3.2350 \(\int x (d+e x)^5 \left (a+b x+c x^2\right ) \, dx\)

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

[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)

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Rubi [A]  time = 0.335587, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{(d+e x)^7 \left (3 c d^2-e (2 b d-a e)\right )}{7 e^4}-\frac{d (d+e x)^6 \left (a e^2-b d e+c d^2\right )}{6 e^4}-\frac{(d+e x)^8 (3 c d-b e)}{8 e^4}+\frac{c (d+e x)^9}{9 e^4} \]

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)

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Rubi in Sympy [A]  time = 33.8894, size = 92, normalized size = 0.89 \[ \frac{c \left (d + e x\right )^{9}}{9 e^{4}} - \frac{d \left (d + e x\right )^{6} \left (a e^{2} - b d e + c d^{2}\right )}{6 e^{4}} + \frac{\left (d + e x\right )^{8} \left (b e - 3 c d\right )}{8 e^{4}} + \frac{\left (d + e x\right )^{7} \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{7 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(e*x+d)**5*(c*x**2+b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.07828, size = 166, normalized size = 1.61 \[ \frac{5}{6} d e^2 x^6 \left (a e^2+2 b d e+2 c d^2\right )+d^2 e x^5 \left (2 a e^2+2 b d e+c d^2\right )+\frac{1}{7} e^3 x^7 \left (a e^2+5 b d e+10 c d^2\right )+\frac{1}{4} d^3 x^4 \left (10 a e^2+5 b d e+c d^2\right )+\frac{1}{3} d^4 x^3 (5 a e+b d)+\frac{1}{2} a d^5 x^2+\frac{1}{8} e^4 x^8 (b e+5 c d)+\frac{1}{9} c e^5 x^9 \]

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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Maple [A]  time = 0.001, size = 172, normalized size = 1.7 \[{\frac{{e}^{5}c{x}^{9}}{9}}+{\frac{ \left ({e}^{5}b+5\,d{e}^{4}c \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{5}a+5\,d{e}^{4}b+10\,{d}^{2}{e}^{3}c \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,d{e}^{4}a+10\,{d}^{2}{e}^{3}b+10\,{d}^{3}{e}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}a+10\,{d}^{3}{e}^{2}b+5\,{d}^{4}ec \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}a+5\,{d}^{4}eb+{d}^{5}c \right ){x}^{4}}{4}}+{\frac{ \left ( 5\,{d}^{4}ea+{d}^{5}b \right ){x}^{3}}{3}}+{\frac{{d}^{5}a{x}^{2}}{2}} \]

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 = 0.698748, size = 227, normalized size = 2.2 \[ \frac{1}{9} \, c e^{5} x^{9} + \frac{1}{8} \,{\left (5 \, c d e^{4} + b e^{5}\right )} x^{8} + \frac{1}{2} \, a d^{5} x^{2} + \frac{1}{7} \,{\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{7} + \frac{5}{6} \,{\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{6} +{\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{5} + 5 \, a d^{4} e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^5*x,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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Fricas [A]  time = 0.24059, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{5} c + \frac{5}{8} x^{8} e^{4} d c + \frac{1}{8} x^{8} e^{5} b + \frac{10}{7} x^{7} e^{3} d^{2} c + \frac{5}{7} x^{7} e^{4} d b + \frac{1}{7} x^{7} e^{5} a + \frac{5}{3} x^{6} e^{2} d^{3} c + \frac{5}{3} x^{6} e^{3} d^{2} b + \frac{5}{6} x^{6} e^{4} d a + x^{5} e d^{4} c + 2 x^{5} e^{2} d^{3} b + 2 x^{5} e^{3} d^{2} a + \frac{1}{4} x^{4} d^{5} c + \frac{5}{4} x^{4} e d^{4} b + \frac{5}{2} x^{4} e^{2} d^{3} a + \frac{1}{3} x^{3} d^{5} b + \frac{5}{3} x^{3} e d^{4} a + \frac{1}{2} x^{2} d^{5} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^5*x,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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Sympy [A]  time = 0.187343, size = 192, normalized size = 1.86 \[ \frac{a d^{5} x^{2}}{2} + \frac{c e^{5} x^{9}}{9} + x^{8} \left (\frac{b e^{5}}{8} + \frac{5 c d e^{4}}{8}\right ) + x^{7} \left (\frac{a e^{5}}{7} + \frac{5 b d e^{4}}{7} + \frac{10 c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{5 a d e^{4}}{6} + \frac{5 b d^{2} e^{3}}{3} + \frac{5 c d^{3} e^{2}}{3}\right ) + x^{5} \left (2 a d^{2} e^{3} + 2 b d^{3} e^{2} + c d^{4} e\right ) + x^{4} \left (\frac{5 a d^{3} e^{2}}{2} + \frac{5 b d^{4} e}{4} + \frac{c d^{5}}{4}\right ) + x^{3} \left (\frac{5 a d^{4} e}{3} + \frac{b d^{5}}{3}\right ) \]

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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GIAC/XCAS [A]  time = 0.260526, size = 239, normalized size = 2.32 \[ \frac{1}{9} \, c x^{9} e^{5} + \frac{5}{8} \, c d x^{8} e^{4} + \frac{10}{7} \, c d^{2} x^{7} e^{3} + \frac{5}{3} \, c d^{3} x^{6} e^{2} + c d^{4} x^{5} e + \frac{1}{4} \, c d^{5} x^{4} + \frac{1}{8} \, b x^{8} e^{5} + \frac{5}{7} \, b d x^{7} e^{4} + \frac{5}{3} \, b d^{2} x^{6} e^{3} + 2 \, b d^{3} x^{5} e^{2} + \frac{5}{4} \, b d^{4} x^{4} e + \frac{1}{3} \, b d^{5} x^{3} + \frac{1}{7} \, a x^{7} e^{5} + \frac{5}{6} \, a d x^{6} e^{4} + 2 \, a d^{2} x^{5} e^{3} + \frac{5}{2} \, a d^{3} x^{4} e^{2} + \frac{5}{3} \, a d^{4} x^{3} e + \frac{1}{2} \, a d^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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