3.22.11 \(\int x (d+e x)^4 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=103 \[ \frac {(d+e x)^6 \left (3 c d^2-e (2 b d-a e)\right )}{6 e^4}-\frac {d (d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {(d+e x)^7 (3 c d-b e)}{7 e^4}+\frac {c (d+e x)^8}{8 e^4} \]

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Rubi [A]  time = 0.13, 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)^6 \left (3 c d^2-e (2 b d-a e)\right )}{6 e^4}-\frac {d (d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {(d+e x)^7 (3 c d-b e)}{7 e^4}+\frac {c (d+e x)^8}{8 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)^4*(a + b*x + c*x^2),x]

[Out]

-(d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5)/(5*e^4) + ((3*c*d^2 - e*(2*b*d - a*e))*(d + e*x)^6)/(6*e^4) - ((3*c*d
- b*e)*(d + e*x)^7)/(7*e^4) + (c*(d + e*x)^8)/(8*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)^4 \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)^4}{e^3}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^5}{e^3}+\frac {(-3 c d+b e) (d+e x)^6}{e^3}+\frac {c (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^6}{6 e^4}-\frac {(3 c d-b e) (d+e x)^7}{7 e^4}+\frac {c (d+e x)^8}{8 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 140, normalized size = 1.36 \begin {gather*} \frac {1}{6} e^2 x^6 \left (a e^2+4 b d e+6 c d^2\right )+\frac {2}{5} d e x^5 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac {1}{4} d^2 x^4 \left (6 a e^2+4 b d e+c d^2\right )+\frac {1}{3} d^3 x^3 (4 a e+b d)+\frac {1}{2} a d^4 x^2+\frac {1}{7} e^3 x^7 (b e+4 c d)+\frac {1}{8} c e^4 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)^4*(a + b*x + c*x^2),x]

[Out]

(a*d^4*x^2)/2 + (d^3*(b*d + 4*a*e)*x^3)/3 + (d^2*(c*d^2 + 4*b*d*e + 6*a*e^2)*x^4)/4 + (2*d*e*(2*c*d^2 + 3*b*d*
e + 2*a*e^2)*x^5)/5 + (e^2*(6*c*d^2 + 4*b*d*e + a*e^2)*x^6)/6 + (e^3*(4*c*d + b*e)*x^7)/7 + (c*e^4*x^8)/8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (d+e x)^4 \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(d + e*x)^4*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[x*(d + e*x)^4*(a + b*x + c*x^2), x]

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fricas [A]  time = 0.35, size = 149, normalized size = 1.45 \begin {gather*} \frac {1}{8} x^{8} e^{4} c + \frac {4}{7} x^{7} e^{3} d c + \frac {1}{7} x^{7} e^{4} b + x^{6} e^{2} d^{2} c + \frac {2}{3} x^{6} e^{3} d b + \frac {1}{6} x^{6} e^{4} a + \frac {4}{5} x^{5} e d^{3} c + \frac {6}{5} x^{5} e^{2} d^{2} b + \frac {4}{5} x^{5} e^{3} d a + \frac {1}{4} x^{4} d^{4} c + x^{4} e d^{3} b + \frac {3}{2} x^{4} e^{2} d^{2} a + \frac {1}{3} x^{3} d^{4} b + \frac {4}{3} x^{3} e d^{3} a + \frac {1}{2} x^{2} d^{4} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/8*x^8*e^4*c + 4/7*x^7*e^3*d*c + 1/7*x^7*e^4*b + x^6*e^2*d^2*c + 2/3*x^6*e^3*d*b + 1/6*x^6*e^4*a + 4/5*x^5*e*
d^3*c + 6/5*x^5*e^2*d^2*b + 4/5*x^5*e^3*d*a + 1/4*x^4*d^4*c + x^4*e*d^3*b + 3/2*x^4*e^2*d^2*a + 1/3*x^3*d^4*b
+ 4/3*x^3*e*d^3*a + 1/2*x^2*d^4*a

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giac [A]  time = 0.16, size = 143, normalized size = 1.39 \begin {gather*} \frac {1}{8} \, c x^{8} e^{4} + \frac {4}{7} \, c d x^{7} e^{3} + c d^{2} x^{6} e^{2} + \frac {4}{5} \, c d^{3} x^{5} e + \frac {1}{4} \, c d^{4} x^{4} + \frac {1}{7} \, b x^{7} e^{4} + \frac {2}{3} \, b d x^{6} e^{3} + \frac {6}{5} \, b d^{2} x^{5} e^{2} + b d^{3} x^{4} e + \frac {1}{3} \, b d^{4} x^{3} + \frac {1}{6} \, a x^{6} e^{4} + \frac {4}{5} \, a d x^{5} e^{3} + \frac {3}{2} \, a d^{2} x^{4} e^{2} + \frac {4}{3} \, a d^{3} x^{3} e + \frac {1}{2} \, a d^{4} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/8*c*x^8*e^4 + 4/7*c*d*x^7*e^3 + c*d^2*x^6*e^2 + 4/5*c*d^3*x^5*e + 1/4*c*d^4*x^4 + 1/7*b*x^7*e^4 + 2/3*b*d*x^
6*e^3 + 6/5*b*d^2*x^5*e^2 + b*d^3*x^4*e + 1/3*b*d^4*x^3 + 1/6*a*x^6*e^4 + 4/5*a*d*x^5*e^3 + 3/2*a*d^2*x^4*e^2
+ 4/3*a*d^3*x^3*e + 1/2*a*d^4*x^2

