3.610 \(\int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx\)

Optimal. Leaf size=55 \[ \frac{a f^3 (d+e x)^4}{4 e}+\frac{b f^3 (d+e x)^6}{6 e}+\frac{c f^3 (d+e x)^8}{8 e} \]

[Out]

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

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Rubi [A]  time = 0.122093, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{a f^3 (d+e x)^4}{4 e}+\frac{b f^3 (d+e x)^6}{6 e}+\frac{c f^3 (d+e x)^8}{8 e} \]

Antiderivative was successfully verified.

[In]  Int[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a f^{3} \int ^{\left (d + e x\right )^{2}} x\, dx}{2 e} + \frac{b f^{3} \left (d + e x\right )^{6}}{6 e} + \frac{c f^{3} \left (d + e x\right )^{8}}{8 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*f*x+d*f)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

a*f**3*Integral(x, (x, (d + e*x)**2))/(2*e) + b*f**3*(d + e*x)**6/(6*e) + c*f**3
*(d + e*x)**8/(8*e)

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Mathematica [B]  time = 0.0611145, size = 154, normalized size = 2.8 \[ f^3 \left (\frac{1}{4} e^3 x^4 \left (a+10 b d^2+35 c d^4\right )+\frac{1}{3} d e^2 x^3 \left (3 a+10 b d^2+21 c d^4\right )+\frac{1}{2} d^2 e x^2 \left (3 a+5 b d^2+7 c d^4\right )+d^3 x \left (a+b d^2+c d^4\right )+\frac{1}{6} e^5 x^6 \left (b+21 c d^2\right )+d e^4 x^5 \left (b+7 c d^2\right )+c d e^6 x^7+\frac{1}{8} c e^7 x^8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4),x]

[Out]

f^3*(d^3*(a + b*d^2 + c*d^4)*x + (d^2*(3*a + 5*b*d^2 + 7*c*d^4)*e*x^2)/2 + (d*(3
*a + 10*b*d^2 + 21*c*d^4)*e^2*x^3)/3 + ((a + 10*b*d^2 + 35*c*d^4)*e^3*x^4)/4 + d
*(b + 7*c*d^2)*e^4*x^5 + ((b + 21*c*d^2)*e^5*x^6)/6 + c*d*e^6*x^7 + (c*e^7*x^8)/
8)

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Maple [B]  time = 0.002, size = 349, normalized size = 6.4 \[{\frac{{e}^{7}{f}^{3}c{x}^{8}}{8}}+d{f}^{3}{e}^{6}c{x}^{7}+{\frac{ \left ( 15\,{d}^{2}{f}^{3}{e}^{5}c+{e}^{3}{f}^{3} \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{d}^{3}{f}^{3}c{e}^{4}+3\,d{f}^{3}{e}^{2} \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) +{e}^{3}{f}^{3} \left ( 4\,c{d}^{3}e+2\,bde \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{d}^{4}{f}^{3}c{e}^{3}+3\,{d}^{2}{f}^{3}e \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) +3\,d{f}^{3}{e}^{2} \left ( 4\,c{d}^{3}e+2\,bde \right ) +{e}^{3}{f}^{3} \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3}{f}^{3} \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ) +3\,{d}^{2}{f}^{3}e \left ( 4\,c{d}^{3}e+2\,bde \right ) +3\,d{f}^{3}{e}^{2} \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{3}{f}^{3} \left ( 4\,c{d}^{3}e+2\,bde \right ) +3\,{d}^{2}{f}^{3}e \left ( c{d}^{4}+b{d}^{2}+a \right ) \right ){x}^{2}}{2}}+{d}^{3}{f}^{3} \left ( c{d}^{4}+b{d}^{2}+a \right ) x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4),x)

[Out]

