3.1 \(\int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx\)

Optimal. Leaf size=149 \[ \frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26} \]

[Out]

(a^5*d*x^4)/4 + (a^5*e*x^6)/6 + (5*a^4*c*d*x^8)/8 + (a^4*c*e*x^10)/2 + (5*a^3*c^
2*d*x^12)/6 + (5*a^3*c^2*e*x^14)/7 + (5*a^2*c^3*d*x^16)/8 + (5*a^2*c^3*e*x^18)/9
+ (a*c^4*d*x^20)/4 + (5*a*c^4*e*x^22)/22 + (c^5*d*x^24)/24 + (c^5*e*x^26)/26

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Rubi [A]  time = 0.461615, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(d + e*x^2)*(a + c*x^4)^5,x]

[Out]

(a^5*d*x^4)/4 + (a^5*e*x^6)/6 + (5*a^4*c*d*x^8)/8 + (a^4*c*e*x^10)/2 + (5*a^3*c^
2*d*x^12)/6 + (5*a^3*c^2*e*x^14)/7 + (5*a^2*c^3*d*x^16)/8 + (5*a^2*c^3*e*x^18)/9
+ (a*c^4*d*x^20)/4 + (5*a*c^4*e*x^22)/22 + (c^5*d*x^24)/24 + (c^5*e*x^26)/26

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5} d \int ^{x^{2}} x\, dx}{2} + \frac{a^{5} e x^{6}}{6} + \frac{5 a^{4} c d x^{8}}{8} + \frac{a^{4} c e x^{10}}{2} + \frac{5 a^{3} c^{2} d x^{12}}{6} + \frac{5 a^{3} c^{2} e x^{14}}{7} + \frac{5 a^{2} c^{3} d x^{16}}{8} + \frac{5 a^{2} c^{3} e x^{18}}{9} + \frac{a c^{4} d x^{20}}{4} + \frac{5 a c^{4} e x^{22}}{22} + \frac{c^{5} d x^{24}}{24} + \frac{c^{5} e x^{26}}{26} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x**2+d)*(c*x**4+a)**5,x)

[Out]

a**5*d*Integral(x, (x, x**2))/2 + a**5*e*x**6/6 + 5*a**4*c*d*x**8/8 + a**4*c*e*x
**10/2 + 5*a**3*c**2*d*x**12/6 + 5*a**3*c**2*e*x**14/7 + 5*a**2*c**3*d*x**16/8 +
5*a**2*c**3*e*x**18/9 + a*c**4*d*x**20/4 + 5*a*c**4*e*x**22/22 + c**5*d*x**24/2
4 + c**5*e*x**26/26

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Mathematica [A]  time = 0.00782934, size = 149, normalized size = 1. \[ \frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(d + e*x^2)*(a + c*x^4)^5,x]

[Out]

(a^5*d*x^4)/4 + (a^5*e*x^6)/6 + (5*a^4*c*d*x^8)/8 + (a^4*c*e*x^10)/2 + (5*a^3*c^
2*d*x^12)/6 + (5*a^3*c^2*e*x^14)/7 + (5*a^2*c^3*d*x^16)/8 + (5*a^2*c^3*e*x^18)/9
+ (a*c^4*d*x^20)/4 + (5*a*c^4*e*x^22)/22 + (c^5*d*x^24)/24 + (c^5*e*x^26)/26

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Maple [A]  time = 0.004, size = 126, normalized size = 0.9 \[{\frac{{a}^{5}d{x}^{4}}{4}}+{\frac{{a}^{5}e{x}^{6}}{6}}+{\frac{5\,{a}^{4}cd{x}^{8}}{8}}+{\frac{{a}^{4}ce{x}^{10}}{2}}+{\frac{5\,{a}^{3}{c}^{2}d{x}^{12}}{6}}+{\frac{5\,{a}^{3}{c}^{2}e{x}^{14}}{7}}+{\frac{5\,{a}^{2}{c}^{3}d{x}^{16}}{8}}+{\frac{5\,{a}^{2}{c}^{3}e{x}^{18}}{9}}+{\frac{a{c}^{4}d{x}^{20}}{4}}+{\frac{5\,a{c}^{4}e{x}^{22}}{22}}+{\frac{{c}^{5}d{x}^{24}}{24}}+{\frac{{c}^{5}e{x}^{26}}{26}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x^2+d)*(c*x^4+a)^5,x)

[Out]

1/4*a^5*d*x^4+1/6*a^5*e*x^6+5/8*a^4*c*d*x^8+1/2*a^4*c*e*x^10+5/6*a^3*c^2*d*x^12+
5/7*a^3*c^2*e*x^14+5/8*a^2*c^3*d*x^16+5/9*a^2*c^3*e*x^18+1/4*a*c^4*d*x^20+5/22*a
*c^4*e*x^22+1/24*c^5*d*x^24+1/26*c^5*e*x^26

