3.7 \(\int \frac{\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^3} \, dx\)

Optimal. Leaf size=142 \[ -\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20} \]

[Out]

-(a^5*d)/(2*x^2) + (5*a^4*c*d*x^2)/2 + (5*a^4*c*e*x^4)/4 + (5*a^3*c^2*d*x^6)/3 +
(5*a^3*c^2*e*x^8)/4 + a^2*c^3*d*x^10 + (5*a^2*c^3*e*x^12)/6 + (5*a*c^4*d*x^14)/
14 + (5*a*c^4*e*x^16)/16 + (c^5*d*x^18)/18 + (c^5*e*x^20)/20 + a^5*e*Log[x]

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Rubi [A]  time = 0.279526, antiderivative size = 142, 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 \[ -\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x^2)*(a + c*x^4)^5)/x^3,x]

[Out]

-(a^5*d)/(2*x^2) + (5*a^4*c*d*x^2)/2 + (5*a^4*c*e*x^4)/4 + (5*a^3*c^2*d*x^6)/3 +
(5*a^3*c^2*e*x^8)/4 + a^2*c^3*d*x^10 + (5*a^2*c^3*e*x^12)/6 + (5*a*c^4*d*x^14)/
14 + (5*a*c^4*e*x^16)/16 + (c^5*d*x^18)/18 + (c^5*e*x^20)/20 + a^5*e*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} d}{2 x^{2}} + \frac{a^{5} e \log{\left (x^{2} \right )}}{2} + \frac{5 a^{4} c d x^{2}}{2} + \frac{5 a^{4} c e \int ^{x^{2}} x\, dx}{2} + \frac{5 a^{3} c^{2} d x^{6}}{3} + \frac{5 a^{3} c^{2} e x^{8}}{4} + a^{2} c^{3} d x^{10} + \frac{5 a^{2} c^{3} e x^{12}}{6} + \frac{5 a c^{4} d x^{14}}{14} + \frac{5 a c^{4} e x^{16}}{16} + \frac{c^{5} d x^{18}}{18} + \frac{c^{5} e x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)*(c*x**4+a)**5/x**3,x)

[Out]

-a**5*d/(2*x**2) + a**5*e*log(x**2)/2 + 5*a**4*c*d*x**2/2 + 5*a**4*c*e*Integral(
x, (x, x**2))/2 + 5*a**3*c**2*d*x**6/3 + 5*a**3*c**2*e*x**8/4 + a**2*c**3*d*x**1
0 + 5*a**2*c**3*e*x**12/6 + 5*a*c**4*d*x**14/14 + 5*a*c**4*e*x**16/16 + c**5*d*x
**18/18 + c**5*e*x**20/20

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Mathematica [A]  time = 0.0131955, size = 142, normalized size = 1. \[ -\frac{a^5 d}{2 x^2}+a^5 e \log (x)+\frac{5}{2} a^4 c d x^2+\frac{5}{4} a^4 c e x^4+\frac{5}{3} a^3 c^2 d x^6+\frac{5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac{5}{6} a^2 c^3 e x^{12}+\frac{5}{14} a c^4 d x^{14}+\frac{5}{16} a c^4 e x^{16}+\frac{1}{18} c^5 d x^{18}+\frac{1}{20} c^5 e x^{20} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x^2)*(a + c*x^4)^5)/x^3,x]

[Out]

-(a^5*d)/(2*x^2) + (5*a^4*c*d*x^2)/2 + (5*a^4*c*e*x^4)/4 + (5*a^3*c^2*d*x^6)/3 +
(5*a^3*c^2*e*x^8)/4 + a^2*c^3*d*x^10 + (5*a^2*c^3*e*x^12)/6 + (5*a*c^4*d*x^14)/
14 + (5*a*c^4*e*x^16)/16 + (c^5*d*x^18)/18 + (c^5*e*x^20)/20 + a^5*e*Log[x]

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Maple [A]  time = 0.009, size = 123, normalized size = 0.9 \[ -{\frac{{a}^{5}d}{2\,{x}^{2}}}+{\frac{5\,{a}^{4}cd{x}^{2}}{2}}+{\frac{5\,{a}^{4}ce{x}^{4}}{4}}+{\frac{5\,{a}^{3}{c}^{2}d{x}^{6}}{3}}+{\frac{5\,{a}^{3}{c}^{2}e{x}^{8}}{4}}+{a}^{2}{c}^{3}d{x}^{10}+{\frac{5\,{a}^{2}{c}^{3}e{x}^{12}}{6}}+{\frac{5\,a{c}^{4}d{x}^{14}}{14}}+{\frac{5\,a{c}^{4}e{x}^{16}}{16}}+{\frac{{c}^{5}d{x}^{18}}{18}}+{\frac{{c}^{5}e{x}^{20}}{20}}+{a}^{5}e\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)*(c*x^4+a)^5/x^3,x)

[Out]

-1/2*a^5*d/x^2+5/2*a^4*c*d*x^2+5/4*a^4*c*e*x^4+5/3*a^3*c^2*d*x^6+5/4*a^3*c^2*e*x
^8+a^2*c^3*d*x^10+5/6*a^2*c^3*e*x^12+5/14*a*c^4*d*x^14+5/16*a*c^4*e*x^16+1/18*c^
5*d*x^18+1/20*c^5*e*x^20+a^5*e*ln(x)