1/8*e^7*f^3*c*x^8+d*f^3*e^6*c*x^7+1/6*(15*d^2*f^3*e^5*c+e^3*f^3*(6*c*d^2*e^2+b*e
^2))*x^6+1/5*(13*d^3*f^3*c*e^4+3*d*f^3*e^2*(6*c*d^2*e^2+b*e^2)+e^3*f^3*(4*c*d^3*
e+2*b*d*e))*x^5+1/4*(4*d^4*f^3*c*e^3+3*d^2*f^3*e*(6*c*d^2*e^2+b*e^2)+3*d*f^3*e^2
*(4*c*d^3*e+2*b*d*e)+e^3*f^3*(c*d^4+b*d^2+a))*x^4+1/3*(d^3*f^3*(6*c*d^2*e^2+b*e^
2)+3*d^2*f^3*e*(4*c*d^3*e+2*b*d*e)+3*d*f^3*e^2*(c*d^4+b*d^2+a))*x^3+1/2*(d^3*f^3
*(4*c*d^3*e+2*b*d*e)+3*d^2*f^3*e*(c*d^4+b*d^2+a))*x^2+d^3*f^3*(c*d^4+b*d^2+a)*x

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Maxima [A]  time = 0.747397, size = 224, normalized size = 4.07 \[ \frac{1}{8} \, c e^{7} f^{3} x^{8} + c d e^{6} f^{3} x^{7} + \frac{1}{6} \,{\left (21 \, c d^{2} + b\right )} e^{5} f^{3} x^{6} +{\left (7 \, c d^{3} + b d\right )} e^{4} f^{3} x^{5} + \frac{1}{4} \,{\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} f^{3} x^{4} + \frac{1}{3} \,{\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} f^{3} x^{3} + \frac{1}{2} \,{\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e f^{3} x^{2} +{\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*f*x + d*f)^3,x, algorithm="maxima")

[Out]

1/8*c*e^7*f^3*x^8 + c*d*e^6*f^3*x^7 + 1/6*(21*c*d^2 + b)*e^5*f^3*x^6 + (7*c*d^3
+ b*d)*e^4*f^3*x^5 + 1/4*(35*c*d^4 + 10*b*d^2 + a)*e^3*f^3*x^4 + 1/3*(21*c*d^5 +
10*b*d^3 + 3*a*d)*e^2*f^3*x^3 + 1/2*(7*c*d^6 + 5*b*d^4 + 3*a*d^2)*e*f^3*x^2 + (
c*d^7 + b*d^5 + a*d^3)*f^3*x

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Fricas [A]  time = 0.242006, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} f^{3} e^{7} c + x^{7} f^{3} e^{6} d c + \frac{7}{2} x^{6} f^{3} e^{5} d^{2} c + 7 x^{5} f^{3} e^{4} d^{3} c + \frac{35}{4} x^{4} f^{3} e^{3} d^{4} c + \frac{1}{6} x^{6} f^{3} e^{5} b + 7 x^{3} f^{3} e^{2} d^{5} c + x^{5} f^{3} e^{4} d b + \frac{7}{2} x^{2} f^{3} e d^{6} c + \frac{5}{2} x^{4} f^{3} e^{3} d^{2} b + x f^{3} d^{7} c + \frac{10}{3} x^{3} f^{3} e^{2} d^{3} b + \frac{5}{2} x^{2} f^{3} e d^{4} b + \frac{1}{4} x^{4} f^{3} e^{3} a + x f^{3} d^{5} b + x^{3} f^{3} e^{2} d a + \frac{3}{2} x^{2} f^{3} e d^{2} a + x f^{3} d^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*f*x + d*f)^3,x, algorithm="fricas")

[Out]

1/8*x^8*f^3*e^7*c + x^7*f^3*e^6*d*c + 7/2*x^6*f^3*e^5*d^2*c + 7*x^5*f^3*e^4*d^3*
c + 35/4*x^4*f^3*e^3*d^4*c + 1/6*x^6*f^3*e^5*b + 7*x^3*f^3*e^2*d^5*c + x^5*f^3*e
^4*d*b + 7/2*x^2*f^3*e*d^6*c + 5/2*x^4*f^3*e^3*d^2*b + x*f^3*d^7*c + 10/3*x^3*f^
3*e^2*d^3*b + 5/2*x^2*f^3*e*d^4*b + 1/4*x^4*f^3*e^3*a + x*f^3*d^5*b + x^3*f^3*e^
2*d*a + 3/2*x^2*f^3*e*d^2*a + x*f^3*d^3*a

