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

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

[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^1
4)/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 = 0.219689, 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, Rules used = {1252, 766} \[ \frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26} \]

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^1
4)/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

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (d+e x) \left (a+c x^2\right )^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^5 d x+a^5 e x^2+5 a^4 c d x^3+5 a^4 c e x^4+10 a^3 c^2 d x^5+10 a^3 c^2 e x^6+10 a^2 c^3 d x^7+10 a^2 c^3 e x^8+5 a c^4 d x^9+5 a c^4 e x^{10}+c^5 d x^{11}+c^5 e x^{12}\right ) \, dx,x,x^2\right )\\ &=\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}\\ \end{align*}

Mathematica [A]  time = 0.0051757, size = 149, normalized size = 1. \[ \frac{5}{8} a^2 c^3 d x^{16}+\frac{5}{6} a^3 c^2 d x^{12}+\frac{5}{9} a^2 c^3 e x^{18}+\frac{5}{7} a^3 c^2 e x^{14}+\frac{5}{8} a^4 c d x^8+\frac{1}{2} a^4 c e x^{10}+\frac{1}{4} a^5 d x^4+\frac{1}{6} a^5 e x^6+\frac{1}{4} a c^4 d x^{20}+\frac{5}{22} a c^4 e x^{22}+\frac{1}{24} c^5 d x^{24}+\frac{1}{26} c^5 e x^{26} \]

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^1
4)/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 = 0.01, size = 126, normalized size = 0.9 \begin{align*}{\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}} \end{align*}

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 = 0.965705, size = 169, normalized size = 1.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(c*x^4+a)^5,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 = 1.7152, size = 306, normalized size = 2.05 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(c*x^4+a)^5,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 = 0.084633, size = 151, normalized size = 1.01 \begin{align*} \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} \end{align*}

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**26/26

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Giac [A]  time = 1.12323, size = 177, normalized size = 1.19 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(c*x^4+a)^5,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