3.18 \(\int (A+B x) (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=55 \[ \frac {1}{3} A b^2 x^3+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{4} b x^4 (2 A c+b B)+\frac {1}{6} B c^2 x^6 \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \[ \frac {1}{3} A b^2 x^3+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{4} b x^4 (2 A c+b B)+\frac {1}{6} B c^2 x^6 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

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

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin {align*} \int (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^2+b (b B+2 A c) x^3+c (2 b B+A c) x^4+B c^2 x^5\right ) \, dx\\ &=\frac {1}{3} A b^2 x^3+\frac {1}{4} b (b B+2 A c) x^4+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{6} B c^2 x^6\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 0.89 \[ \frac {1}{60} x^3 \left (20 A b^2+12 c x^2 (A c+2 b B)+15 b x (2 A c+b B)+10 B c^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(20*A*b^2 + 15*b*(b*B + 2*A*c)*x + 12*c*(2*b*B + A*c)*x^2 + 10*B*c^2*x^3))/60

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fricas [A]  time = 0.78, size = 53, normalized size = 0.96 \[ \frac {1}{6} x^{6} c^{2} B + \frac {2}{5} x^{5} c b B + \frac {1}{5} x^{5} c^{2} A + \frac {1}{4} x^{4} b^{2} B + \frac {1}{2} x^{4} c b A + \frac {1}{3} x^{3} b^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 53, normalized size = 0.96 \[ \frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {1}{3} \, A b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 52, normalized size = 0.95 \[ \frac {B \,c^{2} x^{6}}{6}+\frac {A \,b^{2} x^{3}}{3}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{5}}{5}+\frac {\left (2 A b c +b^{2} B \right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2,x)

[Out]

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

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maxima [A]  time = 0.92, size = 51, normalized size = 0.93 \[ \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{3} \, A b^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, A b c\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 51, normalized size = 0.93 \[ x^4\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+\frac {A\,b^2\,x^3}{3}+\frac {B\,c^2\,x^6}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(A + B*x),x)

[Out]

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

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sympy [A]  time = 0.08, size = 54, normalized size = 0.98 \[ \frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {B b^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2,x)

[Out]

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

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