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

Optimal. Leaf size=54 \[ x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4 \]

[Out]

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

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Rubi [A]  time = 0.0481677, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1802} \[ x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0326805, size = 54, normalized size = 1. \[ x (a C+A b)-\frac{a A}{x}+\frac{1}{2} x^2 (a D+b B)+a B \log (x)+\frac{1}{3} b C x^3+\frac{1}{4} b D x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 50, normalized size = 0.9 \begin{align*}{\frac{bD{x}^{4}}{4}}+{\frac{bC{x}^{3}}{3}}+{\frac{B{x}^{2}b}{2}}+{\frac{D{x}^{2}a}{2}}+Abx+aCx+aB\ln \left ( x \right ) -{\frac{Aa}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21773, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{4} \, D b x^{4} + \frac{1}{3} \, C b x^{3} + \frac{1}{2} \,{\left (D a + B b\right )} x^{2} + B a \log \left (x\right ) +{\left (C a + A b\right )} x - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.302393, size = 49, normalized size = 0.91 \begin{align*} - \frac{A a}{x} + B a \log{\left (x \right )} + \frac{C b x^{3}}{3} + \frac{D b x^{4}}{4} + x^{2} \left (\frac{B b}{2} + \frac{D a}{2}\right ) + x \left (A b + C a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17958, size = 68, normalized size = 1.26 \begin{align*} \frac{1}{4} \, D b x^{4} + \frac{1}{3} \, C b x^{3} + \frac{1}{2} \, D a x^{2} + \frac{1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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