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

Optimal. Leaf size=149 \[ \frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25} \]

[Out]

(a^5*d*x^3)/3 + (a^5*e*x^5)/5 + (5*a^4*c*d*x^7)/7 + (5*a^4*c*e*x^9)/9 + (10*a^3*c^2*d*x^11)/11 + (10*a^3*c^2*e
*x^13)/13 + (2*a^2*c^3*d*x^15)/3 + (10*a^2*c^3*e*x^17)/17 + (5*a*c^4*d*x^19)/19 + (5*a*c^4*e*x^21)/21 + (c^5*d
*x^23)/23 + (c^5*e*x^25)/25

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Rubi [A]  time = 0.0977514, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1262} \[ \frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + e*x^2)*(a + c*x^4)^5,x]

[Out]

(a^5*d*x^3)/3 + (a^5*e*x^5)/5 + (5*a^4*c*d*x^7)/7 + (5*a^4*c*e*x^9)/9 + (10*a^3*c^2*d*x^11)/11 + (10*a^3*c^2*e
*x^13)/13 + (2*a^2*c^3*d*x^15)/3 + (10*a^2*c^3*e*x^17)/17 + (5*a*c^4*d*x^19)/19 + (5*a*c^4*e*x^21)/21 + (c^5*d
*x^23)/23 + (c^5*e*x^25)/25

Rule 1262

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx &=\int \left (a^5 d x^2+a^5 e x^4+5 a^4 c d x^6+5 a^4 c e x^8+10 a^3 c^2 d x^{10}+10 a^3 c^2 e x^{12}+10 a^2 c^3 d x^{14}+10 a^2 c^3 e x^{16}+5 a c^4 d x^{18}+5 a c^4 e x^{20}+c^5 d x^{22}+c^5 e x^{24}\right ) \, dx\\ &=\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25}\\ \end{align*}

Mathematica [A]  time = 0.0041141, size = 149, normalized size = 1. \[ \frac{2}{3} a^2 c^3 d x^{15}+\frac{10}{11} a^3 c^2 d x^{11}+\frac{10}{17} a^2 c^3 e x^{17}+\frac{10}{13} a^3 c^2 e x^{13}+\frac{5}{7} a^4 c d x^7+\frac{5}{9} a^4 c e x^9+\frac{1}{3} a^5 d x^3+\frac{1}{5} a^5 e x^5+\frac{5}{19} a c^4 d x^{19}+\frac{5}{21} a c^4 e x^{21}+\frac{1}{23} c^5 d x^{23}+\frac{1}{25} c^5 e x^{25} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*d*x^3)/3 + (a^5*e*x^5)/5 + (5*a^4*c*d*x^7)/7 + (5*a^4*c*e*x^9)/9 + (10*a^3*c^2*d*x^11)/11 + (10*a^3*c^2*e
*x^13)/13 + (2*a^2*c^3*d*x^15)/3 + (10*a^2*c^3*e*x^17)/17 + (5*a*c^4*d*x^19)/19 + (5*a*c^4*e*x^21)/21 + (c^5*d
*x^23)/23 + (c^5*e*x^25)/25

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Maple [A]  time = 0.001, size = 126, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}d{x}^{3}}{3}}+{\frac{{a}^{5}e{x}^{5}}{5}}+{\frac{5\,{a}^{4}cd{x}^{7}}{7}}+{\frac{5\,{a}^{4}ce{x}^{9}}{9}}+{\frac{10\,{a}^{3}{c}^{2}d{x}^{11}}{11}}+{\frac{10\,{a}^{3}{c}^{2}e{x}^{13}}{13}}+{\frac{2\,{a}^{2}{c}^{3}d{x}^{15}}{3}}+{\frac{10\,{a}^{2}{c}^{3}e{x}^{17}}{17}}+{\frac{5\,a{c}^{4}d{x}^{19}}{19}}+{\frac{5\,a{c}^{4}e{x}^{21}}{21}}+{\frac{{c}^{5}d{x}^{23}}{23}}+{\frac{{c}^{5}e{x}^{25}}{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^5*d*x^3+1/5*a^5*e*x^5+5/7*a^4*c*d*x^7+5/9*a^4*c*e*x^9+10/11*a^3*c^2*d*x^11+10/13*a^3*c^2*e*x^13+2/3*a^2*
c^3*d*x^15+10/17*a^2*c^3*e*x^17+5/19*a*c^4*d*x^19+5/21*a*c^4*e*x^21+1/23*c^5*d*x^23+1/25*c^5*e*x^25