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Maxima [A]  time = 0.702379, size = 169, normalized size = 1.13 \[ \frac{1}{26} \, c^{5} e x^{26} + \frac{1}{24} \, c^{5} d x^{24} + \frac{5}{22} \, a c^{4} e x^{22} + \frac{1}{4} \, a c^{4} d x^{20} + \frac{5}{9} \, a^{2} c^{3} e x^{18} + \frac{5}{8} \, a^{2} c^{3} d x^{16} + \frac{5}{7} \, a^{3} c^{2} e x^{14} + \frac{5}{6} \, a^{3} c^{2} d x^{12} + \frac{1}{2} \, a^{4} c e x^{10} + \frac{5}{8} \, a^{4} c d x^{8} + \frac{1}{6} \, a^{5} e x^{6} + \frac{1}{4} \, a^{5} d x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^5*(e*x^2 + d)*x^3,x, algorithm="maxima")

[Out]

1/26*c^5*e*x^26 + 1/24*c^5*d*x^24 + 5/22*a*c^4*e*x^22 + 1/4*a*c^4*d*x^20 + 5/9*a
^2*c^3*e*x^18 + 5/8*a^2*c^3*d*x^16 + 5/7*a^3*c^2*e*x^14 + 5/6*a^3*c^2*d*x^12 + 1
/2*a^4*c*e*x^10 + 5/8*a^4*c*d*x^8 + 1/6*a^5*e*x^6 + 1/4*a^5*d*x^4

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Fricas [A]  time = 0.229426, size = 1, normalized size = 0.01 \[ \frac{1}{26} x^{26} e c^{5} + \frac{1}{24} x^{24} d c^{5} + \frac{5}{22} x^{22} e c^{4} a + \frac{1}{4} x^{20} d c^{4} a + \frac{5}{9} x^{18} e c^{3} a^{2} + \frac{5}{8} x^{16} d c^{3} a^{2} + \frac{5}{7} x^{14} e c^{2} a^{3} + \frac{5}{6} x^{12} d c^{2} a^{3} + \frac{1}{2} x^{10} e c a^{4} + \frac{5}{8} x^{8} d c a^{4} + \frac{1}{6} x^{6} e a^{5} + \frac{1}{4} x^{4} d a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^5*(e*x^2 + d)*x^3,x, algorithm="fricas")

[Out]

1/26*x^26*e*c^5 + 1/24*x^24*d*c^5 + 5/22*x^22*e*c^4*a + 1/4*x^20*d*c^4*a + 5/9*x
^18*e*c^3*a^2 + 5/8*x^16*d*c^3*a^2 + 5/7*x^14*e*c^2*a^3 + 5/6*x^12*d*c^2*a^3 + 1
/2*x^10*e*c*a^4 + 5/8*x^8*d*c*a^4 + 1/6*x^6*e*a^5 + 1/4*x^4*d*a^5

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Sympy [A]  time = 0.161928, size = 151, normalized size = 1.01 \[ \frac{a^{5} d x^{4}}{4} + \frac{a^{5} e x^{6}}{6} + \frac{5 a^{4} c d x^{8}}{8} + \frac{a^{4} c e x^{10}}{2} + \frac{5 a^{3} c^{2} d x^{12}}{6} + \frac{5 a^{3} c^{2} e x^{14}}{7} + \frac{5 a^{2} c^{3} d x^{16}}{8} + \frac{5 a^{2} c^{3} e x^{18}}{9} + \frac{a c^{4} d x^{20}}{4} + \frac{5 a c^{4} e x^{22}}{22} + \frac{c^{5} d x^{24}}{24} + \frac{c^{5} e x^{26}}{26} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x**2+d)*(c*x**4+a)**5,x)

[Out]

a**5*d*x**4/4 + a**5*e*x**6/6 + 5*a**4*c*d*x**8/8 + a**4*c*e*x**10/2 + 5*a**3*c*
*2*d*x**12/6 + 5*a**3*c**2*e*x**14/7 + 5*a**2*c**3*d*x**16/8 + 5*a**2*c**3*e*x**
18/9 + a*c**4*d*x**20/4 + 5*a*c**4*e*x**22/22 + c**5*d*x**24/24 + c**5*e*x**26/2
6

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GIAC/XCAS [A]  time = 0.260016, size = 177, normalized size = 1.19 \[ \frac{1}{26} \, c^{5} x^{26} e + \frac{1}{24} \, c^{5} d x^{24} + \frac{5}{22} \, a c^{4} x^{22} e + \frac{1}{4} \, a c^{4} d x^{20} + \frac{5}{9} \, a^{2} c^{3} x^{18} e + \frac{5}{8} \, a^{2} c^{3} d x^{16} + \frac{5}{7} \, a^{3} c^{2} x^{14} e + \frac{5}{6} \, a^{3} c^{2} d x^{12} + \frac{1}{2} \, a^{4} c x^{10} e + \frac{5}{8} \, a^{4} c d x^{8} + \frac{1}{6} \, a^{5} x^{6} e + \frac{1}{4} \, a^{5} d x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^5*(e*x^2 + d)*x^3,x, algorithm="giac")

[Out]

1/26*c^5*x^26*e + 1/24*c^5*d*x^24 + 5/22*a*c^4*x^22*e + 1/4*a*c^4*d*x^20 + 5/9*a
^2*c^3*x^18*e + 5/8*a^2*c^3*d*x^16 + 5/7*a^3*c^2*x^14*e + 5/6*a^3*c^2*d*x^12 + 1
/2*a^4*c*x^10*e + 5/8*a^4*c*d*x^8 + 1/6*a^5*x^6*e + 1/4*a^5*d*x^4