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

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

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Rubi [A]  time = 0.09, antiderivative size = 98, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 83, normalized size = 0.85 \begin {gather*} -\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 ) \end {gather*}

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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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^4} \, 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^4,x]

[Out]

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

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fricas [A]  time = 0.88, size = 103, normalized size = 1.05 \begin {gather*} \frac {3 \, D b^{2} x^{7} + 4 \, C b^{2} x^{6} + 6 \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + 12 \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + 12 \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} \log \relax (x) - 6 \, B a^{2} x - 4 \, A a^{2} - 12 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.43, size = 97, normalized size = 0.99 \begin {gather*} \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 {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^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

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maple [A]  time = 0.01, size = 97, normalized size = 0.99 \begin {gather*} \frac {D b^{2} x^{4}}{4}+\frac {C \,b^{2} x^{3}}{3}+\frac {B \,b^{2} x^{2}}{2}+D a b \,x^{2}+A \,b^{2} x +2 B a b \ln \relax (x )+2 C a b x +D a^{2} \ln \relax (x )-\frac {2 A a b}{x}-\frac {C \,a^{2}}{x}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \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^4,x)

[Out]

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

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maxima [A]  time = 1.37, size = 97, normalized size = 0.99 \begin {gather*} \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 \relax (x) - \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 {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^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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mupad [B]  time = 1.28, size = 106, normalized size = 1.08 \begin {gather*} \frac {b^2\,x^4\,D}{4}+\frac {a^2\,\ln \left (x^2\right )\,D}{2}-\frac {A\,\left (a^2+6\,a\,b\,x^2-3\,b^2\,x^4\right )}{3\,x^3}+\frac {B\,\left (b^2\,x^4-a^2+4\,a\,b\,x^2\,\ln \relax (x)\right )}{2\,x^2}+\frac {C\,\left (-3\,a^2+6\,a\,b\,x^2+b^2\,x^4\right )}{3\,x}+a\,b\,x^2\,D \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^4,x)

[Out]

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

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sympy [A]  time = 1.46, size = 100, normalized size = 1.02 \begin {gather*} \frac {C b^{2} x^{3}}{3} + \frac {D b^{2} x^{4}}{4} + a \left (2 B b + D a\right ) \log {\relax (x )} + 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 {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**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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