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

Optimal. Leaf size=136 \[ \frac{(3 a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{x (-2 x (b B-3 a D)+3 a C+A b)}{8 a b^2 \left (a+b x^2\right )}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac{D \log \left (a+b x^2\right )}{2 b^3} \]

[Out]

-(x^2*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(4*a*b*(a + b*x^2)^2) - (x*(A*b + 3*a*C - 2*(b*B - 3*a*D)*x))/(8*a*b^
2*(a + b*x^2)) + ((A*b + 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2)) + (D*Log[a + b*x^2])/(2*b^3)

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Rubi [A]  time = 0.158462, antiderivative size = 136, normalized size of antiderivative = 1., 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 = {1804, 635, 205, 260} \[ \frac{(3 a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{x (-2 x (b B-3 a D)+3 a C+A b)}{8 a b^2 \left (a+b x^2\right )}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac{D \log \left (a+b x^2\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^2*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(4*a*b*(a + b*x^2)^2) - (x*(A*b + 3*a*C - 2*(b*B - 3*a*D)*x))/(8*a*b^
2*(a + b*x^2)) + ((A*b + 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(3/2)*b^(5/2)) + (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 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 )^3} \, dx &=-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{\int \frac{x \left (-2 a \left (B-\frac{a D}{b}\right )-(A b+3 a C) x-4 a D x^2\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{x (A b+3 a C-2 (b B-3 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac{\int \frac{a (A b+3 a C)+8 a^2 D x}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{x (A b+3 a C-2 (b B-3 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac{(A b+3 a C) \int \frac{1}{a+b x^2} \, dx}{8 a b^2}+\frac{D \int \frac{x}{a+b x^2} \, dx}{b^2}\\ &=-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{x (A b+3 a C-2 (b B-3 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac{(A b+3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}+\frac{D \log \left (a+b x^2\right )}{2 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0932766, size = 122, normalized size = 0.9 \[ \frac{\frac{-2 a^2 D+2 a b (B+C x)-2 A b^2 x}{\left (a+b x^2\right )^2}+\frac{8 a^2 D-a b (4 B+5 C x)+A b^2 x}{a \left (a+b x^2\right )}+\frac{\sqrt{b} (3 a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}+4 D \log \left (a+b x^2\right )}{8 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a^2*D - 2*A*b^2*x + 2*a*b*(B + C*x))/(a + b*x^2)^2 + (8*a^2*D + A*b^2*x - a*b*(4*B + 5*C*x))/(a*(a + b*x^
2)) + (Sqrt[b]*(A*b + 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2) + 4*D*Log[a + b*x^2])/(8*b^3)

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Maple [A]  time = 0.009, size = 133, normalized size = 1. \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-5\,aC \right ){x}^{3}}{8\,ab}}-{\frac{ \left ( Bb-2\,aD \right ){x}^{2}}{2\,{b}^{2}}}-{\frac{ \left ( Ab+3\,aC \right ) x}{8\,{b}^{2}}}-{\frac{a \left ( Bb-3\,aD \right ) }{4\,{b}^{3}}} \right ) }+{\frac{D\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}}+{\frac{A}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,C}{8\,{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)^3,x)

[Out]

(1/8*(A*b-5*C*a)/a/b*x^3-1/2*(B*b-2*D*a)/b^2*x^2-1/8*(A*b+3*C*a)/b^2*x-1/4*a*(B*b-3*D*a)/b^3)/(b*x^2+a)^2+1/2*
D*ln(b*x^2+a)/b^3+1/8/a/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*A+3/8/b^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*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)^3,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)^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 18.1106, size = 303, normalized size = 2.23 \begin{align*} \left (\frac{D}{2 b^{3}} - \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right ) \log{\left (x + \frac{- 8 D a^{2} + 16 a^{2} b^{3} \left (\frac{D}{2 b^{3}} - \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right )}{A b^{2} + 3 C a b} \right )} + \left (\frac{D}{2 b^{3}} + \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right ) \log{\left (x + \frac{- 8 D a^{2} + 16 a^{2} b^{3} \left (\frac{D}{2 b^{3}} + \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right )}{A b^{2} + 3 C a b} \right )} - \frac{2 B a^{2} b - 6 D a^{3} + x^{3} \left (- A b^{3} + 5 C a b^{2}\right ) + x^{2} \left (4 B a b^{2} - 8 D a^{2} b\right ) + x \left (A a b^{2} + 3 C a^{2} b\right )}{8 a^{3} b^{3} + 16 a^{2} b^{4} x^{2} + 8 a b^{5} x^{4}} \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)**3,x)

[Out]

(D/(2*b**3) - sqrt(-a**3*b**7)*(A*b + 3*C*a)/(16*a**3*b**6))*log(x + (-8*D*a**2 + 16*a**2*b**3*(D/(2*b**3) - s
qrt(-a**3*b**7)*(A*b + 3*C*a)/(16*a**3*b**6)))/(A*b**2 + 3*C*a*b)) + (D/(2*b**3) + sqrt(-a**3*b**7)*(A*b + 3*C
*a)/(16*a**3*b**6))*log(x + (-8*D*a**2 + 16*a**2*b**3*(D/(2*b**3) + sqrt(-a**3*b**7)*(A*b + 3*C*a)/(16*a**3*b*
*6)))/(A*b**2 + 3*C*a*b)) - (2*B*a**2*b - 6*D*a**3 + x**3*(-A*b**3 + 5*C*a*b**2) + x**2*(4*B*a*b**2 - 8*D*a**2
*b) + x*(A*a*b**2 + 3*C*a**2*b))/(8*a**3*b**3 + 16*a**2*b**4*x**2 + 8*a*b**5*x**4)

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Giac [A]  time = 1.19087, size = 173, normalized size = 1.27 \begin{align*} \frac{D \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{{\left (3 \, C a + A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{2}} - \frac{{\left (5 \, C a b - A b^{2}\right )} x^{3} - 4 \,{\left (2 \, D a^{2} - B a b\right )} x^{2} +{\left (3 \, C a^{2} + A a b\right )} x - \frac{2 \,{\left (3 \, D a^{3} - B a^{2} b\right )}}{b}}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{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)^3,x, algorithm="giac")

[Out]

1/2*D*log(b*x^2 + a)/b^3 + 1/8*(3*C*a + A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^2) - 1/8*((5*C*a*b - A*b^2)*
x^3 - 4*(2*D*a^2 - B*a*b)*x^2 + (3*C*a^2 + A*a*b)*x - 2*(3*D*a^3 - B*a^2*b)/b)/((b*x^2 + a)^2*a*b^2)