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Sympy [A]  time = 0.231997, size = 240, normalized size = 4.36 \[ c d e^{6} f^{3} x^{7} + \frac{c e^{7} f^{3} x^{8}}{8} + x^{6} \left (\frac{b e^{5} f^{3}}{6} + \frac{7 c d^{2} e^{5} f^{3}}{2}\right ) + x^{5} \left (b d e^{4} f^{3} + 7 c d^{3} e^{4} f^{3}\right ) + x^{4} \left (\frac{a e^{3} f^{3}}{4} + \frac{5 b d^{2} e^{3} f^{3}}{2} + \frac{35 c d^{4} e^{3} f^{3}}{4}\right ) + x^{3} \left (a d e^{2} f^{3} + \frac{10 b d^{3} e^{2} f^{3}}{3} + 7 c d^{5} e^{2} f^{3}\right ) + x^{2} \left (\frac{3 a d^{2} e f^{3}}{2} + \frac{5 b d^{4} e f^{3}}{2} + \frac{7 c d^{6} e f^{3}}{2}\right ) + x \left (a d^{3} f^{3} + b d^{5} f^{3} + c d^{7} f^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*f*x+d*f)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

c*d*e**6*f**3*x**7 + c*e**7*f**3*x**8/8 + x**6*(b*e**5*f**3/6 + 7*c*d**2*e**5*f*
*3/2) + x**5*(b*d*e**4*f**3 + 7*c*d**3*e**4*f**3) + x**4*(a*e**3*f**3/4 + 5*b*d*
*2*e**3*f**3/2 + 35*c*d**4*e**3*f**3/4) + x**3*(a*d*e**2*f**3 + 10*b*d**3*e**2*f
**3/3 + 7*c*d**5*e**2*f**3) + x**2*(3*a*d**2*e*f**3/2 + 5*b*d**4*e*f**3/2 + 7*c*
d**6*e*f**3/2) + x*(a*d**3*f**3 + b*d**5*f**3 + c*d**7*f**3)

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GIAC/XCAS [A]  time = 0.270184, size = 297, normalized size = 5.4 \[ \frac{1}{8} \, c f^{3} x^{8} e^{7} + c d f^{3} x^{7} e^{6} + \frac{7}{2} \, c d^{2} f^{3} x^{6} e^{5} + 7 \, c d^{3} f^{3} x^{5} e^{4} + \frac{35}{4} \, c d^{4} f^{3} x^{4} e^{3} + 7 \, c d^{5} f^{3} x^{3} e^{2} + \frac{7}{2} \, c d^{6} f^{3} x^{2} e + c d^{7} f^{3} x + \frac{1}{6} \, b f^{3} x^{6} e^{5} + b d f^{3} x^{5} e^{4} + \frac{5}{2} \, b d^{2} f^{3} x^{4} e^{3} + \frac{10}{3} \, b d^{3} f^{3} x^{3} e^{2} + \frac{5}{2} \, b d^{4} f^{3} x^{2} e + b d^{5} f^{3} x + \frac{1}{4} \, a f^{3} x^{4} e^{3} + a d f^{3} x^{3} e^{2} + \frac{3}{2} \, a d^{2} f^{3} x^{2} e + a d^{3} f^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((e*x + d)^4*c + (e*x + d)^2*b + a)*(e*f*x + d*f)^3,x, algorithm="giac")

[Out]

1/8*c*f^3*x^8*e^7 + c*d*f^3*x^7*e^6 + 7/2*c*d^2*f^3*x^6*e^5 + 7*c*d^3*f^3*x^5*e^
4 + 35/4*c*d^4*f^3*x^4*e^3 + 7*c*d^5*f^3*x^3*e^2 + 7/2*c*d^6*f^3*x^2*e + c*d^7*f
^3*x + 1/6*b*f^3*x^6*e^5 + b*d*f^3*x^5*e^4 + 5/2*b*d^2*f^3*x^4*e^3 + 10/3*b*d^3*
f^3*x^3*e^2 + 5/2*b*d^4*f^3*x^2*e + b*d^5*f^3*x + 1/4*a*f^3*x^4*e^3 + a*d*f^3*x^
3*e^2 + 3/2*a*d^2*f^3*x^2*e + a*d^3*f^3*x