3.22.13 \(\int x (d+e x)^2 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=78 \[ \frac {1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{3} d x^3 (2 a e+b d)+\frac {1}{2} a d^2 x^2+\frac {1}{5} e x^5 (b e+2 c d)+\frac {1}{6} c e^2 x^6 \]

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Rubi [A]  time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {771} \begin {gather*} \frac {1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{3} d x^3 (2 a e+b d)+\frac {1}{2} a d^2 x^2+\frac {1}{5} e x^5 (b e+2 c d)+\frac {1}{6} c e^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)^2*(a + b*x + c*x^2),x]

[Out]

(a*d^2*x^2)/2 + (d*(b*d + 2*a*e)*x^3)/3 + ((c*d^2 + e*(2*b*d + a*e))*x^4)/4 + (e*(2*c*d + b*e)*x^5)/5 + (c*e^2
*x^6)/6

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 70, normalized size = 0.90 \begin {gather*} \frac {1}{60} x^2 \left (15 x^2 \left (e (a e+2 b d)+c d^2\right )+20 d x (2 a e+b d)+30 a d^2+12 e x^3 (b e+2 c d)+10 c e^2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)^2*(a + b*x + c*x^2),x]

[Out]

(x^2*(30*a*d^2 + 20*d*(b*d + 2*a*e)*x + 15*(c*d^2 + e*(2*b*d + a*e))*x^2 + 12*e*(2*c*d + b*e)*x^3 + 10*c*e^2*x
^4))/60

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(d + e*x)^2*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[x*(d + e*x)^2*(a + b*x + c*x^2), x]

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fricas [A]  time = 0.34, size = 79, normalized size = 1.01 \begin {gather*} \frac {1}{6} x^{6} e^{2} c + \frac {2}{5} x^{5} e d c + \frac {1}{5} x^{5} e^{2} b + \frac {1}{4} x^{4} d^{2} c + \frac {1}{2} x^{4} e d b + \frac {1}{4} x^{4} e^{2} a + \frac {1}{3} x^{3} d^{2} b + \frac {2}{3} x^{3} e d a + \frac {1}{2} x^{2} d^{2} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^2*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/6*x^6*e^2*c + 2/5*x^5*e*d*c + 1/5*x^5*e^2*b + 1/4*x^4*d^2*c + 1/2*x^4*e*d*b + 1/4*x^4*e^2*a + 1/3*x^3*d^2*b
+ 2/3*x^3*e*d*a + 1/2*x^2*d^2*a

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giac [A]  time = 0.15, size = 79, normalized size = 1.01 \begin {gather*} \frac {1}{6} \, c x^{6} e^{2} + \frac {2}{5} \, c d x^{5} e + \frac {1}{4} \, c d^{2} x^{4} + \frac {1}{5} \, b x^{5} e^{2} + \frac {1}{2} \, b d x^{4} e + \frac {1}{3} \, b d^{2} x^{3} + \frac {1}{4} \, a x^{4} e^{2} + \frac {2}{3} \, a d x^{3} e + \frac {1}{2} \, a d^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^2*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/6*c*x^6*e^2 + 2/5*c*d*x^5*e + 1/4*c*d^2*x^4 + 1/5*b*x^5*e^2 + 1/2*b*d*x^4*e + 1/3*b*d^2*x^3 + 1/4*a*x^4*e^2
+ 2/3*a*d*x^3*e + 1/2*a*d^2*x^2

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maple [A]  time = 0.04, size = 73, normalized size = 0.94 \begin {gather*} \frac {c \,e^{2} x^{6}}{6}+\frac {a \,d^{2} x^{2}}{2}+\frac {\left (b \,e^{2}+2 c d e \right ) x^{5}}{5}+\frac {\left (a \,e^{2}+2 b d e +c \,d^{2}\right ) x^{4}}{4}+\frac {\left (2 a d e +b \,d^{2}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^2*(c*x^2+b*x+a),x)

[Out]

1/6*c*e^2*x^6+1/5*(b*e^2+2*c*d*e)*x^5+1/4*(a*e^2+2*b*d*e+c*d^2)*x^4+1/3*(2*a*d*e+b*d^2)*x^3+1/2*a*d^2*x^2

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maxima [A]  time = 0.57, size = 72, normalized size = 0.92 \begin {gather*} \frac {1}{6} \, c e^{2} x^{6} + \frac {1}{5} \, {\left (2 \, c d e + b e^{2}\right )} x^{5} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{2} + 2 \, a d e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^2*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/6*c*e^2*x^6 + 1/5*(2*c*d*e + b*e^2)*x^5 + 1/2*a*d^2*x^2 + 1/4*(c*d^2 + 2*b*d*e + a*e^2)*x^4 + 1/3*(b*d^2 + 2
*a*d*e)*x^3

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mupad [B]  time = 0.03, size = 73, normalized size = 0.94 \begin {gather*} x^4\,\left (\frac {c\,d^2}{4}+\frac {b\,d\,e}{2}+\frac {a\,e^2}{4}\right )+x^3\,\left (\frac {b\,d^2}{3}+\frac {2\,a\,e\,d}{3}\right )+x^5\,\left (\frac {b\,e^2}{5}+\frac {2\,c\,d\,e}{5}\right )+\frac {a\,d^2\,x^2}{2}+\frac {c\,e^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d + e*x)^2*(a + b*x + c*x^2),x)

[Out]

x^4*((a*e^2)/4 + (c*d^2)/4 + (b*d*e)/2) + x^3*((b*d^2)/3 + (2*a*d*e)/3) + x^5*((b*e^2)/5 + (2*c*d*e)/5) + (a*d
^2*x^2)/2 + (c*e^2*x^6)/6

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sympy [A]  time = 0.08, size = 80, normalized size = 1.03 \begin {gather*} \frac {a d^{2} x^{2}}{2} + \frac {c e^{2} x^{6}}{6} + x^{5} \left (\frac {b e^{2}}{5} + \frac {2 c d e}{5}\right ) + x^{4} \left (\frac {a e^{2}}{4} + \frac {b d e}{2} + \frac {c d^{2}}{4}\right ) + x^{3} \left (\frac {2 a d e}{3} + \frac {b d^{2}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)**2*(c*x**2+b*x+a),x)

[Out]

a*d**2*x**2/2 + c*e**2*x**6/6 + x**5*(b*e**2/5 + 2*c*d*e/5) + x**4*(a*e**2/4 + b*d*e/2 + c*d**2/4) + x**3*(2*a
*d*e/3 + b*d**2/3)

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