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Maxima [A]  time = 0.952864, size = 169, normalized size = 1.13 \begin{align*} \frac{1}{25} \, c^{5} e x^{25} + \frac{1}{23} \, c^{5} d x^{23} + \frac{5}{21} \, a c^{4} e x^{21} + \frac{5}{19} \, a c^{4} d x^{19} + \frac{10}{17} \, a^{2} c^{3} e x^{17} + \frac{2}{3} \, a^{2} c^{3} d x^{15} + \frac{10}{13} \, a^{3} c^{2} e x^{13} + \frac{10}{11} \, a^{3} c^{2} d x^{11} + \frac{5}{9} \, a^{4} c e x^{9} + \frac{5}{7} \, a^{4} c d x^{7} + \frac{1}{5} \, a^{5} e x^{5} + \frac{1}{3} \, a^{5} d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*c^5*e*x^25 + 1/23*c^5*d*x^23 + 5/21*a*c^4*e*x^21 + 5/19*a*c^4*d*x^19 + 10/17*a^2*c^3*e*x^17 + 2/3*a^2*c^3
*d*x^15 + 10/13*a^3*c^2*e*x^13 + 10/11*a^3*c^2*d*x^11 + 5/9*a^4*c*e*x^9 + 5/7*a^4*c*d*x^7 + 1/5*a^5*e*x^5 + 1/
3*a^5*d*x^3

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Fricas [A]  time = 1.27712, size = 315, normalized size = 2.11 \begin{align*} \frac{1}{25} x^{25} e c^{5} + \frac{1}{23} x^{23} d c^{5} + \frac{5}{21} x^{21} e c^{4} a + \frac{5}{19} x^{19} d c^{4} a + \frac{10}{17} x^{17} e c^{3} a^{2} + \frac{2}{3} x^{15} d c^{3} a^{2} + \frac{10}{13} x^{13} e c^{2} a^{3} + \frac{10}{11} x^{11} d c^{2} a^{3} + \frac{5}{9} x^{9} e c a^{4} + \frac{5}{7} x^{7} d c a^{4} + \frac{1}{5} x^{5} e a^{5} + \frac{1}{3} x^{3} d a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^25*e*c^5 + 1/23*x^23*d*c^5 + 5/21*x^21*e*c^4*a + 5/19*x^19*d*c^4*a + 10/17*x^17*e*c^3*a^2 + 2/3*x^15*d*
c^3*a^2 + 10/13*x^13*e*c^2*a^3 + 10/11*x^11*d*c^2*a^3 + 5/9*x^9*e*c*a^4 + 5/7*x^7*d*c*a^4 + 1/5*x^5*e*a^5 + 1/
3*x^3*d*a^5

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Sympy [A]  time = 0.086129, size = 155, normalized size = 1.04 \begin{align*} \frac{a^{5} d x^{3}}{3} + \frac{a^{5} e x^{5}}{5} + \frac{5 a^{4} c d x^{7}}{7} + \frac{5 a^{4} c e x^{9}}{9} + \frac{10 a^{3} c^{2} d x^{11}}{11} + \frac{10 a^{3} c^{2} e x^{13}}{13} + \frac{2 a^{2} c^{3} d x^{15}}{3} + \frac{10 a^{2} c^{3} e x^{17}}{17} + \frac{5 a c^{4} d x^{19}}{19} + \frac{5 a c^{4} e x^{21}}{21} + \frac{c^{5} d x^{23}}{23} + \frac{c^{5} e x^{25}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d*x**3/3 + a**5*e*x**5/5 + 5*a**4*c*d*x**7/7 + 5*a**4*c*e*x**9/9 + 10*a**3*c**2*d*x**11/11 + 10*a**3*c**2
*e*x**13/13 + 2*a**2*c**3*d*x**15/3 + 10*a**2*c**3*e*x**17/17 + 5*a*c**4*d*x**19/19 + 5*a*c**4*e*x**21/21 + c*
*5*d*x**23/23 + c**5*e*x**25/25

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Giac [A]  time = 1.16342, size = 177, normalized size = 1.19 \begin{align*} \frac{1}{25} \, c^{5} x^{25} e + \frac{1}{23} \, c^{5} d x^{23} + \frac{5}{21} \, a c^{4} x^{21} e + \frac{5}{19} \, a c^{4} d x^{19} + \frac{10}{17} \, a^{2} c^{3} x^{17} e + \frac{2}{3} \, a^{2} c^{3} d x^{15} + \frac{10}{13} \, a^{3} c^{2} x^{13} e + \frac{10}{11} \, a^{3} c^{2} d x^{11} + \frac{5}{9} \, a^{4} c x^{9} e + \frac{5}{7} \, a^{4} c d x^{7} + \frac{1}{5} \, a^{5} x^{5} e + \frac{1}{3} \, a^{5} d x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*c^5*x^25*e + 1/23*c^5*d*x^23 + 5/21*a*c^4*x^21*e + 5/19*a*c^4*d*x^19 + 10/17*a^2*c^3*x^17*e + 2/3*a^2*c^3
*d*x^15 + 10/13*a^3*c^2*x^13*e + 10/11*a^3*c^2*d*x^11 + 5/9*a^4*c*x^9*e + 5/7*a^4*c*d*x^7 + 1/5*a^5*x^5*e + 1/
3*a^5*d*x^3