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Maxima [A]  time = 0.707444, size = 169, normalized size = 1.19 \[ \frac{1}{20} \, c^{5} e x^{20} + \frac{1}{18} \, c^{5} d x^{18} + \frac{5}{16} \, a c^{4} e x^{16} + \frac{5}{14} \, a c^{4} d x^{14} + \frac{5}{6} \, a^{2} c^{3} e x^{12} + a^{2} c^{3} d x^{10} + \frac{5}{4} \, a^{3} c^{2} e x^{8} + \frac{5}{3} \, a^{3} c^{2} d x^{6} + \frac{5}{4} \, a^{4} c e x^{4} + \frac{5}{2} \, a^{4} c d x^{2} + \frac{1}{2} \, a^{5} e \log \left (x^{2}\right ) - \frac{a^{5} d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^5*(e*x^2 + d)/x^3,x, algorithm="maxima")

[Out]

1/20*c^5*e*x^20 + 1/18*c^5*d*x^18 + 5/16*a*c^4*e*x^16 + 5/14*a*c^4*d*x^14 + 5/6*
a^2*c^3*e*x^12 + a^2*c^3*d*x^10 + 5/4*a^3*c^2*e*x^8 + 5/3*a^3*c^2*d*x^6 + 5/4*a^
4*c*e*x^4 + 5/2*a^4*c*d*x^2 + 1/2*a^5*e*log(x^2) - 1/2*a^5*d/x^2

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Fricas [A]  time = 0.250074, size = 174, normalized size = 1.23 \[ \frac{252 \, c^{5} e x^{22} + 280 \, c^{5} d x^{20} + 1575 \, a c^{4} e x^{18} + 1800 \, a c^{4} d x^{16} + 4200 \, a^{2} c^{3} e x^{14} + 5040 \, a^{2} c^{3} d x^{12} + 6300 \, a^{3} c^{2} e x^{10} + 8400 \, a^{3} c^{2} d x^{8} + 6300 \, a^{4} c e x^{6} + 12600 \, a^{4} c d x^{4} + 5040 \, a^{5} e x^{2} \log \left (x\right ) - 2520 \, a^{5} d}{5040 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^5*(e*x^2 + d)/x^3,x, algorithm="fricas")

[Out]

1/5040*(252*c^5*e*x^22 + 280*c^5*d*x^20 + 1575*a*c^4*e*x^18 + 1800*a*c^4*d*x^16
+ 4200*a^2*c^3*e*x^14 + 5040*a^2*c^3*d*x^12 + 6300*a^3*c^2*e*x^10 + 8400*a^3*c^2
*d*x^8 + 6300*a^4*c*e*x^6 + 12600*a^4*c*d*x^4 + 5040*a^5*e*x^2*log(x) - 2520*a^5
*d)/x^2

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Sympy [A]  time = 1.64821, size = 150, normalized size = 1.06 \[ - \frac{a^{5} d}{2 x^{2}} + a^{5} e \log{\left (x \right )} + \frac{5 a^{4} c d x^{2}}{2} + \frac{5 a^{4} c e x^{4}}{4} + \frac{5 a^{3} c^{2} d x^{6}}{3} + \frac{5 a^{3} c^{2} e x^{8}}{4} + a^{2} c^{3} d x^{10} + \frac{5 a^{2} c^{3} e x^{12}}{6} + \frac{5 a c^{4} d x^{14}}{14} + \frac{5 a c^{4} e x^{16}}{16} + \frac{c^{5} d x^{18}}{18} + \frac{c^{5} e x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)*(c*x**4+a)**5/x**3,x)

[Out]

-a**5*d/(2*x**2) + a**5*e*log(x) + 5*a**4*c*d*x**2/2 + 5*a**4*c*e*x**4/4 + 5*a**
3*c**2*d*x**6/3 + 5*a**3*c**2*e*x**8/4 + a**2*c**3*d*x**10 + 5*a**2*c**3*e*x**12
/6 + 5*a*c**4*d*x**14/14 + 5*a*c**4*e*x**16/16 + c**5*d*x**18/18 + c**5*e*x**20/
20

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GIAC/XCAS [A]  time = 0.263759, size = 192, normalized size = 1.35 \[ \frac{1}{20} \, c^{5} x^{20} e + \frac{1}{18} \, c^{5} d x^{18} + \frac{5}{16} \, a c^{4} x^{16} e + \frac{5}{14} \, a c^{4} d x^{14} + \frac{5}{6} \, a^{2} c^{3} x^{12} e + a^{2} c^{3} d x^{10} + \frac{5}{4} \, a^{3} c^{2} x^{8} e + \frac{5}{3} \, a^{3} c^{2} d x^{6} + \frac{5}{4} \, a^{4} c x^{4} e + \frac{5}{2} \, a^{4} c d x^{2} + \frac{1}{2} \, a^{5} e{\rm ln}\left (x^{2}\right ) - \frac{a^{5} x^{2} e + a^{5} d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^5*(e*x^2 + d)/x^3,x, algorithm="giac")

[Out]

1/20*c^5*x^20*e + 1/18*c^5*d*x^18 + 5/16*a*c^4*x^16*e + 5/14*a*c^4*d*x^14 + 5/6*
a^2*c^3*x^12*e + a^2*c^3*d*x^10 + 5/4*a^3*c^2*x^8*e + 5/3*a^3*c^2*d*x^6 + 5/4*a^
4*c*x^4*e + 5/2*a^4*c*d*x^2 + 1/2*a^5*e*ln(x^2) - 1/2*(a^5*x^2*e + a^5*d)/x^2