3.41 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^{10}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{A b^3}{6 x^6}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{3 x^3}-\frac{3 b c (A c+b B)}{4 x^4}-\frac{B c^3}{2 x^2} \]

[Out]

-(A*b^3)/(6*x^6) - (b^2*(b*B + 3*A*c))/(5*x^5) - (3*b*c*(b*B + A*c))/(4*x^4) - (
c^2*(3*b*B + A*c))/(3*x^3) - (B*c^3)/(2*x^2)

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Rubi [A]  time = 0.117448, antiderivative size = 75, 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 \[ -\frac{A b^3}{6 x^6}-\frac{b^2 (3 A c+b B)}{5 x^5}-\frac{c^2 (A c+3 b B)}{3 x^3}-\frac{3 b c (A c+b B)}{4 x^4}-\frac{B c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^10,x]

[Out]

-(A*b^3)/(6*x^6) - (b^2*(b*B + 3*A*c))/(5*x^5) - (3*b*c*(b*B + A*c))/(4*x^4) - (
c^2*(3*b*B + A*c))/(3*x^3) - (B*c^3)/(2*x^2)

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Rubi in Sympy [A]  time = 15.2703, size = 71, normalized size = 0.95 \[ - \frac{A b^{3}}{6 x^{6}} - \frac{B c^{3}}{2 x^{2}} - \frac{b^{2} \left (3 A c + B b\right )}{5 x^{5}} - \frac{3 b c \left (A c + B b\right )}{4 x^{4}} - \frac{c^{2} \left (A c + 3 B b\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**10,x)

[Out]

-A*b**3/(6*x**6) - B*c**3/(2*x**2) - b**2*(3*A*c + B*b)/(5*x**5) - 3*b*c*(A*c +
B*b)/(4*x**4) - c**2*(A*c + 3*B*b)/(3*x**3)

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Mathematica [A]  time = 0.0403953, size = 74, normalized size = 0.99 \[ -\frac{A \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )+3 B x \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )}{60 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^10,x]

[Out]

-(3*B*x*(4*b^3 + 15*b^2*c*x + 20*b*c^2*x^2 + 10*c^3*x^3) + A*(10*b^3 + 36*b^2*c*
x + 45*b*c^2*x^2 + 20*c^3*x^3))/(60*x^6)

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Maple [A]  time = 0.009, size = 66, normalized size = 0.9 \[ -{\frac{A{b}^{3}}{6\,{x}^{6}}}-{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{5\,{x}^{5}}}-{\frac{3\,bc \left ( Ac+Bb \right ) }{4\,{x}^{4}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^10,x)

[Out]

-1/6*A*b^3/x^6-1/5*b^2*(3*A*c+B*b)/x^5-3/4*b*c*(A*c+B*b)/x^4-1/3*c^2*(A*c+3*B*b)
/x^3-1/2*B*c^3/x^2

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Maxima [A]  time = 0.719429, size = 99, normalized size = 1.32 \[ -\frac{30 \, B c^{3} x^{4} + 10 \, A b^{3} + 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 12 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^10,x, algorithm="maxima")

[Out]

-1/60*(30*B*c^3*x^4 + 10*A*b^3 + 20*(3*B*b*c^2 + A*c^3)*x^3 + 45*(B*b^2*c + A*b*
c^2)*x^2 + 12*(B*b^3 + 3*A*b^2*c)*x)/x^6

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Fricas [A]  time = 0.294938, size = 99, normalized size = 1.32 \[ -\frac{30 \, B c^{3} x^{4} + 10 \, A b^{3} + 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 12 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^10,x, algorithm="fricas")

[Out]

-1/60*(30*B*c^3*x^4 + 10*A*b^3 + 20*(3*B*b*c^2 + A*c^3)*x^3 + 45*(B*b^2*c + A*b*
c^2)*x^2 + 12*(B*b^3 + 3*A*b^2*c)*x)/x^6

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Sympy [A]  time = 6.99709, size = 78, normalized size = 1.04 \[ - \frac{10 A b^{3} + 30 B c^{3} x^{4} + x^{3} \left (20 A c^{3} + 60 B b c^{2}\right ) + x^{2} \left (45 A b c^{2} + 45 B b^{2} c\right ) + x \left (36 A b^{2} c + 12 B b^{3}\right )}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**10,x)

[Out]

-(10*A*b**3 + 30*B*c**3*x**4 + x**3*(20*A*c**3 + 60*B*b*c**2) + x**2*(45*A*b*c**
2 + 45*B*b**2*c) + x*(36*A*b**2*c + 12*B*b**3))/(60*x**6)

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GIAC/XCAS [A]  time = 0.266842, size = 101, normalized size = 1.35 \[ -\frac{30 \, B c^{3} x^{4} + 60 \, B b c^{2} x^{3} + 20 \, A c^{3} x^{3} + 45 \, B b^{2} c x^{2} + 45 \, A b c^{2} x^{2} + 12 \, B b^{3} x + 36 \, A b^{2} c x + 10 \, A b^{3}}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^10,x, algorithm="giac")

[Out]

-1/60*(30*B*c^3*x^4 + 60*B*b*c^2*x^3 + 20*A*c^3*x^3 + 45*B*b^2*c*x^2 + 45*A*b*c^
2*x^2 + 12*B*b^3*x + 36*A*b^2*c*x + 10*A*b^3)/x^6