3.22.12 \(\int x (d+e x)^3 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=103 \[ \frac {(d+e x)^5 \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4}-\frac {d (d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {(d+e x)^6 (3 c d-b e)}{6 e^4}+\frac {c (d+e x)^7}{7 e^4} \]

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Rubi [A]  time = 0.10, antiderivative size = 103, 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 {(d+e x)^5 \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4}-\frac {d (d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {(d+e x)^6 (3 c d-b e)}{6 e^4}+\frac {c (d+e x)^7}{7 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)^3*(a + b*x + c*x^2),x]

[Out]

-(d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4)/(4*e^4) + ((3*c*d^2 - e*(2*b*d - a*e))*(d + e*x)^5)/(5*e^4) - ((3*c*d
- b*e)*(d + e*x)^6)/(6*e^4) + (c*(d + e*x)^7)/(7*e^4)

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)^3 \left (a+b x+c x^2\right ) \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^3}{e^3}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^4}{e^3}+\frac {(-3 c d+b e) (d+e x)^5}{e^3}+\frac {c (d+e x)^6}{e^3}\right ) \, dx\\ &=-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^5}{5 e^4}-\frac {(3 c d-b e) (d+e x)^6}{6 e^4}+\frac {c (d+e x)^7}{7 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 109, normalized size = 1.06 \begin {gather*} \frac {1}{5} e x^5 \left (a e^2+3 b d e+3 c d^2\right )+\frac {1}{4} d x^4 \left (3 a e^2+3 b d e+c d^2\right )+\frac {1}{3} d^2 x^3 (3 a e+b d)+\frac {1}{2} a d^3 x^2+\frac {1}{6} e^2 x^6 (b e+3 c d)+\frac {1}{7} c e^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)^3*(a + b*x + c*x^2),x]

[Out]

(a*d^3*x^2)/2 + (d^2*(b*d + 3*a*e)*x^3)/3 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^4)/4 + (e*(3*c*d^2 + 3*b*d*e + a*
e^2)*x^5)/5 + (e^2*(3*c*d + b*e)*x^6)/6 + (c*e^3*x^7)/7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (d+e x)^3 \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(d + e*x)^3*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[x*(d + e*x)^3*(a + b*x + c*x^2), x]

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fricas [A]  time = 0.35, size = 114, normalized size = 1.11 \begin {gather*} \frac {1}{7} x^{7} e^{3} c + \frac {1}{2} x^{6} e^{2} d c + \frac {1}{6} x^{6} e^{3} b + \frac {3}{5} x^{5} e d^{2} c + \frac {3}{5} x^{5} e^{2} d b + \frac {1}{5} x^{5} e^{3} a + \frac {1}{4} x^{4} d^{3} c + \frac {3}{4} x^{4} e d^{2} b + \frac {3}{4} x^{4} e^{2} d a + \frac {1}{3} x^{3} d^{3} b + x^{3} e d^{2} a + \frac {1}{2} x^{2} d^{3} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^3*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/7*x^7*e^3*c + 1/2*x^6*e^2*d*c + 1/6*x^6*e^3*b + 3/5*x^5*e*d^2*c + 3/5*x^5*e^2*d*b + 1/5*x^5*e^3*a + 1/4*x^4*
d^3*c + 3/4*x^4*e*d^2*b + 3/4*x^4*e^2*d*a + 1/3*x^3*d^3*b + x^3*e*d^2*a + 1/2*x^2*d^3*a

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giac [A]  time = 0.17, size = 111, normalized size = 1.08 \begin {gather*} \frac {1}{7} \, c x^{7} e^{3} + \frac {1}{2} \, c d x^{6} e^{2} + \frac {3}{5} \, c d^{2} x^{5} e + \frac {1}{4} \, c d^{3} x^{4} + \frac {1}{6} \, b x^{6} e^{3} + \frac {3}{5} \, b d x^{5} e^{2} + \frac {3}{4} \, b d^{2} x^{4} e + \frac {1}{3} \, b d^{3} x^{3} + \frac {1}{5} \, a x^{5} e^{3} + \frac {3}{4} \, a d x^{4} e^{2} + a d^{2} x^{3} e + \frac {1}{2} \, a d^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^3*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/7*c*x^7*e^3 + 1/2*c*d*x^6*e^2 + 3/5*c*d^2*x^5*e + 1/4*c*d^3*x^4 + 1/6*b*x^6*e^3 + 3/5*b*d*x^5*e^2 + 3/4*b*d^
2*x^4*e + 1/3*b*d^3*x^3 + 1/5*a*x^5*e^3 + 3/4*a*d*x^4*e^2 + a*d^2*x^3*e + 1/2*a*d^3*x^2

