3.22.10 \(\int x (d+e x)^5 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=103 \[ \frac {(d+e x)^7 \left (3 c d^2-e (2 b d-a e)\right )}{7 e^4}-\frac {d (d+e x)^6 \left (a e^2-b d e+c d^2\right )}{6 e^4}-\frac {(d+e x)^8 (3 c d-b e)}{8 e^4}+\frac {c (d+e x)^9}{9 e^4} \]

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Rubi [A]  time = 0.16, 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)^7 \left (3 c d^2-e (2 b d-a e)\right )}{7 e^4}-\frac {d (d+e x)^6 \left (a e^2-b d e+c d^2\right )}{6 e^4}-\frac {(d+e x)^8 (3 c d-b e)}{8 e^4}+\frac {c (d+e x)^9}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)^5*(a + b*x + c*x^2),x]

[Out]

-(d*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^6)/(6*e^4) + ((3*c*d^2 - e*(2*b*d - a*e))*(d + e*x)^7)/(7*e^4) - ((3*c*d
- b*e)*(d + e*x)^8)/(8*e^4) + (c*(d + e*x)^9)/(9*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)^5 \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)^5}{e^3}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^6}{e^3}+\frac {(-3 c d+b e) (d+e x)^7}{e^3}+\frac {c (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{6 e^4}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^7}{7 e^4}-\frac {(3 c d-b e) (d+e x)^8}{8 e^4}+\frac {c (d+e x)^9}{9 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 166, normalized size = 1.61 \begin {gather*} \frac {5}{6} d e^2 x^6 \left (a e^2+2 b d e+2 c d^2\right )+d^2 e x^5 \left (2 a e^2+2 b d e+c d^2\right )+\frac {1}{7} e^3 x^7 \left (a e^2+5 b d e+10 c d^2\right )+\frac {1}{4} d^3 x^4 \left (10 a e^2+5 b d e+c d^2\right )+\frac {1}{3} d^4 x^3 (5 a e+b d)+\frac {1}{2} a d^5 x^2+\frac {1}{8} e^4 x^8 (b e+5 c d)+\frac {1}{9} c e^5 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)^5*(a + b*x + c*x^2),x]

[Out]

(a*d^5*x^2)/2 + (d^4*(b*d + 5*a*e)*x^3)/3 + (d^3*(c*d^2 + 5*b*d*e + 10*a*e^2)*x^4)/4 + d^2*e*(c*d^2 + 2*b*d*e
+ 2*a*e^2)*x^5 + (5*d*e^2*(2*c*d^2 + 2*b*d*e + a*e^2)*x^6)/6 + (e^3*(10*c*d^2 + 5*b*d*e + a*e^2)*x^7)/7 + (e^4
*(5*c*d + b*e)*x^8)/8 + (c*e^5*x^9)/9

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (d+e x)^5 \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(d + e*x)^5*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[x*(d + e*x)^5*(a + b*x + c*x^2), x]

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fricas [A]  time = 0.35, size = 186, normalized size = 1.81 \begin {gather*} \frac {1}{9} x^{9} e^{5} c + \frac {5}{8} x^{8} e^{4} d c + \frac {1}{8} x^{8} e^{5} b + \frac {10}{7} x^{7} e^{3} d^{2} c + \frac {5}{7} x^{7} e^{4} d b + \frac {1}{7} x^{7} e^{5} a + \frac {5}{3} x^{6} e^{2} d^{3} c + \frac {5}{3} x^{6} e^{3} d^{2} b + \frac {5}{6} x^{6} e^{4} d a + x^{5} e d^{4} c + 2 x^{5} e^{2} d^{3} b + 2 x^{5} e^{3} d^{2} a + \frac {1}{4} x^{4} d^{5} c + \frac {5}{4} x^{4} e d^{4} b + \frac {5}{2} x^{4} e^{2} d^{3} a + \frac {1}{3} x^{3} d^{5} b + \frac {5}{3} x^{3} e d^{4} a + \frac {1}{2} x^{2} d^{5} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^5*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/9*x^9*e^5*c + 5/8*x^8*e^4*d*c + 1/8*x^8*e^5*b + 10/7*x^7*e^3*d^2*c + 5/7*x^7*e^4*d*b + 1/7*x^7*e^5*a + 5/3*x
^6*e^2*d^3*c + 5/3*x^6*e^3*d^2*b + 5/6*x^6*e^4*d*a + x^5*e*d^4*c + 2*x^5*e^2*d^3*b + 2*x^5*e^3*d^2*a + 1/4*x^4
*d^5*c + 5/4*x^4*e*d^4*b + 5/2*x^4*e^2*d^3*a + 1/3*x^3*d^5*b + 5/3*x^3*e*d^4*a + 1/2*x^2*d^5*a

