - a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2)) + ((b*B - a*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 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 68, normalized size = 0.93 \[ \frac {\frac {2 \sqrt {b} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+(b B-a D) \log \left (a+b x^2\right )+b x (2 C+D x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.68, size = 157, normalized size = 2.15 \[ \left [\frac {D a b x^{2} + 2 \, C a b x + {\left (C a - A b\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}}, \frac {D a b x^{2} + 2 \, C a b x - 2 \, {\left (C a - A b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}}\right ] \]

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),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.46, size = 66, normalized size = 0.90 \[ -\frac {{\left (C a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {{\left (D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {D b x^{2} + 2 \, C b x}{2 \, b^{2}} \]

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),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 83, normalized size = 1.14 \[ \frac {A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {C a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {D x^{2}}{2 b}+\frac {B \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {C x}{b}-\frac {D a \ln \left (b \,x^{2}+a \right )}{2 b^{2}} \]

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),x)

[Out]

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

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maxima [A]  time = 2.93, size = 64, normalized size = 0.88 \[ -\frac {{\left (C a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {D x^{2} + 2 \, C x}{2 \, b} - \frac {{\left (D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.42, size = 79, normalized size = 1.08 \[ \frac {B\,\ln \left (b\,x^2+a\right )}{2\,b}-\frac {\left (a\,\ln \left (b\,x^2+a\right )-b\,x^2\right )\,D}{2\,b^2}+\frac {C\,x}{b}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}}-\frac {C\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{3/2}} \]

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),x)

[Out]

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

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sympy [B]  time = 0.88, size = 219, normalized size = 3.00 \[ \frac {C x}{b} + \frac {D x^{2}}{2 b} + \left (- \frac {- B b + D a}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right ) \log {\left (x + \frac {B a b - D a^{2} - 2 a b^{2} \left (- \frac {- B b + D a}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right )}{- A b^{2} + C a b} \right )} + \left (- \frac {- B b + D a}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right ) \log {\left (x + \frac {B a b - D a^{2} - 2 a b^{2} \left (- \frac {- B b + D a}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- A b + C a\right )}{2 a b^{4}}\right )}{- A b^{2} + C a b} \right )} \]

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),x)

[Out]

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

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