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

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

[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 = 0.284305, 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 \[ -\frac{a^3 A}{3 x^3}-\frac{a^3 B}{2 x^2}-\frac{a^2 (a C+3 A b)}{x}+a^2 \log (x) (a D+3 b B)+\frac{1}{3} b^2 x^3 (3 a C+A b)+3 a b x (a C+A b)+\frac{1}{4} b^2 x^4 (3 a D+b B)+\frac{3}{2} a b x^2 (a D+b B)+\frac{1}{5} b^3 C x^5+\frac{1}{6} b^3 D x^6 \]

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]

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

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**4,x)

[Out]

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

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Mathematica [A]  time = 0.138119, size = 124, normalized size = 0.89 \[ -\frac{a^3 (2 A+3 x (B+2 C x))}{6 x^3}+\frac{3 a^2 b \left (x^2 (2 C+D x)-2 A\right )}{2 x}+a^2 \log (x) (a D+3 b B)+\frac{1}{4} a b^2 x (12 A+x (6 B+x (4 C+3 D x)))+\frac{1}{60} b^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x))) \]

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 = 0.011, size = 146, normalized size = 1.1 \[{\frac{{b}^{3}D{x}^{6}}{6}}+{\frac{{b}^{3}C{x}^{5}}{5}}+{\frac{B{x}^{4}{b}^{3}}{4}}+{\frac{3\,D{x}^{4}a{b}^{2}}{4}}+{\frac{A{x}^{3}{b}^{3}}{3}}+C{x}^{3}a{b}^{2}+{\frac{3\,B{x}^{2}a{b}^{2}}{2}}+{\frac{3\,D{x}^{2}{a}^{2}b}{2}}+3\,Axa{b}^{2}+3\,Cx{a}^{2}b+3\,B\ln \left ( x \right ){a}^{2}b+D\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{3\,{x}^{3}}}-{\frac{B{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{A{a}^{2}b}{x}}-{\frac{{a}^{3}C}{x}} \]

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*C*x*a^2*b+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 = 1.36271, size = 192, normalized size = 1.38 \[ \frac{1}{6} \, D b^{3} x^{6} + \frac{1}{5} \, C b^{3} x^{5} + \frac{1}{4} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (D a^{2} b + B a b^{2}\right )} x^{2} + 3 \,{\left (C a^{2} b + A a b^{2}\right )} x +{\left (D a^{3} + 3 \, B a^{2} b\right )} \log \left (x\right ) - \frac{3 \, B a^{3} x + 2 \, A a^{3} + 6 \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{6 \, x^{3}} \]

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^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 [A]  time = 0.228882, size = 198, normalized size = 1.42 \[ \frac{10 \, D b^{3} x^{9} + 12 \, C b^{3} x^{8} + 15 \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{7} + 20 \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{6} + 90 \,{\left (D a^{2} b + B a b^{2}\right )} x^{5} - 30 \, B a^{3} x + 180 \,{\left (C a^{2} b + A a b^{2}\right )} x^{4} + 60 \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{3} \log \left (x\right ) - 20 \, A a^{3} - 60 \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{60 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

1/60*(10*D*b^3*x^9 + 12*C*b^3*x^8 + 15*(3*D*a*b^2 + B*b^3)*x^7 + 20*(3*C*a*b^2 +
A*b^3)*x^6 + 90*(D*a^2*b + B*a*b^2)*x^5 - 30*B*a^3*x + 180*(C*a^2*b + A*a*b^2)*
x^4 + 60*(D*a^3 + 3*B*a^2*b)*x^3*log(x) - 20*A*a^3 - 60*(C*a^3 + 3*A*a^2*b)*x^2)
/x^3

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Sympy [A]  time = 2.15736, size = 153, normalized size = 1.1 \[ \frac{C b^{3} x^{5}}{5} + \frac{D b^{3} x^{6}}{6} + a^{2} \left (3 B b + D a\right ) \log{\left (x \right )} + x^{4} \left (\frac{B b^{3}}{4} + \frac{3 D a b^{2}}{4}\right ) + x^{3} \left (\frac{A b^{3}}{3} + C a b^{2}\right ) + x^{2} \left (\frac{3 B a b^{2}}{2} + \frac{3 D a^{2} b}{2}\right ) + x \left (3 A a b^{2} + 3 C a^{2} b\right ) - \frac{2 A a^{3} + 3 B a^{3} x + x^{2} \left (18 A a^{2} b + 6 C a^{3}\right )}{6 x^{3}} \]

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/XCAS [A]  time = 0.223223, size = 197, normalized size = 1.42 \[ \frac{1}{6} \, D b^{3} x^{6} + \frac{1}{5} \, C b^{3} x^{5} + \frac{3}{4} \, D a b^{2} x^{4} + \frac{1}{4} \, B b^{3} x^{4} + C a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + \frac{3}{2} \, D a^{2} b x^{2} + \frac{3}{2} \, B a b^{2} x^{2} + 3 \, C a^{2} b x + 3 \, A a b^{2} x +{\left (D a^{3} + 3 \, B a^{2} b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, B a^{3} x + 2 \, A a^{3} + 6 \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{6 \, x^{3}} \]

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^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)*ln(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