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

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Rubi [A]  time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, size = 54, normalized size = 1.00 \[ x (a C+A b)-\frac {a A}{x}+\frac {1}{2} x^2 (a D+b B)+a B \log (x)+\frac {1}{3} b C x^3+\frac {1}{4} b D x^4 \]

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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fricas [A]  time = 0.73, size = 55, normalized size = 1.02 \[ \frac {3 \, D b x^{5} + 4 \, C b x^{4} + 6 \, {\left (D a + B b\right )} x^{3} + 12 \, B a x \log \relax (x) + 12 \, {\left (C a + A b\right )} x^{2} - 12 \, A a}{12 \, x} \]

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]

1/12*(3*D*b*x^5 + 4*C*b*x^4 + 6*(D*a + B*b)*x^3 + 12*B*a*x*log(x) + 12*(C*a + A*b)*x^2 - 12*A*a)/x

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giac [A]  time = 0.39, size = 50, normalized size = 0.93 \[ \frac {1}{4} \, D b x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{2} \, D a x^{2} + \frac {1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac {A a}{x} \]

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

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maple [A]  time = 0.01, size = 50, normalized size = 0.93 \[ \frac {D b \,x^{4}}{4}+\frac {C b \,x^{3}}{3}+\frac {B b \,x^{2}}{2}+\frac {D a \,x^{2}}{2}+A b x +B a \ln \relax (x )+C a x -\frac {A a}{x} \]

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*A/x+a*B*ln(x)

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maxima [A]  time = 1.34, size = 48, normalized size = 0.89 \[ \frac {1}{4} \, D b x^{4} + \frac {1}{3} \, C b x^{3} + \frac {1}{2} \, {\left (D a + B b\right )} x^{2} + B a \log \relax (x) + {\left (C a + A b\right )} x - \frac {A a}{x} \]

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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mupad [B]  time = 1.14, size = 49, normalized size = 0.91 \[ \frac {a\,x^2\,D}{2}+\frac {b\,x^4\,D}{4}+A\,b\,x+C\,a\,x-\frac {A\,a}{x}+\frac {B\,b\,x^2}{2}+\frac {C\,b\,x^3}{3}+B\,a\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.28, size = 49, normalized size = 0.91 \[ - \frac {A a}{x} + B a \log {\relax (x )} + \frac {C b x^{3}}{3} + \frac {D b x^{4}}{4} + x^{2} \left (\frac {B b}{2} + \frac {D a}{2}\right ) + x \left (A b + C a\right ) \]

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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