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

Optimal. Leaf size=90 \[ -\frac {a^2 A}{x}+a^2 B \log (x)+\frac {1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac {D \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} b^2 B x^4+\frac {1}{5} b^2 C x^5 \]

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Rubi [A]  time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1583, 1628} \begin {gather*} -\frac {a^2 A}{x}+a^2 B \log (x)+\frac {1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac {D \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} b^2 B x^4+\frac {1}{5} b^2 C x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*A)/x) + a*(2*A*b + a*C)*x + a*b*B*x^2 + (b*(A*b + 2*a*C)*x^3)/3 + (b^2*B*x^4)/4 + (b^2*C*x^5)/5 + (D*(a
+ b*x^2)^3)/(6*b) + a^2*B*Log[x]

Rule 1583

Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - m - 1]*(a + b*x^n)^(p
+ 1))/(b*n*(p + 1)), x] + Int[(Px - Coeff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a,
b, m, n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x, n - m - 1], 0]

Rule 1628

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx &=\frac {D \left (a+b x^2\right )^3}{6 b}+\int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2\right )}{x^2} \, dx\\ &=\frac {D \left (a+b x^2\right )^3}{6 b}+\int \left (a (2 A b+a C)+\frac {a^2 A}{x^2}+\frac {a^2 B}{x}+2 a b B x+b (A b+2 a C) x^2+b^2 B x^3+b^2 C x^4\right ) \, dx\\ &=-\frac {a^2 A}{x}+a (2 A b+a C) x+a b B x^2+\frac {1}{3} b (A b+2 a C) x^3+\frac {1}{4} b^2 B x^4+\frac {1}{5} b^2 C x^5+\frac {D \left (a+b x^2\right )^3}{6 b}+a^2 B \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 88, normalized size = 0.98 \begin {gather*} a^2 \left (-\frac {A}{x}+C x+\frac {D x^2}{2}\right )+a^2 B \log (x)+\frac {1}{6} a b x (12 A+x (6 B+x (4 C+3 D x)))+\frac {1}{60} b^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*(-(A/x) + C*x + (D*x^2)/2) + (a*b*x*(12*A + x*(6*B + x*(4*C + 3*D*x))))/6 + (b^2*x^3*(20*A + x*(15*B + 2*x
*(6*C + 5*D*x))))/60 + a^2*B*Log[x]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 103, normalized size = 1.14 \begin {gather*} \frac {10 \, D b^{2} x^{7} + 12 \, C b^{2} x^{6} + 15 \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + 20 \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + 60 \, B a^{2} x \log \relax (x) + 30 \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} - 60 \, A a^{2} + 60 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*D*b^2*x^7 + 12*C*b^2*x^6 + 15*(2*D*a*b + B*b^2)*x^5 + 20*(2*C*a*b + A*b^2)*x^4 + 60*B*a^2*x*log(x) +
30*(D*a^2 + 2*B*a*b)*x^3 - 60*A*a^2 + 60*(C*a^2 + 2*A*a*b)*x^2)/x

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giac [A]  time = 0.38, size = 98, normalized size = 1.09 \begin {gather*} \frac {1}{6} \, D b^{2} x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {1}{2} \, D a b x^{4} + \frac {1}{4} \, B b^{2} x^{4} + \frac {2}{3} \, C a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {1}{2} \, D a^{2} x^{2} + B a b x^{2} + C a^{2} x + 2 \, A a b x + B a^{2} \log \left ({\left | x \right |}\right ) - \frac {A a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*D*b^2*x^6 + 1/5*C*b^2*x^5 + 1/2*D*a*b*x^4 + 1/4*B*b^2*x^4 + 2/3*C*a*b*x^3 + 1/3*A*b^2*x^3 + 1/2*D*a^2*x^2
+ B*a*b*x^2 + C*a^2*x + 2*A*a*b*x + B*a^2*log(abs(x)) - A*a^2/x

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maple [A]  time = 0.01, size = 98, normalized size = 1.09 \begin {gather*} \frac {D b^{2} x^{6}}{6}+\frac {C \,b^{2} x^{5}}{5}+\frac {B \,b^{2} x^{4}}{4}+\frac {D a b \,x^{4}}{2}+\frac {A \,b^{2} x^{3}}{3}+\frac {2 C a b \,x^{3}}{3}+B a b \,x^{2}+\frac {D a^{2} x^{2}}{2}+2 A a b x +B \,a^{2} \ln \relax (x )+C \,a^{2} x -\frac {A \,a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*D*b^2*x^6+1/5*b^2*C*x^5+1/4*B*b^2*x^4+1/2*D*x^4*a*b+1/3*A*x^3*b^2+2/3*C*x^3*a*b+B*a*b*x^2+1/2*D*x^2*a^2+2*
A*a*b*x+a^2*C*x-A*a^2/x+a^2*B*ln(x)

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maxima [A]  time = 1.32, size = 96, normalized size = 1.07 \begin {gather*} \frac {1}{6} \, D b^{2} x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {1}{4} \, {\left (2 \, D a b + B b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, C a b + A b^{2}\right )} x^{3} + B a^{2} \log \relax (x) + \frac {1}{2} \, {\left (D a^{2} + 2 \, B a b\right )} x^{2} - \frac {A a^{2}}{x} + {\left (C a^{2} + 2 \, A a b\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*D*b^2*x^6 + 1/5*C*b^2*x^5 + 1/4*(2*D*a*b + B*b^2)*x^4 + 1/3*(2*C*a*b + A*b^2)*x^3 + B*a^2*log(x) + 1/2*(D*
a^2 + 2*B*a*b)*x^2 - A*a^2/x + (C*a^2 + 2*A*a*b)*x

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mupad [B]  time = 1.11, size = 92, normalized size = 1.02 \begin {gather*} \frac {B\,\left (4\,a^2\,\ln \relax (x)+b^2\,x^4+4\,a\,b\,x^2\right )}{4}+\frac {{\left (b\,x^2+a\right )}^3\,D}{6\,b}+\frac {C\,x\,\left (15\,a^2+10\,a\,b\,x^2+3\,b^2\,x^4\right )}{15}+\frac {A\,\left (-3\,a^2+6\,a\,b\,x^2+b^2\,x^4\right )}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*(4*a^2*log(x) + b^2*x^4 + 4*a*b*x^2))/4 + ((a + b*x^2)^3*D)/(6*b) + (C*x*(15*a^2 + 3*b^2*x^4 + 10*a*b*x^2))
/15 + (A*(b^2*x^4 - 3*a^2 + 6*a*b*x^2))/(3*x)

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sympy [A]  time = 0.35, size = 99, normalized size = 1.10 \begin {gather*} - \frac {A a^{2}}{x} + B a^{2} \log {\relax (x )} + \frac {C b^{2} x^{5}}{5} + \frac {D b^{2} x^{6}}{6} + x^{4} \left (\frac {B b^{2}}{4} + \frac {D a b}{2}\right ) + x^{3} \left (\frac {A b^{2}}{3} + \frac {2 C a b}{3}\right ) + x^{2} \left (B a b + \frac {D a^{2}}{2}\right ) + x \left (2 A a b + C a^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a**2/x + B*a**2*log(x) + C*b**2*x**5/5 + D*b**2*x**6/6 + x**4*(B*b**2/4 + D*a*b/2) + x**3*(A*b**2/3 + 2*C*a
*b/3) + x**2*(B*a*b + D*a**2/2) + x*(2*A*a*b + C*a**2)

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