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

Optimal. Leaf size=55 \[ \frac {1}{4} A b^2 x^4+\frac {1}{6} c x^6 (A c+2 b B)+\frac {1}{5} b x^5 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \]

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Rubi [A]  time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} \frac {1}{4} A b^2 x^4+\frac {1}{6} c x^6 (A c+2 b B)+\frac {1}{5} b x^5 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int x (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^3+b (b B+2 A c) x^4+c (2 b B+A c) x^5+B c^2 x^6\right ) \, dx\\ &=\frac {1}{4} A b^2 x^4+\frac {1}{5} b (b B+2 A c) x^5+\frac {1}{6} c (2 b B+A c) x^6+\frac {1}{7} B c^2 x^7\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{4} A b^2 x^4+\frac {1}{6} c x^6 (A c+2 b B)+\frac {1}{5} b x^5 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (A+B x) \left (b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{7} x^{7} c^{2} B + \frac {1}{3} x^{6} c b B + \frac {1}{6} x^{6} c^{2} A + \frac {1}{5} x^{5} b^{2} B + \frac {2}{5} x^{5} c b A + \frac {1}{4} x^{4} b^{2} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{3} \, B b c x^{6} + \frac {1}{6} \, A c^{2} x^{6} + \frac {1}{5} \, B b^{2} x^{5} + \frac {2}{5} \, A b c x^{5} + \frac {1}{4} \, A b^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 52, normalized size = 0.95 \begin {gather*} \frac {B \,c^{2} x^{7}}{7}+\frac {A \,b^{2} x^{4}}{4}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{6}}{6}+\frac {\left (2 A b c +b^{2} B \right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.84, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{6} \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{2} + 2 \, A b c\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 51, normalized size = 0.93 \begin {gather*} x^5\,\left (\frac {B\,b^2}{5}+\frac {2\,A\,c\,b}{5}\right )+x^6\,\left (\frac {A\,c^2}{6}+\frac {B\,b\,c}{3}\right )+\frac {A\,b^2\,x^4}{4}+\frac {B\,c^2\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x + c*x^2)^2*(A + B*x),x)

[Out]

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

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sympy [A]  time = 0.08, size = 54, normalized size = 0.98 \begin {gather*} \frac {A b^{2} x^{4}}{4} + \frac {B c^{2} x^{7}}{7} + x^{6} \left (\frac {A c^{2}}{6} + \frac {B b c}{3}\right ) + x^{5} \left (\frac {2 A b c}{5} + \frac {B b^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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