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

Optimal. Leaf size=56 \[ \frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5 \]

[Out]

a*B*x + ((A*b + a*C)*x^2)/2 + ((b*B + a*D)*x^3)/3 + (b*C*x^4)/4 + (b*D*x^5)/5 + a*A*Log[x]

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Rubi [A]  time = 0.0400955, antiderivative size = 56, 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} \[ \frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0147159, size = 56, normalized size = 1. \[ \frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*B*x + ((A*b + a*C)*x^2)/2 + ((b*B + a*D)*x^3)/3 + (b*C*x^4)/4 + (b*D*x^5)/5 + a*A*Log[x]

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Maple [A]  time = 0.004, size = 53, normalized size = 1. \begin{align*}{\frac{bD{x}^{5}}{5}}+{\frac{bC{x}^{4}}{4}}+{\frac{bB{x}^{3}}{3}}+{\frac{D{x}^{3}a}{3}}+{\frac{A{x}^{2}b}{2}}+{\frac{C{x}^{2}a}{2}}+Bax+aA\ln \left ( x \right ) \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,x)

[Out]

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

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Maxima [A]  time = 1.0298, size = 65, normalized size = 1.16 \begin{align*} \frac{1}{5} \, D b x^{5} + \frac{1}{4} \, C b x^{4} + \frac{1}{3} \,{\left (D a + B b\right )} x^{3} + B a x + \frac{1}{2} \,{\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

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

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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,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.285238, size = 54, normalized size = 0.96 \begin{align*} A a \log{\left (x \right )} + B a x + \frac{C b x^{4}}{4} + \frac{D b x^{5}}{5} + x^{3} \left (\frac{B b}{3} + \frac{D a}{3}\right ) + x^{2} \left (\frac{A b}{2} + \frac{C a}{2}\right ) \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,x)

[Out]

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

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Giac [A]  time = 1.16539, size = 72, normalized size = 1.29 \begin{align*} \frac{1}{5} \, D b x^{5} + \frac{1}{4} \, C b x^{4} + \frac{1}{3} \, D a x^{3} + \frac{1}{3} \, B b x^{3} + \frac{1}{2} \, C a x^{2} + \frac{1}{2} \, A b x^{2} + B a x + A a \log \left ({\left | x \right |}\right ) \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,x, algorithm="giac")

[Out]

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