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maple [A]  time = 0.04, size = 139, normalized size = 1.35 \begin {gather*} \frac {c \,e^{4} x^{8}}{8}+\frac {a \,d^{4} x^{2}}{2}+\frac {\left (e^{4} b +4 c d \,e^{3}\right ) x^{7}}{7}+\frac {\left (e^{4} a +4 d \,e^{3} b +6 d^{2} e^{2} c \right ) x^{6}}{6}+\frac {\left (4 a d \,e^{3}+6 d^{2} e^{2} b +4 c \,d^{3} e \right ) x^{5}}{5}+\frac {\left (6 a \,d^{2} e^{2}+4 d^{3} e b +c \,d^{4}\right ) x^{4}}{4}+\frac {\left (4 d^{3} e a +b \,d^{4}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^4*(c*x^2+b*x+a),x)

[Out]

1/8*e^4*c*x^8+1/7*(b*e^4+4*c*d*e^3)*x^7+1/6*(a*e^4+4*b*d*e^3+6*c*d^2*e^2)*x^6+1/5*(4*a*d*e^3+6*b*d^2*e^2+4*c*d
^3*e)*x^5+1/4*(6*a*d^2*e^2+4*b*d^3*e+c*d^4)*x^4+1/3*(4*a*d^3*e+b*d^4)*x^3+1/2*d^4*a*x^2

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maxima [A]  time = 0.73, size = 138, normalized size = 1.34 \begin {gather*} \frac {1}{8} \, c e^{4} x^{8} + \frac {1}{7} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{7} + \frac {1}{2} \, a d^{4} x^{2} + \frac {1}{6} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/8*c*e^4*x^8 + 1/7*(4*c*d*e^3 + b*e^4)*x^7 + 1/2*a*d^4*x^2 + 1/6*(6*c*d^2*e^2 + 4*b*d*e^3 + a*e^4)*x^6 + 2/5*
(2*c*d^3*e + 3*b*d^2*e^2 + 2*a*d*e^3)*x^5 + 1/4*(c*d^4 + 4*b*d^3*e + 6*a*d^2*e^2)*x^4 + 1/3*(b*d^4 + 4*a*d^3*e
)*x^3

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mupad [B]  time = 0.06, size = 132, normalized size = 1.28 \begin {gather*} x^3\,\left (\frac {b\,d^4}{3}+\frac {4\,a\,e\,d^3}{3}\right )+x^7\,\left (\frac {b\,e^4}{7}+\frac {4\,c\,d\,e^3}{7}\right )+x^4\,\left (\frac {c\,d^4}{4}+b\,d^3\,e+\frac {3\,a\,d^2\,e^2}{2}\right )+x^6\,\left (c\,d^2\,e^2+\frac {2\,b\,d\,e^3}{3}+\frac {a\,e^4}{6}\right )+\frac {a\,d^4\,x^2}{2}+\frac {c\,e^4\,x^8}{8}+\frac {2\,d\,e\,x^5\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d + e*x)^4*(a + b*x + c*x^2),x)

[Out]

x^3*((b*d^4)/3 + (4*a*d^3*e)/3) + x^7*((b*e^4)/7 + (4*c*d*e^3)/7) + x^4*((c*d^4)/4 + (3*a*d^2*e^2)/2 + b*d^3*e
) + x^6*((a*e^4)/6 + c*d^2*e^2 + (2*b*d*e^3)/3) + (a*d^4*x^2)/2 + (c*e^4*x^8)/8 + (2*d*e*x^5*(2*a*e^2 + 2*c*d^
2 + 3*b*d*e))/5

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sympy [A]  time = 0.09, size = 153, normalized size = 1.49 \begin {gather*} \frac {a d^{4} x^{2}}{2} + \frac {c e^{4} x^{8}}{8} + x^{7} \left (\frac {b e^{4}}{7} + \frac {4 c d e^{3}}{7}\right ) + x^{6} \left (\frac {a e^{4}}{6} + \frac {2 b d e^{3}}{3} + c d^{2} e^{2}\right ) + x^{5} \left (\frac {4 a d e^{3}}{5} + \frac {6 b d^{2} e^{2}}{5} + \frac {4 c d^{3} e}{5}\right ) + x^{4} \left (\frac {3 a d^{2} e^{2}}{2} + b d^{3} e + \frac {c d^{4}}{4}\right ) + x^{3} \left (\frac {4 a d^{3} e}{3} + \frac {b d^{4}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)**4*(c*x**2+b*x+a),x)

[Out]

a*d**4*x**2/2 + c*e**4*x**8/8 + x**7*(b*e**4/7 + 4*c*d*e**3/7) + x**6*(a*e**4/6 + 2*b*d*e**3/3 + c*d**2*e**2)
+ x**5*(4*a*d*e**3/5 + 6*b*d**2*e**2/5 + 4*c*d**3*e/5) + x**4*(3*a*d**2*e**2/2 + b*d**3*e + c*d**4/4) + x**3*(
4*a*d**3*e/3 + b*d**4/3)

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