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

Optimal. Leaf size=50 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{d \left (e+f x^4\right )^3}{12 f} \]

[Out]

(b*e^2*x^2)/2 + (b*e*f*x^6)/3 + (b*f^2*x^10)/10 + (d*(e + f*x^4)^3)/(12*f)

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Rubi [A]  time = 0.142103, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{d \left (e+f x^4\right )^3}{12 f} \]

Antiderivative was successfully verified.

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

[Out]

(b*e^2*x^2)/2 + (b*e*f*x^6)/3 + (b*f^2*x^10)/10 + (d*(e + f*x^4)^3)/(12*f)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} + \frac{b \int ^{x^{2}} e^{2}\, dx}{2} + \frac{d \left (e + f x^{4}\right )^{3}}{12 f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*e*f*x**6/3 + b*f**2*x**10/10 + b*Integral(e**2, (x, x**2))/2 + d*(e + f*x**4)*
*3/(12*f)

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Mathematica [A]  time = 0.00486726, size = 65, normalized size = 1.3 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

(b*e^2*x^2)/2 + (d*e^2*x^4)/4 + (b*e*f*x^6)/3 + (d*e*f*x^8)/4 + (b*f^2*x^10)/10
+ (d*f^2*x^12)/12

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Maple [A]  time = 0.001, size = 54, normalized size = 1.1 \[{\frac{d{f}^{2}{x}^{12}}{12}}+{\frac{b{f}^{2}{x}^{10}}{10}}+{\frac{def{x}^{8}}{4}}+{\frac{bef{x}^{6}}{3}}+{\frac{d{e}^{2}{x}^{4}}{4}}+{\frac{b{e}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+b*x)*(f*x^4+e)^2,x)

[Out]

1/12*d*f^2*x^12+1/10*b*f^2*x^10+1/4*d*e*f*x^8+1/3*b*e*f*x^6+1/4*d*e^2*x^4+1/2*b*
e^2*x^2

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Maxima [A]  time = 1.39247, size = 72, normalized size = 1.44 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{4} \, d e f x^{8} + \frac{1}{3} \, b e f x^{6} + \frac{1}{4} \, d e^{2} x^{4} + \frac{1}{2} \, b e^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*d*f^2*x^12 + 1/10*b*f^2*x^10 + 1/4*d*e*f*x^8 + 1/3*b*e*f*x^6 + 1/4*d*e^2*x^
4 + 1/2*b*e^2*x^2

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Fricas [A]  time = 0.201566, size = 1, normalized size = 0.02 \[ \frac{1}{12} x^{12} f^{2} d + \frac{1}{10} x^{10} f^{2} b + \frac{1}{4} x^{8} f e d + \frac{1}{3} x^{6} f e b + \frac{1}{4} x^{4} e^{2} d + \frac{1}{2} x^{2} e^{2} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x^12*f^2*d + 1/10*x^10*f^2*b + 1/4*x^8*f*e*d + 1/3*x^6*f*e*b + 1/4*x^4*e^2*
d + 1/2*x^2*e^2*b

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Sympy [A]  time = 0.060153, size = 60, normalized size = 1.2 \[ \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} + \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*e**2*x**2/2 + b*e*f*x**6/3 + b*f**2*x**10/10 + d*e**2*x**4/4 + d*e*f*x**8/4 +
d*f**2*x**12/12

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GIAC/XCAS [A]  time = 0.209465, size = 72, normalized size = 1.44 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{4} \, d f x^{8} e + \frac{1}{3} \, b f x^{6} e + \frac{1}{4} \, d x^{4} e^{2} + \frac{1}{2} \, b x^{2} e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*d*f^2*x^12 + 1/10*b*f^2*x^10 + 1/4*d*f*x^8*e + 1/3*b*f*x^6*e + 1/4*d*x^4*e^
2 + 1/2*b*x^2*e^2