Optimal. Leaf size=139 \[ -\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21} \]
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Rubi [A]
time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1276}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 1276
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx &=\int \left (a^5 e+\frac {a^5 d}{x^2}+5 a^4 c d x^2+5 a^4 c e x^4+10 a^3 c^2 d x^6+10 a^3 c^2 e x^8+10 a^2 c^3 d x^{10}+10 a^2 c^3 e x^{12}+5 a c^4 d x^{14}+5 a c^4 e x^{16}+c^5 d x^{18}+c^5 e x^{20}\right ) \, dx\\ &=-\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}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 139, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 122, normalized size = 0.88
method | result | size |
default | \(-\frac {a^{5} d}{x}+a^{5} e x +\frac {5 a^{4} c d \,x^{3}}{3}+a^{4} c e \,x^{5}+\frac {10 a^{3} c^{2} d \,x^{7}}{7}+\frac {10 a^{3} c^{2} e \,x^{9}}{9}+\frac {10 a^{2} c^{3} d \,x^{11}}{11}+\frac {10 a^{2} c^{3} e \,x^{13}}{13}+\frac {a \,c^{4} d \,x^{15}}{3}+\frac {5 a \,c^{4} e \,x^{17}}{17}+\frac {c^{5} d \,x^{19}}{19}+\frac {c^{5} e \,x^{21}}{21}\) | \(122\) |
risch | \(-\frac {a^{5} d}{x}+a^{5} e x +\frac {5 a^{4} c d \,x^{3}}{3}+a^{4} c e \,x^{5}+\frac {10 a^{3} c^{2} d \,x^{7}}{7}+\frac {10 a^{3} c^{2} e \,x^{9}}{9}+\frac {10 a^{2} c^{3} d \,x^{11}}{11}+\frac {10 a^{2} c^{3} e \,x^{13}}{13}+\frac {a \,c^{4} d \,x^{15}}{3}+\frac {5 a \,c^{4} e \,x^{17}}{17}+\frac {c^{5} d \,x^{19}}{19}+\frac {c^{5} e \,x^{21}}{21}\) | \(122\) |
norman | \(\frac {-d \,a^{5}+a^{5} e \,x^{2}+\frac {5}{3} a^{4} c d \,x^{4}+a^{4} c e \,x^{6}+\frac {10}{7} a^{3} c^{2} d \,x^{8}+\frac {10}{9} a^{3} c^{2} e \,x^{10}+\frac {10}{11} a^{2} c^{3} d \,x^{12}+\frac {10}{13} a^{2} c^{3} e \,x^{14}+\frac {1}{3} a \,c^{4} d \,x^{16}+\frac {5}{17} a \,c^{4} e \,x^{18}+\frac {1}{19} c^{5} d \,x^{20}+\frac {1}{21} c^{5} e \,x^{22}}{x}\) | \(125\) |
gosper | \(-\frac {-138567 c^{5} e \,x^{22}-153153 c^{5} d \,x^{20}-855855 a \,c^{4} e \,x^{18}-969969 a \,c^{4} d \,x^{16}-2238390 a^{2} c^{3} e \,x^{14}-2645370 a^{2} c^{3} d \,x^{12}-3233230 a^{3} c^{2} e \,x^{10}-4157010 a^{3} c^{2} d \,x^{8}-2909907 a^{4} c e \,x^{6}-4849845 a^{4} c d \,x^{4}-2909907 a^{5} e \,x^{2}+2909907 d \,a^{5}}{2909907 x}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 127, normalized size = 0.91 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 126, normalized size = 0.91 \begin {gather*} \frac {153153 \, c^{5} d x^{20} + 969969 \, a c^{4} d x^{16} + 2645370 \, a^{2} c^{3} d x^{12} + 4157010 \, a^{3} c^{2} d x^{8} + 4849845 \, a^{4} c d x^{4} - 2909907 \, a^{5} d + 209 \, {\left (663 \, c^{5} x^{22} + 4095 \, a c^{4} x^{18} + 10710 \, a^{2} c^{3} x^{14} + 15470 \, a^{3} c^{2} x^{10} + 13923 \, a^{4} c x^{6} + 13923 \, a^{5} x^{2}\right )} e}{2909907 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 143, normalized size = 1.03 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.19, size = 127, normalized size = 0.91 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 121, normalized size = 0.87 \begin {gather*} \frac {c^5\,d\,x^{19}}{19}-\frac {a^5\,d}{x}+\frac {c^5\,e\,x^{21}}{21}+a^5\,e\,x+\frac {10\,a^3\,c^2\,d\,x^7}{7}+\frac {10\,a^2\,c^3\,d\,x^{11}}{11}+\frac {10\,a^3\,c^2\,e\,x^9}{9}+\frac {10\,a^2\,c^3\,e\,x^{13}}{13}+\frac {5\,a^4\,c\,d\,x^3}{3}+\frac {a\,c^4\,d\,x^{15}}{3}+a^4\,c\,e\,x^5+\frac {5\,a\,c^4\,e\,x^{17}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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