3.85 \(\int \frac{(a+b x^2)^3 (A+B x+C x^2+D x^3)}{x^4} \, dx\)

Optimal. Leaf size=139 \[ -\frac{a^2 (a C+3 A b)}{x}-\frac{a^3 A}{3 x^3}+a^2 \log (x) (a D+3 b B)-\frac{a^3 B}{2 x^2}+\frac{1}{3} b^2 x^3 (3 a C+A b)+3 a b x (a C+A b)+\frac{1}{4} b^2 x^4 (3 a D+b B)+\frac{3}{2} a b x^2 (a D+b B)+\frac{1}{5} b^3 C x^5+\frac{1}{6} b^3 D x^6 \]

[Out]

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

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Rubi [A]  time = 0.113901, antiderivative size = 139, 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} \[ -\frac{a^2 (a C+3 A b)}{x}-\frac{a^3 A}{3 x^3}+a^2 \log (x) (a D+3 b B)-\frac{a^3 B}{2 x^2}+\frac{1}{3} b^2 x^3 (3 a C+A b)+3 a b x (a C+A b)+\frac{1}{4} b^2 x^4 (3 a D+b B)+\frac{3}{2} a b x^2 (a D+b B)+\frac{1}{5} b^3 C x^5+\frac{1}{6} b^3 D x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0483706, size = 124, normalized size = 0.89 \[ \frac{3 a^2 b \left (x^2 (2 C+D x)-2 A\right )}{2 x}-\frac{a^3 (2 A+3 x (B+2 C x))}{6 x^3}+a^2 \log (x) (a D+3 b B)+\frac{1}{4} a b^2 x (12 A+x (6 B+x (4 C+3 D x)))+\frac{1}{60} b^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x))) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*(2*A + 3*x*(B + 2*C*x)))/(6*x^3) + (3*a^2*b*(-2*A + x^2*(2*C + D*x)))/(2*x) + (a*b^2*x*(12*A + x*(6*B +
x*(4*C + 3*D*x))))/4 + (b^3*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))))/60 + a^2*(3*b*B + a*D)*Log[x]

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Maple [A]  time = 0.008, size = 146, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}D{x}^{6}}{6}}+{\frac{{b}^{3}C{x}^{5}}{5}}+{\frac{B{x}^{4}{b}^{3}}{4}}+{\frac{3\,D{x}^{4}a{b}^{2}}{4}}+{\frac{A{x}^{3}{b}^{3}}{3}}+C{x}^{3}a{b}^{2}+{\frac{3\,B{x}^{2}a{b}^{2}}{2}}+{\frac{3\,D{x}^{2}{a}^{2}b}{2}}+3\,Axa{b}^{2}+3\,{a}^{2}bCx+3\,B\ln \left ( x \right ){a}^{2}b+D\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{3\,{x}^{3}}}-{\frac{B{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{A{a}^{2}b}{x}}-{\frac{{a}^{3}C}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3*(D*x^3+C*x^2+B*x+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.00121, size = 192, normalized size = 1.38 \begin{align*} \frac{1}{6} \, D b^{3} x^{6} + \frac{1}{5} \, C b^{3} x^{5} + \frac{1}{4} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (D a^{2} b + B a b^{2}\right )} x^{2} + 3 \,{\left (C a^{2} b + A a b^{2}\right )} x +{\left (D a^{3} + 3 \, B a^{2} b\right )} \log \left (x\right ) - \frac{3 \, B a^{3} x + 2 \, A a^{3} + 6 \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(D*x^3+C*x^2+B*x+A)/x^4,x, algorithm="maxima")

[Out]

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

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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)^3*(D*x^3+C*x^2+B*x+A)/x^4,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 1.06086, size = 153, normalized size = 1.1 \begin{align*} \frac{C b^{3} x^{5}}{5} + \frac{D b^{3} x^{6}}{6} + a^{2} \left (3 B b + D a\right ) \log{\left (x \right )} + x^{4} \left (\frac{B b^{3}}{4} + \frac{3 D a b^{2}}{4}\right ) + x^{3} \left (\frac{A b^{3}}{3} + C a b^{2}\right ) + x^{2} \left (\frac{3 B a b^{2}}{2} + \frac{3 D a^{2} b}{2}\right ) + x \left (3 A a b^{2} + 3 C a^{2} b\right ) - \frac{2 A a^{3} + 3 B a^{3} x + x^{2} \left (18 A a^{2} b + 6 C a^{3}\right )}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3*(D*x**3+C*x**2+B*x+A)/x**4,x)

[Out]

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

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Giac [A]  time = 1.16839, size = 197, normalized size = 1.42 \begin{align*} \frac{1}{6} \, D b^{3} x^{6} + \frac{1}{5} \, C b^{3} x^{5} + \frac{3}{4} \, D a b^{2} x^{4} + \frac{1}{4} \, B b^{3} x^{4} + C a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + \frac{3}{2} \, D a^{2} b x^{2} + \frac{3}{2} \, B a b^{2} x^{2} + 3 \, C a^{2} b x + 3 \, A a b^{2} x +{\left (D a^{3} + 3 \, B a^{2} b\right )} \log \left ({\left | x \right |}\right ) - \frac{3 \, B a^{3} x + 2 \, A a^{3} + 6 \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(D*x^3+C*x^2+B*x+A)/x^4,x, algorithm="giac")

[Out]

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