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

Optimal. Leaf size=141 \[ \frac{10}{13} a^2 c^3 d x^{13}+\frac{10}{9} a^3 c^2 d x^9+\frac{2}{3} a^2 c^3 e x^{15}+\frac{10}{11} a^3 c^2 e x^{11}+a^4 c d x^5+\frac{5}{7} a^4 c e x^7+a^5 d x+\frac{1}{3} a^5 e x^3+\frac{5}{17} a c^4 d x^{17}+\frac{5}{19} a c^4 e x^{19}+\frac{1}{21} c^5 d x^{21}+\frac{1}{23} c^5 e x^{23} \]

[Out]

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

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Rubi [A]  time = 0.0819144, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {1154} \[ \frac{10}{13} a^2 c^3 d x^{13}+\frac{10}{9} a^3 c^2 d x^9+\frac{2}{3} a^2 c^3 e x^{15}+\frac{10}{11} a^3 c^2 e x^{11}+a^4 c d x^5+\frac{5}{7} a^4 c e x^7+a^5 d x+\frac{1}{3} a^5 e x^3+\frac{5}{17} a c^4 d x^{17}+\frac{5}{19} a c^4 e x^{19}+\frac{1}{21} c^5 d x^{21}+\frac{1}{23} c^5 e x^{23} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + c*x^4)^5,x]

[Out]

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

Rule 1154

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

Rubi steps

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

Mathematica [A]  time = 0.0039031, size = 141, normalized size = 1. \[ \frac{10}{13} a^2 c^3 d x^{13}+\frac{10}{9} a^3 c^2 d x^9+\frac{2}{3} a^2 c^3 e x^{15}+\frac{10}{11} a^3 c^2 e x^{11}+a^4 c d x^5+\frac{5}{7} a^4 c e x^7+a^5 d x+\frac{1}{3} a^5 e x^3+\frac{5}{17} a c^4 d x^{17}+\frac{5}{19} a c^4 e x^{19}+\frac{1}{21} c^5 d x^{21}+\frac{1}{23} c^5 e x^{23} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 122, normalized size = 0.9 \begin{align*}{a}^{5}dx+{\frac{{a}^{5}e{x}^{3}}{3}}+{a}^{4}cd{x}^{5}+{\frac{5\,{a}^{4}ce{x}^{7}}{7}}+{\frac{10\,{a}^{3}{c}^{2}d{x}^{9}}{9}}+{\frac{10\,{a}^{3}{c}^{2}e{x}^{11}}{11}}+{\frac{10\,{a}^{2}{c}^{3}d{x}^{13}}{13}}+{\frac{2\,{a}^{2}{c}^{3}e{x}^{15}}{3}}+{\frac{5\,a{c}^{4}d{x}^{17}}{17}}+{\frac{5\,a{c}^{4}e{x}^{19}}{19}}+{\frac{{c}^{5}d{x}^{21}}{21}}+{\frac{{c}^{5}e{x}^{23}}{23}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.941618, size = 163, normalized size = 1.16 \begin{align*} \frac{1}{23} \, c^{5} e x^{23} + \frac{1}{21} \, c^{5} d x^{21} + \frac{5}{19} \, a c^{4} e x^{19} + \frac{5}{17} \, a c^{4} d x^{17} + \frac{2}{3} \, a^{2} c^{3} e x^{15} + \frac{10}{13} \, a^{2} c^{3} d x^{13} + \frac{10}{11} \, a^{3} c^{2} e x^{11} + \frac{10}{9} \, a^{3} c^{2} d x^{9} + \frac{5}{7} \, a^{4} c e x^{7} + a^{4} c d x^{5} + \frac{1}{3} \, a^{5} e x^{3} + a^{5} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50945, size = 298, normalized size = 2.11 \begin{align*} \frac{1}{23} x^{23} e c^{5} + \frac{1}{21} x^{21} d c^{5} + \frac{5}{19} x^{19} e c^{4} a + \frac{5}{17} x^{17} d c^{4} a + \frac{2}{3} x^{15} e c^{3} a^{2} + \frac{10}{13} x^{13} d c^{3} a^{2} + \frac{10}{11} x^{11} e c^{2} a^{3} + \frac{10}{9} x^{9} d c^{2} a^{3} + \frac{5}{7} x^{7} e c a^{4} + x^{5} d c a^{4} + \frac{1}{3} x^{3} e a^{5} + x d a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.082565, size = 148, normalized size = 1.05 \begin{align*} a^{5} d x + \frac{a^{5} e x^{3}}{3} + a^{4} c d x^{5} + \frac{5 a^{4} c e x^{7}}{7} + \frac{10 a^{3} c^{2} d x^{9}}{9} + \frac{10 a^{3} c^{2} e x^{11}}{11} + \frac{10 a^{2} c^{3} d x^{13}}{13} + \frac{2 a^{2} c^{3} e x^{15}}{3} + \frac{5 a c^{4} d x^{17}}{17} + \frac{5 a c^{4} e x^{19}}{19} + \frac{c^{5} d x^{21}}{21} + \frac{c^{5} e x^{23}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12518, size = 171, normalized size = 1.21 \begin{align*} \frac{1}{23} \, c^{5} x^{23} e + \frac{1}{21} \, c^{5} d x^{21} + \frac{5}{19} \, a c^{4} x^{19} e + \frac{5}{17} \, a c^{4} d x^{17} + \frac{2}{3} \, a^{2} c^{3} x^{15} e + \frac{10}{13} \, a^{2} c^{3} d x^{13} + \frac{10}{11} \, a^{3} c^{2} x^{11} e + \frac{10}{9} \, a^{3} c^{2} d x^{9} + \frac{5}{7} \, a^{4} c x^{7} e + a^{4} c d x^{5} + \frac{1}{3} \, a^{5} x^{3} e + a^{5} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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