3.22 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^4} \, dx\)

Optimal. Leaf size=44 \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]

[Out]

-((A*b^2)/x) + c*(2*b*B + A*c)*x + (B*c^2*x^2)/2 + b*(b*B + 2*A*c)*Log[x]

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Rubi [A]  time = 0.0889057, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]

[Out]

-((A*b^2)/x) + c*(2*b*B + A*c)*x + (B*c^2*x^2)/2 + b*(b*B + 2*A*c)*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**4,x)

[Out]

-A*b**2/x + B*c**2*Integral(x, x) + b*(2*A*c + B*b)*log(x) + c*(A*c + 2*B*b)*Int
egral(A, x)/A

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Mathematica [A]  time = 0.0629413, size = 43, normalized size = 0.98 \[ A \left (c^2 x-\frac{b^2}{x}\right )+b \log (x) (2 A c+b B)+\frac{1}{2} B c x (4 b+c x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]

[Out]

(B*c*x*(4*b + c*x))/2 + A*(-(b^2/x) + c^2*x) + b*(b*B + 2*A*c)*Log[x]

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Maple [A]  time = 0.009, size = 46, normalized size = 1.1 \[{\frac{B{c}^{2}{x}^{2}}{2}}+Ax{c}^{2}+2\,Bxbc+2\,A\ln \left ( x \right ) bc+{b}^{2}B\ln \left ( x \right ) -{\frac{{b}^{2}A}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x^4,x)

[Out]

1/2*B*c^2*x^2+A*x*c^2+2*B*x*b*c+2*A*ln(x)*b*c+b^2*B*ln(x)-A*b^2/x

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Maxima [A]  time = 0.692852, size = 62, normalized size = 1.41 \[ \frac{1}{2} \, B c^{2} x^{2} - \frac{A b^{2}}{x} +{\left (2 \, B b c + A c^{2}\right )} x +{\left (B b^{2} + 2 \, A b c\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

1/2*B*c^2*x^2 - A*b^2/x + (2*B*b*c + A*c^2)*x + (B*b^2 + 2*A*b*c)*log(x)

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Fricas [A]  time = 0.270737, size = 70, normalized size = 1.59 \[ \frac{B c^{2} x^{3} - 2 \, A b^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x \log \left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

1/2*(B*c^2*x^3 - 2*A*b^2 + 2*(2*B*b*c + A*c^2)*x^2 + 2*(B*b^2 + 2*A*b*c)*x*log(x
))/x

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Sympy [A]  time = 1.40745, size = 42, normalized size = 0.95 \[ - \frac{A b^{2}}{x} + \frac{B c^{2} x^{2}}{2} + b \left (2 A c + B b\right ) \log{\left (x \right )} + x \left (A c^{2} + 2 B b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/x**4,x)

[Out]

-A*b**2/x + B*c**2*x**2/2 + b*(2*A*c + B*b)*log(x) + x*(A*c**2 + 2*B*b*c)

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GIAC/XCAS [A]  time = 0.268348, size = 62, normalized size = 1.41 \[ \frac{1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x - \frac{A b^{2}}{x} +{\left (B b^{2} + 2 \, A b c\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^4,x, algorithm="giac")

[Out]

1/2*B*c^2*x^2 + 2*B*b*c*x + A*c^2*x - A*b^2/x + (B*b^2 + 2*A*b*c)*ln(abs(x))