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

Optimal. Leaf size=89 \[ \frac{5}{7} a^2 c^3 d x^{14}+a^3 c^2 d x^{10}+\frac{5}{6} a^4 c d x^6+\frac{1}{2} a^5 d x^2+\frac{5}{18} a c^4 d x^{18}+\frac{e \left (a+c x^4\right )^6}{24 c}+\frac{1}{22} c^5 d x^{22} \]

[Out]

(a^5*d*x^2)/2 + (5*a^4*c*d*x^6)/6 + a^3*c^2*d*x^10 + (5*a^2*c^3*d*x^14)/7 + (5*a*c^4*d*x^18)/18 + (c^5*d*x^22)
/22 + (e*(a + c*x^4)^6)/(24*c)

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Rubi [A]  time = 0.0770721, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1248, 641, 194} \[ \frac{5}{7} a^2 c^3 d x^{14}+a^3 c^2 d x^{10}+\frac{5}{6} a^4 c d x^6+\frac{1}{2} a^5 d x^2+\frac{5}{18} a c^4 d x^{18}+\frac{e \left (a+c x^4\right )^6}{24 c}+\frac{1}{22} c^5 d x^{22} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x^2)*(a + c*x^4)^5,x]

[Out]

(a^5*d*x^2)/2 + (5*a^4*c*d*x^6)/6 + a^3*c^2*d*x^10 + (5*a^2*c^3*d*x^14)/7 + (5*a*c^4*d*x^18)/18 + (c^5*d*x^22)
/22 + (e*(a + c*x^4)^6)/(24*c)

Rule 1248

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (d+e x) \left (a+c x^2\right )^5 \, dx,x,x^2\right )\\ &=\frac{e \left (a+c x^4\right )^6}{24 c}+\frac{1}{2} d \operatorname{Subst}\left (\int \left (a+c x^2\right )^5 \, dx,x,x^2\right )\\ &=\frac{e \left (a+c x^4\right )^6}{24 c}+\frac{1}{2} d \operatorname{Subst}\left (\int \left (a^5+5 a^4 c x^2+10 a^3 c^2 x^4+10 a^2 c^3 x^6+5 a c^4 x^8+c^5 x^{10}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} a^5 d x^2+\frac{5}{6} a^4 c d x^6+a^3 c^2 d x^{10}+\frac{5}{7} a^2 c^3 d x^{14}+\frac{5}{18} a c^4 d x^{18}+\frac{1}{22} c^5 d x^{22}+\frac{e \left (a+c x^4\right )^6}{24 c}\\ \end{align*}

Mathematica [A]  time = 0.0039613, size = 146, normalized size = 1.64 \[ \frac{5}{7} a^2 c^3 d x^{14}+a^3 c^2 d x^{10}+\frac{5}{8} a^2 c^3 e x^{16}+\frac{5}{6} a^3 c^2 e x^{12}+\frac{5}{6} a^4 c d x^6+\frac{5}{8} a^4 c e x^8+\frac{1}{2} a^5 d x^2+\frac{1}{4} a^5 e x^4+\frac{5}{18} a c^4 d x^{18}+\frac{1}{4} a c^4 e x^{20}+\frac{1}{22} c^5 d x^{22}+\frac{1}{24} c^5 e x^{24} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 125, normalized size = 1.4 \begin{align*}{\frac{{c}^{5}e{x}^{24}}{24}}+{\frac{{c}^{5}d{x}^{22}}{22}}+{\frac{a{c}^{4}e{x}^{20}}{4}}+{\frac{5\,a{c}^{4}d{x}^{18}}{18}}+{\frac{5\,{a}^{2}{c}^{3}e{x}^{16}}{8}}+{\frac{5\,{a}^{2}{c}^{3}d{x}^{14}}{7}}+{\frac{5\,{a}^{3}{c}^{2}e{x}^{12}}{6}}+{a}^{3}{c}^{2}d{x}^{10}+{\frac{5\,{a}^{4}ce{x}^{8}}{8}}+{\frac{5\,{a}^{4}cd{x}^{6}}{6}}+{\frac{{a}^{5}e{x}^{4}}{4}}+{\frac{{a}^{5}d{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.970514, size = 167, normalized size = 1.88 \begin{align*} \frac{1}{24} \, c^{5} e x^{24} + \frac{1}{22} \, c^{5} d x^{22} + \frac{1}{4} \, a c^{4} e x^{20} + \frac{5}{18} \, a c^{4} d x^{18} + \frac{5}{8} \, a^{2} c^{3} e x^{16} + \frac{5}{7} \, a^{2} c^{3} d x^{14} + \frac{5}{6} \, a^{3} c^{2} e x^{12} + a^{3} c^{2} d x^{10} + \frac{5}{8} \, a^{4} c e x^{8} + \frac{5}{6} \, a^{4} c d x^{6} + \frac{1}{4} \, a^{5} e x^{4} + \frac{1}{2} \, a^{5} d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.21557, size = 300, normalized size = 3.37 \begin{align*} \frac{1}{24} x^{24} e c^{5} + \frac{1}{22} x^{22} d c^{5} + \frac{1}{4} x^{20} e c^{4} a + \frac{5}{18} x^{18} d c^{4} a + \frac{5}{8} x^{16} e c^{3} a^{2} + \frac{5}{7} x^{14} d c^{3} a^{2} + \frac{5}{6} x^{12} e c^{2} a^{3} + x^{10} d c^{2} a^{3} + \frac{5}{8} x^{8} e c a^{4} + \frac{5}{6} x^{6} d c a^{4} + \frac{1}{4} x^{4} e a^{5} + \frac{1}{2} x^{2} d a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.082939, size = 150, normalized size = 1.69 \begin{align*} \frac{a^{5} d x^{2}}{2} + \frac{a^{5} e x^{4}}{4} + \frac{5 a^{4} c d x^{6}}{6} + \frac{5 a^{4} c e x^{8}}{8} + a^{3} c^{2} d x^{10} + \frac{5 a^{3} c^{2} e x^{12}}{6} + \frac{5 a^{2} c^{3} d x^{14}}{7} + \frac{5 a^{2} c^{3} e x^{16}}{8} + \frac{5 a c^{4} d x^{18}}{18} + \frac{a c^{4} e x^{20}}{4} + \frac{c^{5} d x^{22}}{22} + \frac{c^{5} e x^{24}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15953, size = 176, normalized size = 1.98 \begin{align*} \frac{1}{24} \, c^{5} x^{24} e + \frac{1}{22} \, c^{5} d x^{22} + \frac{1}{4} \, a c^{4} x^{20} e + \frac{5}{18} \, a c^{4} d x^{18} + \frac{5}{8} \, a^{2} c^{3} x^{16} e + \frac{5}{7} \, a^{2} c^{3} d x^{14} + \frac{5}{6} \, a^{3} c^{2} x^{12} e + a^{3} c^{2} d x^{10} + \frac{5}{8} \, a^{4} c x^{8} e + \frac{5}{6} \, a^{4} c d x^{6} + \frac{1}{4} \, a^{5} x^{4} e + \frac{1}{2} \, a^{5} d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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