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

Optimal. Leaf size=111 \[ -\frac {\sqrt {a} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x (A b-a C)}{b^2}-\frac {a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}+\frac {x^2 (b B-a D)}{2 b^2}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b} \]

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Rubi [A]  time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1802, 635, 205, 260} \begin {gather*} \frac {x (A b-a C)}{b^2}-\frac {\sqrt {a} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x^2 (b B-a D)}{2 b^2}-\frac {a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*C)*x)/b^2 + ((b*B - a*D)*x^2)/(2*b^2) + (C*x^3)/(3*b) + (D*x^4)/(4*b) - (Sqrt[a]*(A*b - a*C)*ArcTan[
(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) - (a*(b*B - a*D)*Log[a + b*x^2])/(2*b^3)

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

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Mathematica [A]  time = 0.06, size = 95, normalized size = 0.86 \begin {gather*} \frac {b x \left (-6 a (2 C+D x)+12 A b+b x \left (6 B+4 C x+3 D x^2\right )\right )+12 \sqrt {a} \sqrt {b} (a C-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+6 a (a D-b B) \log \left (a+b x^2\right )}{12 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(12*A*b - 6*a*(2*C + D*x) + b*x*(6*B + 4*C*x + 3*D*x^2)) + 12*Sqrt[a]*Sqrt[b]*(-(A*b) + a*C)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]] + 6*a*(-(b*B) + a*D)*Log[a + b*x^2])/(12*b^3)

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

[Out]

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

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fricas [A]  time = 0.92, size = 238, normalized size = 2.14 \begin {gather*} \left [\frac {3 \, D b^{2} x^{4} + 4 \, C b^{2} x^{3} - 6 \, {\left (D a b - B b^{2}\right )} x^{2} - 6 \, {\left (C a b - A b^{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 ) - 12 \, {\left (C a b - A b^{2}\right )} x + 6 \, {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{3}}, \frac {3 \, D b^{2} x^{4} + 4 \, C b^{2} x^{3} - 6 \, {\left (D a b - B b^{2}\right )} x^{2} + 12 \, {\left (C a b - A b^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 12 \, {\left (C a b - A b^{2}\right )} x + 6 \, {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*D*b^2*x^4 + 4*C*b^2*x^3 - 6*(D*a*b - B*b^2)*x^2 - 6*(C*a*b - A*b^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqr
t(-a/b) - a)/(b*x^2 + a)) - 12*(C*a*b - A*b^2)*x + 6*(D*a^2 - B*a*b)*log(b*x^2 + a))/b^3, 1/12*(3*D*b^2*x^4 +
4*C*b^2*x^3 - 6*(D*a*b - B*b^2)*x^2 + 12*(C*a*b - A*b^2)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 12*(C*a*b - A*b^2
)*x + 6*(D*a^2 - B*a*b)*log(b*x^2 + a))/b^3]

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giac [A]  time = 0.40, size = 112, normalized size = 1.01 \begin {gather*} \frac {{\left (C a^{2} - A a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {{\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {3 \, D b^{3} x^{4} + 4 \, C b^{3} x^{3} - 6 \, D a b^{2} x^{2} + 6 \, B b^{3} x^{2} - 12 \, C a b^{2} x + 12 \, A b^{3} x}{12 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(C*a^2 - A*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/2*(D*a^2 - B*a*b)*log(b*x^2 + a)/b^3 + 1/12*(3*D*b^3
*x^4 + 4*C*b^3*x^3 - 6*D*a*b^2*x^2 + 6*B*b^3*x^2 - 12*C*a*b^2*x + 12*A*b^3*x)/b^4

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maple [A]  time = 0.00, size = 128, normalized size = 1.15 \begin {gather*} \frac {D x^{4}}{4 b}+\frac {C \,x^{3}}{3 b}-\frac {A a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {B \,x^{2}}{2 b}+\frac {C \,a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {D a \,x^{2}}{2 b^{2}}+\frac {A x}{b}-\frac {B a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}-\frac {C a x}{b^{2}}+\frac {D a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*D*x^4/b+1/3*C*x^3/b+1/2*B/b*x^2-1/2/b^2*D*x^2*a+A/b*x-1/b^2*a*C*x-1/2*B*a/b^2*ln(b*x^2+a)+1/2*a^2/b^3*ln(b
*x^2+a)*D-1/(a*b)^(1/2)*A*a/b*arctan(1/(a*b)^(1/2)*b*x)+a^2/b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*C

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maxima [A]  time = 3.00, size = 98, normalized size = 0.88 \begin {gather*} \frac {{\left (C a^{2} - A a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {3 \, D b x^{4} + 4 \, C b x^{3} - 6 \, {\left (D a - B b\right )} x^{2} - 12 \, {\left (C a - A b\right )} x}{12 \, b^{2}} + \frac {{\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(C*a^2 - A*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/12*(3*D*b*x^4 + 4*C*b*x^3 - 6*(D*a - B*b)*x^2 - 12*(
C*a - A*b)*x)/b^2 + 1/2*(D*a^2 - B*a*b)*log(b*x^2 + a)/b^3

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

[Out]

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

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sympy [B]  time = 1.65, size = 245, normalized size = 2.21 \begin {gather*} \frac {C x^{3}}{3 b} + \frac {D x^{4}}{4 b} + x^{2} \left (\frac {B}{2 b} - \frac {D a}{2 b^{2}}\right ) + x \left (\frac {A}{b} - \frac {C a}{b^{2}}\right ) + \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} - \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right ) \log {\left (x + \frac {B a b - D a^{2} + 2 b^{3} \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} - \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right )}{- A b^{2} + C a b} \right )} + \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} + \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right ) \log {\left (x + \frac {B a b - D a^{2} + 2 b^{3} \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} + \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right )}{- A b^{2} + C a b} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*x**3/(3*b) + D*x**4/(4*b) + x**2*(B/(2*b) - D*a/(2*b**2)) + x*(A/b - C*a/b**2) + (a*(-B*b + D*a)/(2*b**3) -
sqrt(-a*b**7)*(-A*b + C*a)/(2*b**6))*log(x + (B*a*b - D*a**2 + 2*b**3*(a*(-B*b + D*a)/(2*b**3) - sqrt(-a*b**7)
*(-A*b + C*a)/(2*b**6)))/(-A*b**2 + C*a*b)) + (a*(-B*b + D*a)/(2*b**3) + sqrt(-a*b**7)*(-A*b + C*a)/(2*b**6))*
log(x + (B*a*b - D*a**2 + 2*b**3*(a*(-B*b + D*a)/(2*b**3) + sqrt(-a*b**7)*(-A*b + C*a)/(2*b**6)))/(-A*b**2 + C
*a*b))

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