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maple [A]  time = 0.04, size = 106, normalized size = 1.03 \begin {gather*} \frac {c \,e^{3} x^{7}}{7}+\frac {a \,d^{3} x^{2}}{2}+\frac {\left (b \,e^{3}+3 d \,e^{2} c \right ) x^{6}}{6}+\frac {\left (a \,e^{3}+3 b d \,e^{2}+3 c \,d^{2} e \right ) x^{5}}{5}+\frac {\left (3 a d \,e^{2}+3 b \,d^{2} e +c \,d^{3}\right ) x^{4}}{4}+\frac {\left (3 d^{2} e a +b \,d^{3}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^3*(c*x^2+b*x+a),x)

[Out]

1/7*e^3*c*x^7+1/6*(b*e^3+3*c*d*e^2)*x^6+1/5*(a*e^3+3*b*d*e^2+3*c*d^2*e)*x^5+1/4*(3*a*d*e^2+3*b*d^2*e+c*d^3)*x^
4+1/3*(3*a*d^2*e+b*d^3)*x^3+1/2*d^3*a*x^2

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maxima [A]  time = 0.50, size = 105, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, c e^{3} x^{7} + \frac {1}{6} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{6} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{5} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^3*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/7*c*e^3*x^7 + 1/6*(3*c*d*e^2 + b*e^3)*x^6 + 1/2*a*d^3*x^2 + 1/5*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*x^5 + 1/4*(c
*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^4 + 1/3*(b*d^3 + 3*a*d^2*e)*x^3

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mupad [B]  time = 2.24, size = 104, normalized size = 1.01 \begin {gather*} x^3\,\left (\frac {b\,d^3}{3}+a\,e\,d^2\right )+x^6\,\left (\frac {b\,e^3}{6}+\frac {c\,d\,e^2}{2}\right )+x^4\,\left (\frac {c\,d^3}{4}+\frac {3\,b\,d^2\,e}{4}+\frac {3\,a\,d\,e^2}{4}\right )+x^5\,\left (\frac {3\,c\,d^2\,e}{5}+\frac {3\,b\,d\,e^2}{5}+\frac {a\,e^3}{5}\right )+\frac {a\,d^3\,x^2}{2}+\frac {c\,e^3\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d + e*x)^3*(a + b*x + c*x^2),x)

[Out]

x^3*((b*d^3)/3 + a*d^2*e) + x^6*((b*e^3)/6 + (c*d*e^2)/2) + x^4*((c*d^3)/4 + (3*a*d*e^2)/4 + (3*b*d^2*e)/4) +
x^5*((a*e^3)/5 + (3*b*d*e^2)/5 + (3*c*d^2*e)/5) + (a*d^3*x^2)/2 + (c*e^3*x^7)/7

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sympy [A]  time = 0.08, size = 116, normalized size = 1.13 \begin {gather*} \frac {a d^{3} x^{2}}{2} + \frac {c e^{3} x^{7}}{7} + x^{6} \left (\frac {b e^{3}}{6} + \frac {c d e^{2}}{2}\right ) + x^{5} \left (\frac {a e^{3}}{5} + \frac {3 b d e^{2}}{5} + \frac {3 c d^{2} e}{5}\right ) + x^{4} \left (\frac {3 a d e^{2}}{4} + \frac {3 b d^{2} e}{4} + \frac {c d^{3}}{4}\right ) + x^{3} \left (a d^{2} e + \frac {b d^{3}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)**3*(c*x**2+b*x+a),x)

[Out]

a*d**3*x**2/2 + c*e**3*x**7/7 + x**6*(b*e**3/6 + c*d*e**2/2) + x**5*(a*e**3/5 + 3*b*d*e**2/5 + 3*c*d**2*e/5) +
x**4*(3*a*d*e**2/4 + 3*b*d**2*e/4 + c*d**3/4) + x**3*(a*d**2*e + b*d**3/3)

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