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

Optimal. Leaf size=134 \[ -\frac{x (A b-3 a C)}{2 a b^2}+\frac{(A b-3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-2 a D) \log \left (a+b x^2\right )}{2 b^3}+\frac{D x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.226388, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1804, 1802, 635, 205, 260} \[ -\frac{x (A b-3 a C)}{2 a b^2}+\frac{(A b-3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-2 a D) \log \left (a+b x^2\right )}{2 b^3}+\frac{D x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1804

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

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]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0749142, size = 100, normalized size = 0.75 \[ \frac{\frac{a^2 (-D)+a b (B+C x)-A b^2 x}{a+b x^2}+\frac{\sqrt{b} (A b-3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}+(b B-2 a D) \log \left (a+b x^2\right )+2 b C x+b D x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 154, normalized size = 1.2 \begin{align*}{\frac{D{x}^{2}}{2\,{b}^{2}}}+{\frac{Cx}{{b}^{2}}}-{\frac{Ax}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{aCx}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Ba}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}D}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) aD}{{b}^{3}}}+{\frac{A}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,aC}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

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)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 3.33235, size = 284, normalized size = 2.12 \begin{align*} \frac{C x}{b^{2}} + \frac{D x^{2}}{2 b^{2}} + \left (- \frac{- B b + 2 D a}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right ) \log{\left (x + \frac{2 B a b - 4 D a^{2} - 4 a b^{3} \left (- \frac{- B b + 2 D a}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right )}{- A b^{2} + 3 C a b} \right )} + \left (- \frac{- B b + 2 D a}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right ) \log{\left (x + \frac{2 B a b - 4 D a^{2} - 4 a b^{3} \left (- \frac{- B b + 2 D a}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right )}{- A b^{2} + 3 C a b} \right )} + \frac{B a b - D a^{2} + x \left (- A b^{2} + C a b\right )}{2 a b^{3} + 2 b^{4} x^{2}} \end{align*}

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)**2,x)

[Out]

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

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Giac [A]  time = 1.19379, size = 150, normalized size = 1.12 \begin{align*} -\frac{{\left (3 \, C a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} - \frac{{\left (2 \, D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{D b^{2} x^{2} + 2 \, C b^{2} x}{2 \, b^{4}} - \frac{D a^{2} - B a b -{\left (C a b - A b^{2}\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{3}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

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