3.98 \(\int \frac{\left (1+(a+b x)^2\right )^2}{x} \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{2} \left (a^2+2\right ) (a+b x)^2+a \left (a^2+2\right ) b x+\left (a^2+1\right )^2 \log (x)+\frac{1}{4} (a+b x)^4+\frac{1}{3} a (a+b x)^3 \]

[Out]

a*(2 + a^2)*b*x + ((2 + a^2)*(a + b*x)^2)/2 + (a*(a + b*x)^3)/3 + (a + b*x)^4/4
+ (1 + a^2)^2*Log[x]

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Rubi [A]  time = 0.115344, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{2} \left (a^2+2\right ) (a+b x)^2+a \left (a^2+2\right ) b x+\left (a^2+1\right )^2 \log (x)+\frac{1}{4} (a+b x)^4+\frac{1}{3} a (a+b x)^3 \]

Antiderivative was successfully verified.

[In]  Int[(1 + (a + b*x)^2)^2/x,x]

[Out]

a*(2 + a^2)*b*x + ((2 + a^2)*(a + b*x)^2)/2 + (a*(a + b*x)^3)/3 + (a + b*x)^4/4
+ (1 + a^2)^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a \left (a + b x\right )^{3}}{3} + \frac{\left (a + b x\right )^{4}}{4} + \left (a^{2} + 1\right )^{2} \log{\left (- b x \right )} + \left (a^{2} + 2\right ) \int ^{a + b x} x\, dx + \frac{\left (a^{2} + 2\right ) \int ^{a + b x} a^{3}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+(b*x+a)**2)**2/x,x)

[Out]

a*(a + b*x)**3/3 + (a + b*x)**4/4 + (a**2 + 1)**2*log(-b*x) + (a**2 + 2)*Integra
l(x, (x, a + b*x)) + (a**2 + 2)*Integral(a**3, (x, a + b*x))/a**2

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Mathematica [A]  time = 0.0374015, size = 64, normalized size = 1.08 \[ \frac{1}{2} \left (a^2+2\right ) (a+b x)^2+a \left (a^2+2\right ) (a+b x)+\left (a^2+1\right )^2 \log (b x)+\frac{1}{4} (a+b x)^4+\frac{1}{3} a (a+b x)^3 \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + (a + b*x)^2)^2/x,x]

[Out]

a*(2 + a^2)*(a + b*x) + ((2 + a^2)*(a + b*x)^2)/2 + (a*(a + b*x)^3)/3 + (a + b*x
)^4/4 + (1 + a^2)^2*Log[b*x]

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Maple [A]  time = 0.002, size = 64, normalized size = 1.1 \[{\frac{{b}^{4}{x}^{4}}{4}}+{\frac{4\,a{b}^{3}{x}^{3}}{3}}+3\,{a}^{2}{b}^{2}{x}^{2}+4\,{a}^{3}bx+{b}^{2}{x}^{2}+4\,abx+{a}^{4}\ln \left ( x \right ) +2\,{a}^{2}\ln \left ( x \right ) +\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+(b*x+a)^2)^2/x,x)

[Out]

1/4*b^4*x^4+4/3*a*b^3*x^3+3*a^2*b^2*x^2+4*a^3*b*x+b^2*x^2+4*a*b*x+a^4*ln(x)+2*a^
2*ln(x)+ln(x)

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Maxima [A]  time = 0.797313, size = 73, normalized size = 1.24 \[ \frac{1}{4} \, b^{4} x^{4} + \frac{4}{3} \, a b^{3} x^{3} +{\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + 4 \,{\left (a^{3} + a\right )} b x +{\left (a^{4} + 2 \, a^{2} + 1\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^2 + 1)^2/x,x, algorithm="maxima")

[Out]

1/4*b^4*x^4 + 4/3*a*b^3*x^3 + (3*a^2 + 1)*b^2*x^2 + 4*(a^3 + a)*b*x + (a^4 + 2*a
^2 + 1)*log(x)

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Fricas [A]  time = 0.271526, size = 73, normalized size = 1.24 \[ \frac{1}{4} \, b^{4} x^{4} + \frac{4}{3} \, a b^{3} x^{3} +{\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + 4 \,{\left (a^{3} + a\right )} b x +{\left (a^{4} + 2 \, a^{2} + 1\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^2 + 1)^2/x,x, algorithm="fricas")

[Out]

1/4*b^4*x^4 + 4/3*a*b^3*x^3 + (3*a^2 + 1)*b^2*x^2 + 4*(a^3 + a)*b*x + (a^4 + 2*a
^2 + 1)*log(x)

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Sympy [A]  time = 1.27172, size = 58, normalized size = 0.98 \[ \frac{4 a b^{3} x^{3}}{3} + \frac{b^{4} x^{4}}{4} + x^{2} \left (3 a^{2} b^{2} + b^{2}\right ) + x \left (4 a^{3} b + 4 a b\right ) + \left (a^{2} + 1\right )^{2} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+(b*x+a)**2)**2/x,x)

[Out]

4*a*b**3*x**3/3 + b**4*x**4/4 + x**2*(3*a**2*b**2 + b**2) + x*(4*a**3*b + 4*a*b)
+ (a**2 + 1)**2*log(x)

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GIAC/XCAS [A]  time = 0.262096, size = 84, normalized size = 1.42 \[ \frac{1}{4} \, b^{4} x^{4} + \frac{4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + b^{2} x^{2} + 4 \, a b x +{\left (a^{4} + 2 \, a^{2} + 1\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^2 + 1)^2/x,x, algorithm="giac")

[Out]

1/4*b^4*x^4 + 4/3*a*b^3*x^3 + 3*a^2*b^2*x^2 + 4*a^3*b*x + b^2*x^2 + 4*a*b*x + (a
^4 + 2*a^2 + 1)*ln(abs(x))