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

Optimal. Leaf size=98 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac{1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac{1}{4} b^2 C x^4+\frac{1}{5} b^2 D x^5 \]

[Out]

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

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Rubi [A]  time = 0.186014, antiderivative size = 98, 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 \[ -\frac{a^2 A}{2 x^2}-\frac{a^2 B}{x}+\frac{1}{2} b x^2 (2 a C+A b)+a \log (x) (a C+2 A b)+\frac{1}{3} b x^3 (2 a D+b B)+a x (a D+2 b B)+\frac{1}{4} b^2 C x^4+\frac{1}{5} b^2 D x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0850998, size = 87, normalized size = 0.89 \[ -\frac{a^2 \left (A+2 B x-2 D x^3\right )}{2 x^2}+a \log (x) (a C+2 A b)+\frac{1}{3} a b x (6 B+x (3 C+2 D x))+\frac{1}{60} b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x))) \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*(A + 2*B*x - 2*D*x^3))/(2*x^2) + (a*b*x*(6*B + x*(3*C + 2*D*x)))/3 + (b^2*
x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))))/60 + a*(2*A*b + a*C)*Log[x]

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Maple [A]  time = 0.01, size = 97, normalized size = 1. \[{\frac{{b}^{2}D{x}^{5}}{5}}+{\frac{{b}^{2}C{x}^{4}}{4}}+{\frac{B{b}^{2}{x}^{3}}{3}}+{\frac{2\,D{x}^{3}ab}{3}}+{\frac{A{x}^{2}{b}^{2}}{2}}+C{x}^{2}ab+2\,Bxab+Dx{a}^{2}+2\,A\ln \left ( x \right ) ab+C\ln \left ( x \right ){a}^{2}-{\frac{A{a}^{2}}{2\,{x}^{2}}}-{\frac{{a}^{2}B}{x}} \]

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^3,x)

[Out]

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

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Maxima [A]  time = 1.35124, size = 130, normalized size = 1.33 \[ \frac{1}{5} \, D b^{2} x^{5} + \frac{1}{4} \, C b^{2} x^{4} + \frac{1}{3} \,{\left (2 \, D a b + B b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, C a b + A b^{2}\right )} x^{2} +{\left (D a^{2} + 2 \, B a b\right )} x +{\left (C a^{2} + 2 \, A a b\right )} \log \left (x\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]

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)^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219041, size = 139, normalized size = 1.42 \[ \frac{12 \, D b^{2} x^{7} + 15 \, C b^{2} x^{6} + 20 \,{\left (2 \, D a b + B b^{2}\right )} x^{5} + 30 \,{\left (2 \, C a b + A b^{2}\right )} x^{4} - 60 \, B a^{2} x + 60 \,{\left (D a^{2} + 2 \, B a b\right )} x^{3} + 60 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2} \log \left (x\right ) - 30 \, A a^{2}}{60 \, x^{2}} \]

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)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.02945, size = 99, normalized size = 1.01 \[ \frac{C b^{2} x^{4}}{4} + \frac{D b^{2} x^{5}}{5} + a \left (2 A b + C a\right ) \log{\left (x \right )} + x^{3} \left (\frac{B b^{2}}{3} + \frac{2 D a b}{3}\right ) + x^{2} \left (\frac{A b^{2}}{2} + C a b\right ) + x \left (2 B a b + D a^{2}\right ) - \frac{A a^{2} + 2 B a^{2} x}{2 x^{2}} \]

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**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223473, size = 131, normalized size = 1.34 \[ \frac{1}{5} \, D b^{2} x^{5} + \frac{1}{4} \, C b^{2} x^{4} + \frac{2}{3} \, D a b x^{3} + \frac{1}{3} \, B b^{2} x^{3} + C a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + D a^{2} x + 2 \, B a b x +{\left (C a^{2} + 2 \, A a b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \]

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)^2/x^3,x, algorithm="giac")

[Out]

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