3.38 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^7} \, dx\)

Optimal. Leaf size=64 \[ -\frac{A b^3}{3 x^3}-\frac{b^2 (3 A c+b B)}{2 x^2}+c^2 \log (x) (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+B c^3 x \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.109834, antiderivative size = 64, 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 \[ -\frac{A b^3}{3 x^3}-\frac{b^2 (3 A c+b B)}{2 x^2}+c^2 \log (x) (A c+3 b B)-\frac{3 b c (A c+b B)}{x}+B c^3 x \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^7,x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**7,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0606326, size = 73, normalized size = 1.14 \[ -\frac{A b^3}{3 x^3}-\frac{3 \left (A b c^2+b^2 B c\right )}{x}+\frac{b^3 (-B)-3 A b^2 c}{2 x^2}+\log (x) \left (A c^3+3 b B c^2\right )+B c^3 x \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^7,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 72, normalized size = 1.1 \[ B{c}^{3}x+A\ln \left ( x \right ){c}^{3}+3\,B\ln \left ( x \right ) b{c}^{2}-{\frac{A{b}^{3}}{3\,{x}^{3}}}-{\frac{3\,A{b}^{2}c}{2\,{x}^{2}}}-{\frac{B{b}^{3}}{2\,{x}^{2}}}-3\,{\frac{Ab{c}^{2}}{x}}-3\,{\frac{B{b}^{2}c}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^7,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.713149, size = 93, normalized size = 1.45 \[ B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )} \log \left (x\right ) - \frac{2 \, A b^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

B*c^3*x + (3*B*b*c^2 + A*c^3)*log(x) - 1/6*(2*A*b^3 + 18*(B*b^2*c + A*b*c^2)*x^2
+ 3*(B*b^3 + 3*A*b^2*c)*x)/x^3

_______________________________________________________________________________________

Fricas [A]  time = 0.269895, size = 101, normalized size = 1.58 \[ \frac{6 \, B c^{3} x^{4} + 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} \log \left (x\right ) - 2 \, A b^{3} - 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} - 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

1/6*(6*B*c^3*x^4 + 6*(3*B*b*c^2 + A*c^3)*x^3*log(x) - 2*A*b^3 - 18*(B*b^2*c + A*
b*c^2)*x^2 - 3*(B*b^3 + 3*A*b^2*c)*x)/x^3

_______________________________________________________________________________________

Sympy [A]  time = 3.33924, size = 70, normalized size = 1.09 \[ B c^{3} x + c^{2} \left (A c + 3 B b\right ) \log{\left (x \right )} - \frac{2 A b^{3} + x^{2} \left (18 A b c^{2} + 18 B b^{2} c\right ) + x \left (9 A b^{2} c + 3 B b^{3}\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**7,x)

[Out]

B*c**3*x + c**2*(A*c + 3*B*b)*log(x) - (2*A*b**3 + x**2*(18*A*b*c**2 + 18*B*b**2
*c) + x*(9*A*b**2*c + 3*B*b**3))/(6*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271745, size = 95, normalized size = 1.48 \[ B c^{3} x +{\left (3 \, B b c^{2} + A c^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A b^{3} + 18 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^7,x, algorithm="giac")

[Out]

B*c^3*x + (3*B*b*c^2 + A*c^3)*ln(abs(x)) - 1/6*(2*A*b^3 + 18*(B*b^2*c + A*b*c^2)
*x^2 + 3*(B*b^3 + 3*A*b^2*c)*x)/x^3