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

Optimal. Leaf size=54 \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]

[Out]

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

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Rubi [A]  time = 0.0990235, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \log (x) (a C+A b)-\frac{a A}{2 x^2}+x (a D+b B)-\frac{a B}{x}+\frac{1}{2} b C x^2+\frac{1}{3} b D x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0524996, size = 51, normalized size = 0.94 \[ \log (x) (a C+A b)-\frac{a \left (A+2 B x-2 D x^3\right )}{2 x^2}+\frac{1}{6} b x \left (6 B+3 C x+2 D x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 48, normalized size = 0.9 \[{\frac{bD{x}^{3}}{3}}+{\frac{bC{x}^{2}}{2}}+bBx+Dxa+A\ln \left ( x \right ) b+C\ln \left ( x \right ) a-{\frac{Aa}{2\,{x}^{2}}}-{\frac{Ba}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34612, size = 65, normalized size = 1.2 \[ \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} +{\left (D a + B b\right )} x +{\left (C a + A b\right )} \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \]

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)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.263849, size = 74, normalized size = 1.37 \[ \frac{2 \, D b x^{5} + 3 \, C b x^{4} + 6 \,{\left (D a + B b\right )} x^{3} + 6 \,{\left (C a + A b\right )} x^{2} \log \left (x\right ) - 6 \, B a x - 3 \, A a}{6 \, x^{2}} \]

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)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.850776, size = 49, normalized size = 0.91 \[ \frac{C b x^{2}}{2} + \frac{D b x^{3}}{3} + x \left (B b + D a\right ) + \left (A b + C a\right ) \log{\left (x \right )} - \frac{A a + 2 B a x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214842, size = 65, normalized size = 1.2 \[ \frac{1}{3} \, D b x^{3} + \frac{1}{2} \, C b x^{2} + D a x + B b x +{\left (C a + A b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \]

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)/x^3,x, algorithm="giac")

[Out]

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