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

Optimal. Leaf size=92 \[ \frac {(A b-a C) \log \left (a+b x^2\right )}{2 b^2}-\frac {\sqrt {a} (b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x (b B-a D)}{b^2}+\frac {C x^2}{2 b}+\frac {D x^3}{3 b} \]

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Rubi [A]  time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1802, 635, 205, 260} \begin {gather*} \frac {(A b-a C) \log \left (a+b x^2\right )}{2 b^2}+\frac {x (b B-a D)}{b^2}-\frac {\sqrt {a} (b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+\frac {C x^2}{2 b}+\frac {D x^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*B - a*D)*x)/b^2 + (C*x^2)/(2*b) + (D*x^3)/(3*b) - (Sqrt[a]*(b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2
) + ((A*b - a*C)*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 1802

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 81, normalized size = 0.88 \begin {gather*} \frac {3 (A b-a C) \log \left (a+b x^2\right )+x (-6 a D+6 b B+b x (3 C+2 D x))}{6 b^2}+\frac {\sqrt {a} (a D-b B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*(-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) + (x*(6*b*B - 6*a*D + b*x*(3*C + 2*D*x)) + 3*(A*b
- a*C)*Log[a + b*x^2])/(6*b^2)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.98, size = 180, normalized size = 1.96 \begin {gather*} \left [\frac {2 \, D b x^{3} + 3 \, C b x^{2} + 3 \, {\left (D a - B b\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 (D a - B b\right )} x - 3 \, {\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{6 \, b^{2}}, \frac {2 \, D b x^{3} + 3 \, C b x^{2} + 6 \, {\left (D a - B b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 6 \, {\left (D a - B b\right )} x - 3 \, {\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{6 \, b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(2*D*b*x^3 + 3*C*b*x^2 + 3*(D*a - B*b)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 6*(D*
a - B*b)*x - 3*(C*a - A*b)*log(b*x^2 + a))/b^2, 1/6*(2*D*b*x^3 + 3*C*b*x^2 + 6*(D*a - B*b)*sqrt(a/b)*arctan(b*
x*sqrt(a/b)/a) - 6*(D*a - B*b)*x - 3*(C*a - A*b)*log(b*x^2 + a))/b^2]

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giac [A]  time = 0.42, size = 88, normalized size = 0.96 \begin {gather*} -\frac {{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (D a^{2} - B a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, D b^{2} x^{3} + 3 \, C b^{2} x^{2} - 6 \, D a b x + 6 \, B b^{2} x}{6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(C*a - A*b)*log(b*x^2 + a)/b^2 + (D*a^2 - B*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/6*(2*D*b^2*x^3
+ 3*C*b^2*x^2 - 6*D*a*b*x + 6*B*b^2*x)/b^3

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maple [A]  time = 0.01, size = 106, normalized size = 1.15 \begin {gather*} \frac {D x^{3}}{3 b}-\frac {B a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {C \,x^{2}}{2 b}+\frac {D a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {A \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {B x}{b}-\frac {C a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}-\frac {D a x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, size = 82, normalized size = 0.89 \begin {gather*} -\frac {{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (D a^{2} - B a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, D b x^{3} + 3 \, C b x^{2} - 6 \, {\left (D a - B b\right )} x}{6 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(C*a - A*b)*log(b*x^2 + a)/b^2 + (D*a^2 - B*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/6*(2*D*b*x^3 +
3*C*b*x^2 - 6*(D*a - B*b)*x)/b^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{b\,x^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.99, size = 211, normalized size = 2.29 \begin {gather*} \frac {C x^{2}}{2 b} + \frac {D x^{3}}{3 b} + x \left (\frac {B}{b} - \frac {D a}{b^{2}}\right ) + \left (- \frac {- A b + C a}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right ) \log {\left (x + \frac {- A b + C a + 2 b^{2} \left (- \frac {- A b + C a}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right )}{- B b + D a} \right )} + \left (- \frac {- A b + C a}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right ) \log {\left (x + \frac {- A b + C a + 2 b^{2} \left (- \frac {- A b + C a}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- B b + D a\right )}{2 b^{5}}\right )}{- B b + D a} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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