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

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

[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]

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Rubi [A]  time = 0.0863914, 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} \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac{1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac{1}{4} b^2 C x^4+\frac{1}{5} b^2 D x^5 \]

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*}

Mathematica [A]  time = 0.0384175, size = 87, normalized size = 0.89 \[ -\frac{a^2 \left (A+2 B x-2 D x^3\right )}{2 x^2}+a \log (x) (a C+2 A b)+\frac{1}{3} a b x (6 B+x (3 C+2 D x))+\frac{1}{60} b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x))) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*(A + 2*B*x - 2*D*x^3))/(2*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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Maple [A]  time = 0.006, size = 97, normalized size = 1. \begin{align*}{\frac{{b}^{2}D{x}^{5}}{5}}+{\frac{{b}^{2}C{x}^{4}}{4}}+{\frac{B{x}^{3}{b}^{2}}{3}}+{\frac{2\,D{x}^{3}ab}{3}}+{\frac{A{x}^{2}{b}^{2}}{2}}+C{x}^{2}ab+2\,Bxab+{a}^{2}Dx+2\,A\ln \left ( x \right ) ab+C\ln \left ( x \right ){a}^{2}-{\frac{A{a}^{2}}{2\,{x}^{2}}}-{\frac{B{a}^{2}}{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^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*x^2*b^2+C*x^2*a*b+2*B*x*a*b+a^2*D*x+2*A*ln(x)*a*
b+C*ln(x)*a^2-1/2*a^2*A/x^2-a^2*B/x

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Maxima [A]  time = 0.988491, size = 130, normalized size = 1.33 \begin{align*} \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 \left (x\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \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^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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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^3,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.511069, size = 99, normalized size = 1.01 \begin{align*} \frac{C b^{2} x^{4}}{4} + \frac{D b^{2} x^{5}}{5} + a \left (2 A b + C a\right ) \log{\left (x \right )} + 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{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**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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Giac [A]  time = 1.16058, size = 131, normalized size = 1.34 \begin{align*} \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{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^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