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

Optimal. Leaf size=56 \[ \frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5 \]

[Out]

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

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Rubi [A]  time = 0.080597, antiderivative size = 56, 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 \[ \frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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

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,x)

[Out]

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

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Mathematica [A]  time = 0.0298128, size = 56, normalized size = 1. \[ \frac{1}{2} x^2 (a C+A b)+a A \log (x)+\frac{1}{3} x^3 (a D+b B)+a B x+\frac{1}{4} b C x^4+\frac{1}{5} b D x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 53, normalized size = 1. \[{\frac{bD{x}^{5}}{5}}+{\frac{bC{x}^{4}}{4}}+{\frac{bB{x}^{3}}{3}}+{\frac{D{x}^{3}a}{3}}+{\frac{A{x}^{2}b}{2}}+{\frac{C{x}^{2}a}{2}}+Bxa+aA\ln \left ( x \right ) \]

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,x)

[Out]

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

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Maxima [A]  time = 1.34488, size = 65, normalized size = 1.16 \[ \frac{1}{5} \, D b x^{5} + \frac{1}{4} \, C b x^{4} + \frac{1}{3} \,{\left (D a + B b\right )} x^{3} + B a x + \frac{1}{2} \,{\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \]

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.246517, size = 65, normalized size = 1.16 \[ \frac{1}{5} \, D b x^{5} + \frac{1}{4} \, C b x^{4} + \frac{1}{3} \,{\left (D a + B b\right )} x^{3} + B a x + \frac{1}{2} \,{\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \]

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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.596622, size = 54, normalized size = 0.96 \[ A a \log{\left (x \right )} + B a x + \frac{C b x^{4}}{4} + \frac{D b x^{5}}{5} + x^{3} \left (\frac{B b}{3} + \frac{D a}{3}\right ) + x^{2} \left (\frac{A b}{2} + \frac{C a}{2}\right ) \]

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,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.225917, size = 72, normalized size = 1.29 \[ \frac{1}{5} \, D b x^{5} + \frac{1}{4} \, C b x^{4} + \frac{1}{3} \, D a x^{3} + \frac{1}{3} \, B b x^{3} + \frac{1}{2} \, C a x^{2} + \frac{1}{2} \, A b x^{2} + B a x + A a{\rm ln}\left ({\left | x \right |}\right ) \]

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,x, algorithm="giac")

[Out]

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