Optimal. Leaf size=142 \[ a^5 d \log (x)+\frac{1}{2} a^5 e x^2+\frac{5}{4} a^4 c d x^4+\frac{5}{6} a^4 c e x^6+\frac{5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac{5}{6} a^2 c^3 d x^{12}+\frac{5}{7} a^2 c^3 e x^{14}+\frac{5}{16} a c^4 d x^{16}+\frac{5}{18} a c^4 e x^{18}+\frac{1}{20} c^5 d x^{20}+\frac{1}{22} c^5 e x^{22} \]
[Out]
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Rubi [A] time = 0.217483, 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 \[ a^5 d \log (x)+\frac{1}{2} a^5 e x^2+\frac{5}{4} a^4 c d x^4+\frac{5}{6} a^4 c e x^6+\frac{5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac{5}{6} a^2 c^3 d x^{12}+\frac{5}{7} a^2 c^3 e x^{14}+\frac{5}{16} a c^4 d x^{16}+\frac{5}{18} a c^4 e x^{18}+\frac{1}{20} c^5 d x^{20}+\frac{1}{22} c^5 e x^{22} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x^2)*(a + c*x^4)^5)/x,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{5} d \log{\left (x^{2} \right )}}{2} + \frac{a^{5} \int ^{x^{2}} e\, dx}{2} + \frac{5 a^{4} c d \int ^{x^{2}} x\, dx}{2} + \frac{5 a^{4} c e x^{6}}{6} + \frac{5 a^{3} c^{2} d x^{8}}{4} + a^{3} c^{2} e x^{10} + \frac{5 a^{2} c^{3} d x^{12}}{6} + \frac{5 a^{2} c^{3} e x^{14}}{7} + \frac{5 a c^{4} d x^{16}}{16} + \frac{5 a c^{4} e x^{18}}{18} + \frac{c^{5} d x^{20}}{20} + \frac{c^{5} e x^{22}}{22} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)*(c*x**4+a)**5/x,x)
[Out]
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Mathematica [A] time = 0.0135551, size = 142, normalized size = 1. \[ a^5 d \log (x)+\frac{1}{2} a^5 e x^2+\frac{5}{4} a^4 c d x^4+\frac{5}{6} a^4 c e x^6+\frac{5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac{5}{6} a^2 c^3 d x^{12}+\frac{5}{7} a^2 c^3 e x^{14}+\frac{5}{16} a c^4 d x^{16}+\frac{5}{18} a c^4 e x^{18}+\frac{1}{20} c^5 d x^{20}+\frac{1}{22} c^5 e x^{22} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x^2)*(a + c*x^4)^5)/x,x]
[Out]
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Maple [A] time = 0.005, size = 123, normalized size = 0.9 \[{\frac{{a}^{5}e{x}^{2}}{2}}+{\frac{5\,{a}^{4}cd{x}^{4}}{4}}+{\frac{5\,{a}^{4}ce{x}^{6}}{6}}+{\frac{5\,{a}^{3}{c}^{2}d{x}^{8}}{4}}+{a}^{3}{c}^{2}e{x}^{10}+{\frac{5\,{a}^{2}{c}^{3}d{x}^{12}}{6}}+{\frac{5\,{a}^{2}{c}^{3}e{x}^{14}}{7}}+{\frac{5\,a{c}^{4}d{x}^{16}}{16}}+{\frac{5\,a{c}^{4}e{x}^{18}}{18}}+{\frac{{c}^{5}d{x}^{20}}{20}}+{\frac{{c}^{5}e{x}^{22}}{22}}+{a}^{5}d\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)*(c*x^4+a)^5/x,x)
[Out]
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Maxima [A] time = 0.710135, size = 169, normalized size = 1.19 \[ \frac{1}{22} \, c^{5} e x^{22} + \frac{1}{20} \, c^{5} d x^{20} + \frac{5}{18} \, a c^{4} e x^{18} + \frac{5}{16} \, a c^{4} d x^{16} + \frac{5}{7} \, a^{2} c^{3} e x^{14} + \frac{5}{6} \, a^{2} c^{3} d x^{12} + a^{3} c^{2} e x^{10} + \frac{5}{4} \, a^{3} c^{2} d x^{8} + \frac{5}{6} \, a^{4} c e x^{6} + \frac{5}{4} \, a^{4} c d x^{4} + \frac{1}{2} \, a^{5} e x^{2} + \frac{1}{2} \, a^{5} d \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^5*(e*x^2 + d)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269625, size = 165, normalized size = 1.16 \[ \frac{1}{22} \, c^{5} e x^{22} + \frac{1}{20} \, c^{5} d x^{20} + \frac{5}{18} \, a c^{4} e x^{18} + \frac{5}{16} \, a c^{4} d x^{16} + \frac{5}{7} \, a^{2} c^{3} e x^{14} + \frac{5}{6} \, a^{2} c^{3} d x^{12} + a^{3} c^{2} e x^{10} + \frac{5}{4} \, a^{3} c^{2} d x^{8} + \frac{5}{6} \, a^{4} c e x^{6} + \frac{5}{4} \, a^{4} c d x^{4} + \frac{1}{2} \, a^{5} e x^{2} + a^{5} d \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^5*(e*x^2 + d)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.51428, size = 150, normalized size = 1.06 \[ a^{5} d \log{\left (x \right )} + \frac{a^{5} e x^{2}}{2} + \frac{5 a^{4} c d x^{4}}{4} + \frac{5 a^{4} c e x^{6}}{6} + \frac{5 a^{3} c^{2} d x^{8}}{4} + a^{3} c^{2} e x^{10} + \frac{5 a^{2} c^{3} d x^{12}}{6} + \frac{5 a^{2} c^{3} e x^{14}}{7} + \frac{5 a c^{4} d x^{16}}{16} + \frac{5 a c^{4} e x^{18}}{18} + \frac{c^{5} d x^{20}}{20} + \frac{c^{5} e x^{22}}{22} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)*(c*x**4+a)**5/x,x)
[Out]
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GIAC/XCAS [A] time = 0.264024, size = 177, normalized size = 1.25 \[ \frac{1}{22} \, c^{5} x^{22} e + \frac{1}{20} \, c^{5} d x^{20} + \frac{5}{18} \, a c^{4} x^{18} e + \frac{5}{16} \, a c^{4} d x^{16} + \frac{5}{7} \, a^{2} c^{3} x^{14} e + \frac{5}{6} \, a^{2} c^{3} d x^{12} + a^{3} c^{2} x^{10} e + \frac{5}{4} \, a^{3} c^{2} d x^{8} + \frac{5}{6} \, a^{4} c x^{6} e + \frac{5}{4} \, a^{4} c d x^{4} + \frac{1}{2} \, a^{5} x^{2} e + \frac{1}{2} \, a^{5} d{\rm ln}\left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^5*(e*x^2 + d)/x,x, algorithm="giac")
[Out]