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

Optimal. Leaf size=92 \[ a^2 A \log (x)+a^2 B x+a A b x^2+\frac{1}{5} b x^5 (2 a D+b B)+\frac{1}{3} a x^3 (a D+2 b B)+\frac{C \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} A b^2 x^4+\frac{1}{7} b^2 D x^7 \]

[Out]

a^2*B*x + a*A*b*x^2 + (a*(2*b*B + a*D)*x^3)/3 + (A*b^2*x^4)/4 + (b*(b*B + 2*a*D)*x^5)/5 + (b^2*D*x^7)/7 + (C*(
a + b*x^2)^3)/(6*b) + a^2*A*Log[x]

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Rubi [A]  time = 0.0685905, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1583, 1802} \[ a^2 A \log (x)+a^2 B x+a A b x^2+\frac{1}{5} b x^5 (2 a D+b B)+\frac{1}{3} a x^3 (a D+2 b B)+\frac{C \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} A b^2 x^4+\frac{1}{7} b^2 D x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*B*x + a*A*b*x^2 + (a*(2*b*B + a*D)*x^3)/3 + (A*b^2*x^4)/4 + (b*(b*B + 2*a*D)*x^5)/5 + (b^2*D*x^7)/7 + (C*(
a + b*x^2)^3)/(6*b) + a^2*A*Log[x]

Rule 1583

Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - m - 1]*(a + b*x^n)^(p
+ 1))/(b*n*(p + 1)), x] + Int[(Px - Coeff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a,
b, m, n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x, n - m - 1], 0]

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} \, dx &=\frac{C \left (a+b x^2\right )^3}{6 b}+\int \frac{\left (a+b x^2\right )^2 \left (A+B x+D x^3\right )}{x} \, dx\\ &=\frac{C \left (a+b x^2\right )^3}{6 b}+\int \left (a^2 B+\frac{a^2 A}{x}+2 a A b x+a (2 b B+a D) x^2+A b^2 x^3+b (b B+2 a D) x^4+b^2 D x^6\right ) \, dx\\ &=a^2 B x+a A b x^2+\frac{1}{3} a (2 b B+a D) x^3+\frac{1}{4} A b^2 x^4+\frac{1}{5} b (b B+2 a D) x^5+\frac{1}{7} b^2 D x^7+\frac{C \left (a+b x^2\right )^3}{6 b}+a^2 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0434073, size = 88, normalized size = 0.96 \[ \frac{1}{420} x \left (70 a^2 (6 B+x (3 C+2 D x))+14 a b x (30 A+x (20 B+3 x (5 C+4 D x)))+b^2 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))\right )+a^2 A \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(70*a^2*(6*B + x*(3*C + 2*D*x)) + 14*a*b*x*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + b^2*x^3*(105*A + 2*x*(42
*B + 5*x*(7*C + 6*D*x)))))/420 + a^2*A*Log[x]

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Maple [A]  time = 0.003, size = 100, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}D{x}^{7}}{7}}+{\frac{C{b}^{2}{x}^{6}}{6}}+{\frac{B{x}^{5}{b}^{2}}{5}}+{\frac{2\,D{x}^{5}ab}{5}}+{\frac{A{b}^{2}{x}^{4}}{4}}+{\frac{C{x}^{4}ab}{2}}+{\frac{2\,B{x}^{3}ab}{3}}+{\frac{D{x}^{3}{a}^{2}}{3}}+aAb{x}^{2}+{\frac{C{x}^{2}{a}^{2}}{2}}+{a}^{2}Bx+{a}^{2}A\ln \left ( x \right ) \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,x)

[Out]

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

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Maxima [A]  time = 0.97097, size = 130, normalized size = 1.41 \begin{align*} \frac{1}{7} \, D b^{2} x^{7} + \frac{1}{6} \, C b^{2} x^{6} + \frac{1}{5} \,{\left (2 \, D a b + B b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, C a b + A b^{2}\right )} x^{4} + B a^{2} x + \frac{1}{3} \,{\left (D a^{2} + 2 \, B a b\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (C a^{2} + 2 \, A a b\right )} 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,x, algorithm="maxima")

[Out]

1/7*D*b^2*x^7 + 1/6*C*b^2*x^6 + 1/5*(2*D*a*b + B*b^2)*x^5 + 1/4*(2*C*a*b + A*b^2)*x^4 + B*a^2*x + 1/3*(D*a^2 +
2*B*a*b)*x^3 + A*a^2*log(x) + 1/2*(C*a^2 + 2*A*a*b)*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,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0.36586, size = 104, normalized size = 1.13 \begin{align*} A a^{2} \log{\left (x \right )} + B a^{2} x + \frac{C b^{2} x^{6}}{6} + \frac{D b^{2} x^{7}}{7} + x^{5} \left (\frac{B b^{2}}{5} + \frac{2 D a b}{5}\right ) + x^{4} \left (\frac{A b^{2}}{4} + \frac{C a b}{2}\right ) + x^{3} \left (\frac{2 B a b}{3} + \frac{D a^{2}}{3}\right ) + x^{2} \left (A a b + \frac{C a^{2}}{2}\right ) \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,x)

[Out]

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

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Giac [A]  time = 1.16297, size = 135, normalized size = 1.47 \begin{align*} \frac{1}{7} \, D b^{2} x^{7} + \frac{1}{6} \, C b^{2} x^{6} + \frac{2}{5} \, D a b x^{5} + \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{2} \, C a b x^{4} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{3} \, D a^{2} x^{3} + \frac{2}{3} \, B a b x^{3} + \frac{1}{2} \, C a^{2} x^{2} + A a b x^{2} + B a^{2} x + A a^{2} \log \left ({\left | x \right |}\right ) \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,x, algorithm="giac")

[Out]

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