3.77 \(\int \frac{(a+b x^2)^2 (A+B x+C x^2+D x^3)}{x^4} \, dx\)

Optimal. Leaf size=98 \[ -\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+b x (2 a C+A b)-\frac{a (a C+2 A b)}{x}+\frac{1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac{1}{3} b^2 C x^3+\frac{1}{4} b^2 D x^4 \]

[Out]

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

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Rubi [A]  time = 0.0857923, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1802} \[ -\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+b x (2 a C+A b)-\frac{a (a C+2 A b)}{x}+\frac{1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac{1}{3} b^2 C x^3+\frac{1}{4} b^2 D x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0472645, size = 83, normalized size = 0.85 \[ -\frac{a^2 (2 A+3 x (B+2 C x))}{6 x^3}-\frac{2 a A b}{x}+a \log (x) (a D+2 b B)+a b x (2 C+D x)+\frac{1}{12} b^2 x \left (12 A+x \left (6 B+4 C x+3 D x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 97, normalized size = 1. \begin{align*}{\frac{{b}^{2}D{x}^{4}}{4}}+{\frac{{b}^{2}C{x}^{3}}{3}}+{\frac{B{x}^{2}{b}^{2}}{2}}+D{x}^{2}ab+A{b}^{2}x+2\,abCx+2\,B\ln \left ( x \right ) ab+D\ln \left ( x \right ){a}^{2}-{\frac{A{a}^{2}}{3\,{x}^{3}}}-{\frac{B{a}^{2}}{2\,{x}^{2}}}-2\,{\frac{Aab}{x}}-{\frac{{a}^{2}C}{x}} \end{align*}

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^4,x)

[Out]

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

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Maxima [A]  time = 0.98409, size = 131, normalized size = 1.34 \begin{align*} \frac{1}{4} \, D b^{2} x^{4} + \frac{1}{3} \, C b^{2} x^{3} + \frac{1}{2} \,{\left (2 \, D a b + B b^{2}\right )} x^{2} +{\left (2 \, C a b + A b^{2}\right )} x +{\left (D a^{2} + 2 \, B a b\right )} \log \left (x\right ) - \frac{3 \, B a^{2} x + 2 \, A a^{2} + 6 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \end{align*}

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^4,x, algorithm="maxima")

[Out]

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

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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((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x^4,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.974943, size = 99, normalized size = 1.01 \begin{align*} \frac{C b^{2} x^{3}}{3} + \frac{D b^{2} x^{4}}{4} + a \left (2 B b + D a\right ) \log{\left (x \right )} + x^{2} \left (\frac{B b^{2}}{2} + D a b\right ) + x \left (A b^{2} + 2 C a b\right ) - \frac{2 A a^{2} + 3 B a^{2} x + x^{2} \left (12 A a b + 6 C a^{2}\right )}{6 x^{3}} \end{align*}

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**4,x)

[Out]

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

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Giac [A]  time = 1.16514, size = 131, normalized size = 1.34 \begin{align*} \frac{1}{4} \, D b^{2} x^{4} + \frac{1}{3} \, C b^{2} x^{3} + D a b x^{2} + \frac{1}{2} \, B b^{2} x^{2} + 2 \, C a b x + A b^{2} x +{\left (D a^{2} + 2 \, B a b\right )} \log \left ({\left | x \right |}\right ) - \frac{3 \, B a^{2} x + 2 \, A a^{2} + 6 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \end{align*}

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^4,x, algorithm="giac")

[Out]

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