3.6 \(\int \frac{\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx\)

Optimal. Leaf size=139 \[ -\frac{a^5 d}{x}+a^5 e x+\frac{5}{3} a^4 c d x^3+a^4 c e x^5+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{9} a^3 c^2 e x^9+\frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{13} a^2 c^3 e x^{13}+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21} \]

[Out]

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

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Rubi [A]  time = 0.182175, antiderivative size = 139, 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 \[ -\frac{a^5 d}{x}+a^5 e x+\frac{5}{3} a^4 c d x^3+a^4 c e x^5+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{9} a^3 c^2 e x^9+\frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{13} a^2 c^3 e x^{13}+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} d}{x} + a^{5} \int e\, dx + \frac{5 a^{4} c d x^{3}}{3} + a^{4} c e x^{5} + \frac{10 a^{3} c^{2} d x^{7}}{7} + \frac{10 a^{3} c^{2} e x^{9}}{9} + \frac{10 a^{2} c^{3} d x^{11}}{11} + \frac{10 a^{2} c^{3} e x^{13}}{13} + \frac{a c^{4} d x^{15}}{3} + \frac{5 a c^{4} e x^{17}}{17} + \frac{c^{5} d x^{19}}{19} + \frac{c^{5} e x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0129878, size = 139, normalized size = 1. \[ -\frac{a^5 d}{x}+a^5 e x+\frac{5}{3} a^4 c d x^3+a^4 c e x^5+\frac{10}{7} a^3 c^2 d x^7+\frac{10}{9} a^3 c^2 e x^9+\frac{10}{11} a^2 c^3 d x^{11}+\frac{10}{13} a^2 c^3 e x^{13}+\frac{1}{3} a c^4 d x^{15}+\frac{5}{17} a c^4 e x^{17}+\frac{1}{19} c^5 d x^{19}+\frac{1}{21} c^5 e x^{21} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 122, normalized size = 0.9 \[ -{\frac{{a}^{5}d}{x}}+{a}^{5}ex+{\frac{5\,{a}^{4}cd{x}^{3}}{3}}+{a}^{4}ce{x}^{5}+{\frac{10\,{a}^{3}{c}^{2}d{x}^{7}}{7}}+{\frac{10\,{a}^{3}{c}^{2}e{x}^{9}}{9}}+{\frac{10\,{a}^{2}{c}^{3}d{x}^{11}}{11}}+{\frac{10\,{a}^{2}{c}^{3}e{x}^{13}}{13}}+{\frac{a{c}^{4}d{x}^{15}}{3}}+{\frac{5\,a{c}^{4}e{x}^{17}}{17}}+{\frac{{c}^{5}d{x}^{19}}{19}}+{\frac{{c}^{5}e{x}^{21}}{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.705077, size = 163, normalized size = 1.17 \[ \frac{1}{21} \, c^{5} e x^{21} + \frac{1}{19} \, c^{5} d x^{19} + \frac{5}{17} \, a c^{4} e x^{17} + \frac{1}{3} \, a c^{4} d x^{15} + \frac{10}{13} \, a^{2} c^{3} e x^{13} + \frac{10}{11} \, a^{2} c^{3} d x^{11} + \frac{10}{9} \, a^{3} c^{2} e x^{9} + \frac{10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c e x^{5} + \frac{5}{3} \, a^{4} c d x^{3} + a^{5} e x - \frac{a^{5} d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.249982, size = 171, normalized size = 1.23 \[ \frac{138567 \, c^{5} e x^{22} + 153153 \, c^{5} d x^{20} + 855855 \, a c^{4} e x^{18} + 969969 \, a c^{4} d x^{16} + 2238390 \, a^{2} c^{3} e x^{14} + 2645370 \, a^{2} c^{3} d x^{12} + 3233230 \, a^{3} c^{2} e x^{10} + 4157010 \, a^{3} c^{2} d x^{8} + 2909907 \, a^{4} c e x^{6} + 4849845 \, a^{4} c d x^{4} + 2909907 \, a^{5} e x^{2} - 2909907 \, a^{5} d}{2909907 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2909907*(138567*c^5*e*x^22 + 153153*c^5*d*x^20 + 855855*a*c^4*e*x^18 + 969969*
a*c^4*d*x^16 + 2238390*a^2*c^3*e*x^14 + 2645370*a^2*c^3*d*x^12 + 3233230*a^3*c^2
*e*x^10 + 4157010*a^3*c^2*d*x^8 + 2909907*a^4*c*e*x^6 + 4849845*a^4*c*d*x^4 + 29
09907*a^5*e*x^2 - 2909907*a^5*d)/x

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Sympy [A]  time = 1.51429, size = 143, normalized size = 1.03 \[ - \frac{a^{5} d}{x} + a^{5} e x + \frac{5 a^{4} c d x^{3}}{3} + a^{4} c e x^{5} + \frac{10 a^{3} c^{2} d x^{7}}{7} + \frac{10 a^{3} c^{2} e x^{9}}{9} + \frac{10 a^{2} c^{3} d x^{11}}{11} + \frac{10 a^{2} c^{3} e x^{13}}{13} + \frac{a c^{4} d x^{15}}{3} + \frac{5 a c^{4} e x^{17}}{17} + \frac{c^{5} d x^{19}}{19} + \frac{c^{5} e x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260405, size = 171, normalized size = 1.23 \[ \frac{1}{21} \, c^{5} x^{21} e + \frac{1}{19} \, c^{5} d x^{19} + \frac{5}{17} \, a c^{4} x^{17} e + \frac{1}{3} \, a c^{4} d x^{15} + \frac{10}{13} \, a^{2} c^{3} x^{13} e + \frac{10}{11} \, a^{2} c^{3} d x^{11} + \frac{10}{9} \, a^{3} c^{2} x^{9} e + \frac{10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c x^{5} e + \frac{5}{3} \, a^{4} c d x^{3} + a^{5} x e - \frac{a^{5} d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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