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

Optimal. Leaf size=98 \[ -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+\frac {1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac {1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac {1}{4} b^2 C x^4+\frac {1}{5} b^2 D x^5 \]

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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}{2 x^2}-\frac {a^2 B}{x}+\frac {1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac {1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac {1}{4} b^2 C x^4+\frac {1}{5} b^2 D 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^3,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 87, normalized size = 0.89 \begin {gather*} -\frac {a^2 \left (A+2 B x-2 D x^3\right )}{2 x^2}+a \log (x) (a C+2 A b)+\frac {1}{3} a b x (6 B+x (3 C+2 D x))+\frac {1}{60} b^2 x^2 (30 A+x (20 B+3 x (5 C+4 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^3,x]

[Out]

-1/2*(a^2*(A + 2*B*x - 2*D*x^3))/x^2 + (a*b*x*(6*B + x*(3*C + 2*D*x)))/3 + (b^2*x^2*(30*A + x*(20*B + 3*x*(5*C
+ 4*D*x))))/60 + a*(2*A*b + a*C)*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^3} \, 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^3,x]

[Out]

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

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fricas [A]  time = 0.66, size = 103, normalized size = 1.05 \begin {gather*} \frac {12 \, D b^{2} x^{7} + 15 \, C b^{2} x^{6} + 20 \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + 30 \, {\left (2 \, C a b + A b^{2}\right )} x^{4} - 60 \, B a^{2} x + 60 \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} + 60 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2} \log \relax (x) - 30 \, A a^{2}}{60 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.34, size = 97, normalized size = 0.99 \begin {gather*} \frac {1}{5} \, D b^{2} x^{5} + \frac {1}{4} \, C b^{2} x^{4} + \frac {2}{3} \, D a b x^{3} + \frac {1}{3} \, B b^{2} x^{3} + C a b x^{2} + \frac {1}{2} \, A b^{2} x^{2} + D a^{2} x + 2 \, B a b x + {\left (C a^{2} + 2 \, A a b\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 97, normalized size = 0.99 \begin {gather*} \frac {D b^{2} x^{5}}{5}+\frac {C \,b^{2} x^{4}}{4}+\frac {B \,b^{2} x^{3}}{3}+\frac {2 D a b \,x^{3}}{3}+\frac {A \,b^{2} x^{2}}{2}+C a b \,x^{2}+2 A a b \ln \relax (x )+2 B a b x +C \,a^{2} \ln \relax (x )+D a^{2} x -\frac {B \,a^{2}}{x}-\frac {A \,a^{2}}{2 x^{2}} \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^3,x)

[Out]

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

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maxima [A]  time = 1.32, size = 96, normalized size = 0.98 \begin {gather*} \frac {1}{5} \, D b^{2} x^{5} + \frac {1}{4} \, C b^{2} x^{4} + \frac {1}{3} \, {\left (2 \, D a b + B b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, C a b + A b^{2}\right )} x^{2} + {\left (D a^{2} + 2 \, B a b\right )} x + {\left (C a^{2} + 2 \, A a b\right )} \log \relax (x) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.11, size = 103, normalized size = 1.05 \begin {gather*} \frac {C\,\left (4\,a^2\,\ln \relax (x)+b^2\,x^4+4\,a\,b\,x^2\right )}{4}+a^2\,x\,D+\frac {b^2\,x^5\,D}{5}+\frac {A\,\left (b^2\,x^4-a^2+4\,a\,b\,x^2\,\ln \relax (x)\right )}{2\,x^2}+\frac {B\,\left (-3\,a^2+6\,a\,b\,x^2+b^2\,x^4\right )}{3\,x}+\frac {2\,a\,b\,x^3\,D}{3} \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^3,x)

[Out]

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

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sympy [A]  time = 0.58, size = 100, normalized size = 1.02 \begin {gather*} \frac {C b^{2} x^{4}}{4} + \frac {D b^{2} x^{5}}{5} + a \left (2 A b + C a\right ) \log {\relax (x )} + x^{3} \left (\frac {B b^{2}}{3} + \frac {2 D a b}{3}\right ) + x^{2} \left (\frac {A b^{2}}{2} + C a b\right ) + x \left (2 B a b + D a^{2}\right ) + \frac {- A a^{2} - 2 B a^{2} x}{2 x^{2}} \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**3,x)

[Out]

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

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