3.37 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^6} \, dx\)

Optimal. Leaf size=65 \[ -\frac{A b^3}{2 x^2}-\frac{b^2 (3 A c+b B)}{x}+c^2 x (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac{1}{2} B c^3 x^2 \]

[Out]

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

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Rubi [A]  time = 0.122677, antiderivative size = 65, 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 \[ -\frac{A b^3}{2 x^2}-\frac{b^2 (3 A c+b B)}{x}+c^2 x (A c+3 b B)+3 b c \log (x) (A c+b B)+\frac{1}{2} B c^3 x^2 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^6,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.0377193, size = 71, normalized size = 1.09 \[ -\frac{A b^3}{2 x^2}+3 \log (x) \left (A b c^2+b^2 B c\right )+\frac{b^3 (-B)-3 A b^2 c}{x}+c^2 x (A c+3 b B)+\frac{1}{2} B c^3 x^2 \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^6,x]

[Out]

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

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Maple [A]  time = 0.01, size = 71, normalized size = 1.1 \[{\frac{B{c}^{3}{x}^{2}}{2}}+Ax{c}^{3}+3\,Bxb{c}^{2}+3\,A\ln \left ( x \right ) b{c}^{2}+3\,B\ln \left ( x \right ){b}^{2}c-{\frac{A{b}^{3}}{2\,{x}^{2}}}-3\,{\frac{A{b}^{2}c}{x}}-{\frac{B{b}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^6,x)

[Out]

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

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Maxima [A]  time = 0.703972, size = 93, normalized size = 1.43 \[ \frac{1}{2} \, B c^{3} x^{2} +{\left (3 \, B b c^{2} + A c^{3}\right )} x + 3 \,{\left (B b^{2} c + A b c^{2}\right )} \log \left (x\right ) - \frac{A b^{3} + 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.274669, size = 100, normalized size = 1.54 \[ \frac{B c^{3} x^{4} - A b^{3} + 2 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 6 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

1/2*(B*c^3*x^4 - A*b^3 + 2*(3*B*b*c^2 + A*c^3)*x^3 + 6*(B*b^2*c + A*b*c^2)*x^2*l
og(x) - 2*(B*b^3 + 3*A*b^2*c)*x)/x^2

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Sympy [A]  time = 2.09662, size = 66, normalized size = 1.02 \[ \frac{B c^{3} x^{2}}{2} + 3 b c \left (A c + B b\right ) \log{\left (x \right )} + x \left (A c^{3} + 3 B b c^{2}\right ) - \frac{A b^{3} + x \left (6 A b^{2} c + 2 B b^{3}\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269633, size = 93, normalized size = 1.43 \[ \frac{1}{2} \, B c^{3} x^{2} + 3 \, B b c^{2} x + A c^{3} x + 3 \,{\left (B b^{2} c + A b c^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A b^{3} + 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^6,x, algorithm="giac")

[Out]

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