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giac [A]  time = 0.15, size = 177, normalized size = 1.72 \begin {gather*} \frac {1}{9} \, c x^{9} e^{5} + \frac {5}{8} \, c d x^{8} e^{4} + \frac {10}{7} \, c d^{2} x^{7} e^{3} + \frac {5}{3} \, c d^{3} x^{6} e^{2} + c d^{4} x^{5} e + \frac {1}{4} \, c d^{5} x^{4} + \frac {1}{8} \, b x^{8} e^{5} + \frac {5}{7} \, b d x^{7} e^{4} + \frac {5}{3} \, b d^{2} x^{6} e^{3} + 2 \, b d^{3} x^{5} e^{2} + \frac {5}{4} \, b d^{4} x^{4} e + \frac {1}{3} \, b d^{5} x^{3} + \frac {1}{7} \, a x^{7} e^{5} + \frac {5}{6} \, a d x^{6} e^{4} + 2 \, a d^{2} x^{5} e^{3} + \frac {5}{2} \, a d^{3} x^{4} e^{2} + \frac {5}{3} \, a d^{4} x^{3} e + \frac {1}{2} \, a d^{5} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^5*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/9*c*x^9*e^5 + 5/8*c*d*x^8*e^4 + 10/7*c*d^2*x^7*e^3 + 5/3*c*d^3*x^6*e^2 + c*d^4*x^5*e + 1/4*c*d^5*x^4 + 1/8*b
*x^8*e^5 + 5/7*b*d*x^7*e^4 + 5/3*b*d^2*x^6*e^3 + 2*b*d^3*x^5*e^2 + 5/4*b*d^4*x^4*e + 1/3*b*d^5*x^3 + 1/7*a*x^7
*e^5 + 5/6*a*d*x^6*e^4 + 2*a*d^2*x^5*e^3 + 5/2*a*d^3*x^4*e^2 + 5/3*a*d^4*x^3*e + 1/2*a*d^5*x^2

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maple [A]  time = 0.04, size = 172, normalized size = 1.67 \begin {gather*} \frac {c \,e^{5} x^{9}}{9}+\frac {a \,d^{5} x^{2}}{2}+\frac {\left (e^{5} b +5 d \,e^{4} c \right ) x^{8}}{8}+\frac {\left (e^{5} a +5 d \,e^{4} b +10 d^{2} e^{3} c \right ) x^{7}}{7}+\frac {\left (5 d \,e^{4} a +10 b \,d^{2} e^{3}+10 d^{3} e^{2} c \right ) x^{6}}{6}+\frac {\left (10 d^{2} e^{3} a +10 d^{3} e^{2} b +5 d^{4} e c \right ) x^{5}}{5}+\frac {\left (10 d^{3} e^{2} a +5 d^{4} e b +d^{5} c \right ) x^{4}}{4}+\frac {\left (5 d^{4} e a +d^{5} b \right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^5*(c*x^2+b*x+a),x)

[Out]

1/9*e^5*c*x^9+1/8*(b*e^5+5*c*d*e^4)*x^8+1/7*(a*e^5+5*b*d*e^4+10*c*d^2*e^3)*x^7+1/6*(5*a*d*e^4+10*b*d^2*e^3+10*
c*d^3*e^2)*x^6+1/5*(10*a*d^2*e^3+10*b*d^3*e^2+5*c*d^4*e)*x^5+1/4*(10*a*d^3*e^2+5*b*d^4*e+c*d^5)*x^4+1/3*(5*a*d
^4*e+b*d^5)*x^3+1/2*d^5*a*x^2

