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

Optimal. Leaf size=135 \[ -\frac{a^3 A}{2 x^2}-\frac{a^3 B}{x}+a^2 \log (x) (a C+3 A b)+a^2 x (a D+3 b B)+\frac{1}{4} b^2 x^4 (3 a C+A b)+\frac{3}{2} a b x^2 (a C+A b)+\frac{1}{5} b^2 x^5 (3 a D+b B)+a b x^3 (a D+b B)+\frac{1}{6} b^3 C x^6+\frac{1}{7} b^3 D x^7 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.284122, antiderivative size = 135, 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 \[ -\frac{a^3 A}{2 x^2}-\frac{a^3 B}{x}+a^2 \log (x) (a C+3 A b)+a^2 x (a D+3 b B)+\frac{1}{4} b^2 x^4 (3 a C+A b)+\frac{3}{2} a b x^2 (a C+A b)+\frac{1}{5} b^2 x^5 (3 a D+b B)+a b x^3 (a D+b B)+\frac{1}{6} b^3 C x^6+\frac{1}{7} b^3 D x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{3}}{2 x^{2}} - \frac{B a^{3}}{x} + \frac{C b^{3} x^{6}}{6} + \frac{D b^{3} x^{7}}{7} + a^{2} \left (3 A b + C a\right ) \log{\left (x \right )} + a b x^{3} \left (B b + D a\right ) + 3 a b \left (A b + C a\right ) \int x\, dx + \frac{b^{2} x^{5} \left (B b + 3 D a\right )}{5} + \frac{b^{2} x^{4} \left (A b + 3 C a\right )}{4} + \frac{a^{2} \left (3 B b + D a\right ) \int D\, dx}{D} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.142639, size = 124, normalized size = 0.92 \[ -\frac{a^3 \left (A+2 B x-2 D x^3\right )}{2 x^2}+a^2 \log (x) (a C+3 A b)+\frac{1}{2} a^2 b x (6 B+x (3 C+2 D x))+\frac{1}{20} a b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+\frac{1}{420} b^3 x^4 (105 A+2 x (42 B+5 x (7 C+6 D x))) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 144, normalized size = 1.1 \[{\frac{{b}^{3}D{x}^{7}}{7}}+{\frac{{b}^{3}C{x}^{6}}{6}}+{\frac{B{x}^{5}{b}^{3}}{5}}+{\frac{3\,D{x}^{5}a{b}^{2}}{5}}+{\frac{A{x}^{4}{b}^{3}}{4}}+{\frac{3\,C{x}^{4}a{b}^{2}}{4}}+B{x}^{3}a{b}^{2}+D{x}^{3}{a}^{2}b+{\frac{3\,A{x}^{2}a{b}^{2}}{2}}+{\frac{3\,C{x}^{2}{a}^{2}b}{2}}+3\,Bx{a}^{2}b+Dx{a}^{3}+3\,A\ln \left ( x \right ){a}^{2}b+C\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{2\,{x}^{2}}}-{\frac{B{a}^{3}}{x}} \]

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^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34474, size = 188, normalized size = 1.39 \[ \frac{1}{7} \, D b^{3} x^{7} + \frac{1}{6} \, C b^{3} x^{6} + \frac{1}{5} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{4} +{\left (D a^{2} b + B a b^{2}\right )} x^{3} + \frac{3}{2} \,{\left (C a^{2} b + A a b^{2}\right )} x^{2} +{\left (D a^{3} + 3 \, B a^{2} b\right )} x +{\left (C a^{3} + 3 \, A a^{2} b\right )} \log \left (x\right ) - \frac{2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.223617, size = 198, normalized size = 1.47 \[ \frac{60 \, D b^{3} x^{9} + 70 \, C b^{3} x^{8} + 84 \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{7} + 105 \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{6} + 420 \,{\left (D a^{2} b + B a b^{2}\right )} x^{5} - 420 \, B a^{3} x + 630 \,{\left (C a^{2} b + A a b^{2}\right )} x^{4} - 210 \, A a^{3} + 420 \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{3} + 420 \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2} \log \left (x\right )}{420 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^3/x^3,x, algorithm="fricas")

[Out]

1/420*(60*D*b^3*x^9 + 70*C*b^3*x^8 + 84*(3*D*a*b^2 + B*b^3)*x^7 + 105*(3*C*a*b^2
+ A*b^3)*x^6 + 420*(D*a^2*b + B*a*b^2)*x^5 - 420*B*a^3*x + 630*(C*a^2*b + A*a*b
^2)*x^4 - 210*A*a^3 + 420*(D*a^3 + 3*B*a^2*b)*x^3 + 420*(C*a^3 + 3*A*a^2*b)*x^2*
log(x))/x^2

_______________________________________________________________________________________

Sympy [A]  time = 1.19946, size = 150, normalized size = 1.11 \[ \frac{C b^{3} x^{6}}{6} + \frac{D b^{3} x^{7}}{7} + a^{2} \left (3 A b + C a\right ) \log{\left (x \right )} + x^{5} \left (\frac{B b^{3}}{5} + \frac{3 D a b^{2}}{5}\right ) + x^{4} \left (\frac{A b^{3}}{4} + \frac{3 C a b^{2}}{4}\right ) + x^{3} \left (B a b^{2} + D a^{2} b\right ) + x^{2} \left (\frac{3 A a b^{2}}{2} + \frac{3 C a^{2} b}{2}\right ) + x \left (3 B a^{2} b + D a^{3}\right ) - \frac{A a^{3} + 2 B a^{3} x}{2 x^{2}} \]

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**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220849, size = 194, normalized size = 1.44 \[ \frac{1}{7} \, D b^{3} x^{7} + \frac{1}{6} \, C b^{3} x^{6} + \frac{3}{5} \, D a b^{2} x^{5} + \frac{1}{5} \, B b^{3} x^{5} + \frac{3}{4} \, C a b^{2} x^{4} + \frac{1}{4} \, A b^{3} x^{4} + D a^{2} b x^{3} + B a b^{2} x^{3} + \frac{3}{2} \, C a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} x^{2} + D a^{3} x + 3 \, B a^{2} b x +{\left (C a^{3} + 3 \, A a^{2} b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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