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

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

[Out]

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

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Rubi [A]  time = 0.048458, 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} \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0261517, size = 51, normalized size = 0.94 \[ \log (x) (a C+A b)-\frac{a \left (A+2 B x-2 D x^3\right )}{2 x^2}+\frac{1}{6} b x \left (6 B+3 C x+2 D x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(6*B + 3*C*x + 2*D*x^2))/6 - (a*(A + 2*B*x - 2*D*x^3))/(2*x^2) + (A*b + a*C)*Log[x]

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Maple [A]  time = 0.005, size = 48, normalized size = 0.9 \begin{align*}{\frac{bD{x}^{3}}{3}}+{\frac{bC{x}^{2}}{2}}+bBx+aDx+A\ln \left ( x \right ) b+C\ln \left ( x \right ) a-{\frac{Aa}{2\,{x}^{2}}}-{\frac{Ba}{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^3,x)

[Out]

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

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Maxima [A]  time = 1.06238, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} +{\left (D a + B b\right )} x +{\left (C a + A b\right )} \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

1/3*D*b*x^3 + 1/2*C*b*x^2 + (D*a + B*b)*x + (C*a + A*b)*log(x) - 1/2*(2*B*a*x + A*a)/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)*(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 = 0.415662, size = 49, normalized size = 0.91 \begin{align*} \frac{C b x^{2}}{2} + \frac{D b x^{3}}{3} + x \left (B b + D a\right ) + \left (A b + C a\right ) \log{\left (x \right )} - \frac{A a + 2 B a x}{2 x^{2}} \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**3,x)

[Out]

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

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Giac [A]  time = 1.15191, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} + D a x + B b x +{\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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