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maxima [A]  time = 0.57, size = 168, normalized size = 1.63 \begin {gather*} \frac {1}{9} \, c e^{5} x^{9} + \frac {1}{8} \, {\left (5 \, c d e^{4} + b e^{5}\right )} x^{8} + \frac {1}{2} \, a d^{5} x^{2} + \frac {1}{7} \, {\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{6} + {\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{5} + 5 \, a d^{4} e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^5*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/9*c*e^5*x^9 + 1/8*(5*c*d*e^4 + b*e^5)*x^8 + 1/2*a*d^5*x^2 + 1/7*(10*c*d^2*e^3 + 5*b*d*e^4 + a*e^5)*x^7 + 5/6
*(2*c*d^3*e^2 + 2*b*d^2*e^3 + a*d*e^4)*x^6 + (c*d^4*e + 2*b*d^3*e^2 + 2*a*d^2*e^3)*x^5 + 1/4*(c*d^5 + 5*b*d^4*
e + 10*a*d^3*e^2)*x^4 + 1/3*(b*d^5 + 5*a*d^4*e)*x^3

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mupad [B]  time = 2.36, size = 160, normalized size = 1.55 \begin {gather*} x^3\,\left (\frac {b\,d^5}{3}+\frac {5\,a\,e\,d^4}{3}\right )+x^8\,\left (\frac {b\,e^5}{8}+\frac {5\,c\,d\,e^4}{8}\right )+x^4\,\left (\frac {c\,d^5}{4}+\frac {5\,b\,d^4\,e}{4}+\frac {5\,a\,d^3\,e^2}{2}\right )+x^7\,\left (\frac {10\,c\,d^2\,e^3}{7}+\frac {5\,b\,d\,e^4}{7}+\frac {a\,e^5}{7}\right )+\frac {a\,d^5\,x^2}{2}+\frac {c\,e^5\,x^9}{9}+d^2\,e\,x^5\,\left (c\,d^2+2\,b\,d\,e+2\,a\,e^2\right )+\frac {5\,d\,e^2\,x^6\,\left (2\,c\,d^2+2\,b\,d\,e+a\,e^2\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d + e*x)^5*(a + b*x + c*x^2),x)

[Out]

x^3*((b*d^5)/3 + (5*a*d^4*e)/3) + x^8*((b*e^5)/8 + (5*c*d*e^4)/8) + x^4*((c*d^5)/4 + (5*a*d^3*e^2)/2 + (5*b*d^
4*e)/4) + x^7*((a*e^5)/7 + (10*c*d^2*e^3)/7 + (5*b*d*e^4)/7) + (a*d^5*x^2)/2 + (c*e^5*x^9)/9 + d^2*e*x^5*(2*a*
e^2 + c*d^2 + 2*b*d*e) + (5*d*e^2*x^6*(a*e^2 + 2*c*d^2 + 2*b*d*e))/6

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sympy [B]  time = 0.10, size = 192, normalized size = 1.86 \begin {gather*} \frac {a d^{5} x^{2}}{2} + \frac {c e^{5} x^{9}}{9} + x^{8} \left (\frac {b e^{5}}{8} + \frac {5 c d e^{4}}{8}\right ) + x^{7} \left (\frac {a e^{5}}{7} + \frac {5 b d e^{4}}{7} + \frac {10 c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {5 a d e^{4}}{6} + \frac {5 b d^{2} e^{3}}{3} + \frac {5 c d^{3} e^{2}}{3}\right ) + x^{5} \left (2 a d^{2} e^{3} + 2 b d^{3} e^{2} + c d^{4} e\right ) + x^{4} \left (\frac {5 a d^{3} e^{2}}{2} + \frac {5 b d^{4} e}{4} + \frac {c d^{5}}{4}\right ) + x^{3} \left (\frac {5 a d^{4} e}{3} + \frac {b d^{5}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)**5*(c*x**2+b*x+a),x)

[Out]

a*d**5*x**2/2 + c*e**5*x**9/9 + x**8*(b*e**5/8 + 5*c*d*e**4/8) + x**7*(a*e**5/7 + 5*b*d*e**4/7 + 10*c*d**2*e**
3/7) + x**6*(5*a*d*e**4/6 + 5*b*d**2*e**3/3 + 5*c*d**3*e**2/3) + x**5*(2*a*d**2*e**3 + 2*b*d**3*e**2 + c*d**4*
e) + x**4*(5*a*d**3*e**2/2 + 5*b*d**4*e/4 + c*d**5/4) + x**3*(5*a*d**4*e/3 + b*d**5/3)

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