3.528 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x} \, dx\)

Optimal. Leaf size=66 \[ a^4 A \log (x)+4 a^3 A b x+3 a^2 A b^2 x^2+\frac{4}{3} a A b^3 x^3+\frac{B (a+b x)^5}{5 b}+\frac{1}{4} A b^4 x^4 \]

[Out]

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

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Rubi [A]  time = 0.0611273, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ a^4 A \log (x)+4 a^3 A b x+3 a^2 A b^2 x^2+\frac{4}{3} a A b^3 x^3+\frac{B (a+b x)^5}{5 b}+\frac{1}{4} A b^4 x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0425827, size = 83, normalized size = 1.26 \[ a^4 A \log (x)+a^4 B x+2 a^3 b x (2 A+B x)+a^2 b^2 x^2 (3 A+2 B x)+\frac{1}{3} a b^3 x^3 (4 A+3 B x)+\frac{1}{20} b^4 x^4 (5 A+4 B x) \]

Antiderivative was successfully verified.

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

[Out]

a^4*B*x + 2*a^3*b*x*(2*A + B*x) + a^2*b^2*x^2*(3*A + 2*B*x) + (a*b^3*x^3*(4*A +
3*B*x))/3 + (b^4*x^4*(5*A + 4*B*x))/20 + a^4*A*Log[x]

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Maple [A]  time = 0.005, size = 94, normalized size = 1.4 \[{\frac{{b}^{4}B{x}^{5}}{5}}+{\frac{A{b}^{4}{x}^{4}}{4}}+B{x}^{4}a{b}^{3}+{\frac{4\,aA{b}^{3}{x}^{3}}{3}}+2\,B{x}^{3}{a}^{2}{b}^{2}+3\,{a}^{2}A{b}^{2}{x}^{2}+2\,B{x}^{2}{a}^{3}b+4\,{a}^{3}Abx+{a}^{4}Bx+{a}^{4}A\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.678168, size = 126, normalized size = 1.91 \[ \frac{1}{5} \, B b^{4} x^{5} + A a^{4} \log \left (x\right ) + \frac{1}{4} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.301912, size = 126, normalized size = 1.91 \[ \frac{1}{5} \, B b^{4} x^{5} + A a^{4} \log \left (x\right ) + \frac{1}{4} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.4388, size = 95, normalized size = 1.44 \[ A a^{4} \log{\left (x \right )} + \frac{B b^{4} x^{5}}{5} + x^{4} \left (\frac{A b^{4}}{4} + B a b^{3}\right ) + x^{3} \left (\frac{4 A a b^{3}}{3} + 2 B a^{2} b^{2}\right ) + x^{2} \left (3 A a^{2} b^{2} + 2 B a^{3} b\right ) + x \left (4 A a^{3} b + B a^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272813, size = 127, normalized size = 1.92 \[ \frac{1}{5} \, B b^{4} x^{5} + B a b^{3} x^{4} + \frac{1}{4} \, A b^{4} x^{4} + 2 \, B a^{2} b^{2} x^{3} + \frac{4}{3} \, A a b^{3} x^{3} + 2 \, B a^{3} b x^{2} + 3 \, A a^{2} b^{2} x^{2} + B a^{4} x + 4 \, A a^{3} b x + A a^{4}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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