3.42 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^{11}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{6 x^6}-\frac{c^2 (A c+3 b B)}{4 x^4}-\frac{3 b c (A c+b B)}{5 x^5}-\frac{B c^3}{3 x^3} \]

[Out]

-(A*b^3)/(7*x^7) - (b^2*(b*B + 3*A*c))/(6*x^6) - (3*b*c*(b*B + A*c))/(5*x^5) - (
c^2*(3*b*B + A*c))/(4*x^4) - (B*c^3)/(3*x^3)

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Rubi [A]  time = 0.109512, 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 \[ -\frac{A b^3}{7 x^7}-\frac{b^2 (3 A c+b B)}{6 x^6}-\frac{c^2 (A c+3 b B)}{4 x^4}-\frac{3 b c (A c+b B)}{5 x^5}-\frac{B c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^11,x]

[Out]

-(A*b^3)/(7*x^7) - (b^2*(b*B + 3*A*c))/(6*x^6) - (3*b*c*(b*B + A*c))/(5*x^5) - (
c^2*(3*b*B + A*c))/(4*x^4) - (B*c^3)/(3*x^3)

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Rubi in Sympy [A]  time = 15.1682, size = 71, normalized size = 0.95 \[ - \frac{A b^{3}}{7 x^{7}} - \frac{B c^{3}}{3 x^{3}} - \frac{b^{2} \left (3 A c + B b\right )}{6 x^{6}} - \frac{3 b c \left (A c + B b\right )}{5 x^{5}} - \frac{c^{2} \left (A c + 3 B b\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**11,x)

[Out]

-A*b**3/(7*x**7) - B*c**3/(3*x**3) - b**2*(3*A*c + B*b)/(6*x**6) - 3*b*c*(A*c +
B*b)/(5*x**5) - c**2*(A*c + 3*B*b)/(4*x**4)

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Mathematica [A]  time = 0.0395493, size = 75, normalized size = 1. \[ -\frac{3 A \left (20 b^3+70 b^2 c x+84 b c^2 x^2+35 c^3 x^3\right )+7 B x \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )}{420 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^11,x]

[Out]

-(7*B*x*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 + 20*c^3*x^3) + 3*A*(20*b^3 + 70*b^2
*c*x + 84*b*c^2*x^2 + 35*c^3*x^3))/(420*x^7)

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Maple [A]  time = 0.006, size = 66, normalized size = 0.9 \[ -{\frac{A{b}^{3}}{7\,{x}^{7}}}-{\frac{{b}^{2} \left ( 3\,Ac+Bb \right ) }{6\,{x}^{6}}}-{\frac{3\,bc \left ( Ac+Bb \right ) }{5\,{x}^{5}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{4\,{x}^{4}}}-{\frac{B{c}^{3}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^11,x)

[Out]

-1/7*A*b^3/x^7-1/6*b^2*(3*A*c+B*b)/x^6-3/5*b*c*(A*c+B*b)/x^5-1/4*c^2*(A*c+3*B*b)
/x^4-1/3*B*c^3/x^3

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Maxima [A]  time = 0.703882, size = 99, normalized size = 1.32 \[ -\frac{140 \, B c^{3} x^{4} + 60 \, A b^{3} + 105 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 252 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 70 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{420 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^11,x, algorithm="maxima")

[Out]

-1/420*(140*B*c^3*x^4 + 60*A*b^3 + 105*(3*B*b*c^2 + A*c^3)*x^3 + 252*(B*b^2*c +
A*b*c^2)*x^2 + 70*(B*b^3 + 3*A*b^2*c)*x)/x^7

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Fricas [A]  time = 0.279149, size = 99, normalized size = 1.32 \[ -\frac{140 \, B c^{3} x^{4} + 60 \, A b^{3} + 105 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 252 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 70 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x}{420 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^11,x, algorithm="fricas")

[Out]

-1/420*(140*B*c^3*x^4 + 60*A*b^3 + 105*(3*B*b*c^2 + A*c^3)*x^3 + 252*(B*b^2*c +
A*b*c^2)*x^2 + 70*(B*b^3 + 3*A*b^2*c)*x)/x^7

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Sympy [A]  time = 8.61083, size = 78, normalized size = 1.04 \[ - \frac{60 A b^{3} + 140 B c^{3} x^{4} + x^{3} \left (105 A c^{3} + 315 B b c^{2}\right ) + x^{2} \left (252 A b c^{2} + 252 B b^{2} c\right ) + x \left (210 A b^{2} c + 70 B b^{3}\right )}{420 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**11,x)

[Out]

-(60*A*b**3 + 140*B*c**3*x**4 + x**3*(105*A*c**3 + 315*B*b*c**2) + x**2*(252*A*b
*c**2 + 252*B*b**2*c) + x*(210*A*b**2*c + 70*B*b**3))/(420*x**7)

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GIAC/XCAS [A]  time = 0.267453, size = 101, normalized size = 1.35 \[ -\frac{140 \, B c^{3} x^{4} + 315 \, B b c^{2} x^{3} + 105 \, A c^{3} x^{3} + 252 \, B b^{2} c x^{2} + 252 \, A b c^{2} x^{2} + 70 \, B b^{3} x + 210 \, A b^{2} c x + 60 \, A b^{3}}{420 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^11,x, algorithm="giac")

[Out]

-1/420*(140*B*c^3*x^4 + 315*B*b*c^2*x^3 + 105*A*c^3*x^3 + 252*B*b^2*c*x^2 + 252*
A*b*c^2*x^2 + 70*B*b^3*x + 210*A*b^2*c*x + 60*A*b^3)/x^7