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

Optimal. Leaf size=62 \[ -\frac{(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac{b (b+c x)^4 (b B-A c)}{4 c^3}+\frac{B (b+c x)^6}{6 c^3} \]

[Out]

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

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Rubi [A]  time = 0.129202, antiderivative size = 62, 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 \[ -\frac{(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac{b (b+c x)^4 (b B-A c)}{4 c^3}+\frac{B (b+c x)^6}{6 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9539, size = 53, normalized size = 0.85 \[ \frac{B \left (b + c x\right )^{6}}{6 c^{3}} - \frac{b \left (b + c x\right )^{4} \left (A c - B b\right )}{4 c^{3}} + \frac{\left (b + c x\right )^{5} \left (A c - 2 B b\right )}{5 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0263576, size = 67, normalized size = 1.08 \[ \frac{1}{60} x^2 \left (30 A b^3+20 b^2 x (3 A c+b B)+12 c^2 x^3 (A c+3 b B)+45 b c x^2 (A c+b B)+10 B c^3 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 76, normalized size = 1.2 \[{\frac{B{c}^{3}{x}^{6}}{6}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{b}^{2}c+B{b}^{3} \right ){x}^{3}}{3}}+{\frac{A{b}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.699852, size = 99, normalized size = 1.6 \[ \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{2} \, A b^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.259984, size = 99, normalized size = 1.6 \[ \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{2} \, A b^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.143101, size = 80, normalized size = 1.29 \[ \frac{A b^{3} x^{2}}{2} + \frac{B c^{3} x^{6}}{6} + x^{5} \left (\frac{A c^{3}}{5} + \frac{3 B b c^{2}}{5}\right ) + x^{4} \left (\frac{3 A b c^{2}}{4} + \frac{3 B b^{2} c}{4}\right ) + x^{3} \left (A b^{2} c + \frac{B b^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271504, size = 103, normalized size = 1.66 \[ \frac{1}{6} \, B c^{3} x^{6} + \frac{3}{5} \, B b c^{2} x^{5} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B b^{2} c x^{4} + \frac{3}{4} \, A b c^{2} x^{4} + \frac{1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + \frac{1}{2} \, A b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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