3.24 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^6} \, dx\)

Optimal. Leaf size=49 \[ -\frac{A b^2}{3 x^3}-\frac{b (2 A c+b B)}{2 x^2}-\frac{c (A c+2 b B)}{x}+B c^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0713169, antiderivative size = 49, 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 \[ -\frac{A b^2}{3 x^3}-\frac{b (2 A c+b B)}{2 x^2}-\frac{c (A c+2 b B)}{x}+B c^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0938, size = 44, normalized size = 0.9 \[ - \frac{A b^{2}}{3 x^{3}} + B c^{2} \log{\left (x \right )} - \frac{b \left (2 A c + B b\right )}{2 x^{2}} - \frac{c \left (A c + 2 B b\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.0533069, size = 47, normalized size = 0.96 \[ B c^2 \log (x)-\frac{2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^6,x]

[Out]

-(3*b*B*x*(b + 4*c*x) + 2*A*(b^2 + 3*b*c*x + 3*c^2*x^2))/(6*x^3) + B*c^2*Log[x]

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Maple [A]  time = 0.008, size = 52, normalized size = 1.1 \[ B{c}^{2}\ln \left ( x \right ) -{\frac{{b}^{2}A}{3\,{x}^{3}}}-{\frac{Abc}{{x}^{2}}}-{\frac{{b}^{2}B}{2\,{x}^{2}}}-{\frac{A{c}^{2}}{x}}-2\,{\frac{Bbc}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x^6,x)

[Out]

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

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Maxima [A]  time = 0.698827, size = 68, normalized size = 1.39 \[ B c^{2} \log \left (x\right ) - \frac{2 \, A b^{2} + 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

B*c^2*log(x) - 1/6*(2*A*b^2 + 6*(2*B*b*c + A*c^2)*x^2 + 3*(B*b^2 + 2*A*b*c)*x)/x
^3

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Fricas [A]  time = 0.27179, size = 72, normalized size = 1.47 \[ \frac{6 \, B c^{2} x^{3} \log \left (x\right ) - 2 \, A b^{2} - 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \,{\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

1/6*(6*B*c^2*x^3*log(x) - 2*A*b^2 - 6*(2*B*b*c + A*c^2)*x^2 - 3*(B*b^2 + 2*A*b*c
)*x)/x^3

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Sympy [A]  time = 2.74243, size = 51, normalized size = 1.04 \[ B c^{2} \log{\left (x \right )} - \frac{2 A b^{2} + x^{2} \left (6 A c^{2} + 12 B b c\right ) + x \left (6 A b c + 3 B b^{2}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/x**6,x)

[Out]

B*c**2*log(x) - (2*A*b**2 + x**2*(6*A*c**2 + 12*B*b*c) + x*(6*A*b*c + 3*B*b**2))
/(6*x**3)

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GIAC/XCAS [A]  time = 0.267533, size = 69, normalized size = 1.41 \[ B c^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A b^{2} + 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^6,x, algorithm="giac")

[Out]

B*c^2*ln(abs(x)) - 1/6*(2*A*b^2 + 6*(2*B*b*c + A*c^2)*x^2 + 3*(B*b^2 + 2*A*b*c)*
x)/x^3