Optimal. Leaf size=198 \[ \frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23}+\frac {c \left (a+b x^4\right )^5}{20 b} \]
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Rubi [A]
time = 0.10, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1596, 1864}
\begin {gather*} \frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {c \left (a+b x^4\right )^5}{20 b}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23} \end {gather*}
Antiderivative was successfully verified.
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Rule 1596
Rule 1864
Rubi steps
\begin {align*} \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx &=\frac {c \left (a+b x^4\right )^5}{20 b}+\int \left (a+b x^4\right )^4 \left (-c x^3+x^3 \left (c+d x+e x^2+f x^3\right )\right ) \, dx\\ &=\frac {c \left (a+b x^4\right )^5}{20 b}+\int \left (a^4 d x^4+a^4 e x^5+a^4 f x^6+4 a^3 b d x^8+4 a^3 b e x^9+4 a^3 b f x^{10}+6 a^2 b^2 d x^{12}+6 a^2 b^2 e x^{13}+6 a^2 b^2 f x^{14}+4 a b^3 d x^{16}+4 a b^3 e x^{17}+4 a b^3 f x^{18}+b^4 d x^{20}+b^4 e x^{21}+b^4 f x^{22}\right ) \, dx\\ &=\frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23}+\frac {c \left (a+b x^4\right )^5}{20 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 241, normalized size = 1.22 \begin {gather*} \frac {1}{4} a^4 c x^4+\frac {1}{5} a^4 d x^5+\frac {1}{6} a^4 e x^6+\frac {1}{7} a^4 f x^7+\frac {1}{2} a^3 b c x^8+\frac {4}{9} a^3 b d x^9+\frac {2}{5} a^3 b e x^{10}+\frac {4}{11} a^3 b f x^{11}+\frac {1}{2} a^2 b^2 c x^{12}+\frac {6}{13} a^2 b^2 d x^{13}+\frac {3}{7} a^2 b^2 e x^{14}+\frac {2}{5} a^2 b^2 f x^{15}+\frac {1}{4} a b^3 c x^{16}+\frac {4}{17} a b^3 d x^{17}+\frac {2}{9} a b^3 e x^{18}+\frac {4}{19} a b^3 f x^{19}+\frac {1}{20} b^4 c x^{20}+\frac {1}{21} b^4 d x^{21}+\frac {1}{22} b^4 e x^{22}+\frac {1}{23} b^4 f x^{23} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 202, normalized size = 1.02
method | result | size |
gosper | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} c \,b^{4} x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
default | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} c \,b^{4} x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
norman | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} c \,b^{4} x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
risch | \(\frac {1}{4} c \,a^{4} x^{4}+\frac {1}{5} a^{4} d \,x^{5}+\frac {1}{6} a^{4} e \,x^{6}+\frac {1}{7} a^{4} f \,x^{7}+\frac {1}{2} c \,a^{3} b \,x^{8}+\frac {4}{9} a^{3} b d \,x^{9}+\frac {2}{5} a^{3} b e \,x^{10}+\frac {4}{11} a^{3} b f \,x^{11}+\frac {1}{2} c \,a^{2} b^{2} x^{12}+\frac {6}{13} a^{2} b^{2} d \,x^{13}+\frac {3}{7} a^{2} b^{2} e \,x^{14}+\frac {2}{5} a^{2} b^{2} f \,x^{15}+\frac {1}{4} a \,b^{3} c \,x^{16}+\frac {4}{17} a \,b^{3} d \,x^{17}+\frac {2}{9} a \,b^{3} e \,x^{18}+\frac {4}{19} a \,b^{3} f \,x^{19}+\frac {1}{20} c \,b^{4} x^{20}+\frac {1}{21} b^{4} d \,x^{21}+\frac {1}{22} b^{4} e \,x^{22}+\frac {1}{23} b^{4} f \,x^{23}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 206, normalized size = 1.04 \begin {gather*} \frac {1}{23} \, b^{4} f x^{23} + \frac {1}{22} \, b^{4} x^{22} e + \frac {1}{21} \, b^{4} d x^{21} + \frac {1}{20} \, b^{4} c x^{20} + \frac {4}{19} \, a b^{3} f x^{19} + \frac {2}{9} \, a b^{3} x^{18} e + \frac {4}{17} \, a b^{3} d x^{17} + \frac {1}{4} \, a b^{3} c x^{16} + \frac {2}{5} \, a^{2} b^{2} f x^{15} + \frac {3}{7} \, a^{2} b^{2} x^{14} e + \frac {6}{13} \, a^{2} b^{2} d x^{13} + \frac {1}{2} \, a^{2} b^{2} c x^{12} + \frac {4}{11} \, a^{3} b f x^{11} + \frac {2}{5} \, a^{3} b x^{10} e + \frac {4}{9} \, a^{3} b d x^{9} + \frac {1}{2} \, a^{3} b c x^{8} + \frac {1}{7} \, a^{4} f x^{7} + \frac {1}{6} \, a^{4} x^{6} e + \frac {1}{5} \, a^{4} d x^{5} + \frac {1}{4} \, a^{4} c x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 201, normalized size = 1.02 \begin {gather*} \frac {1}{23} \, b^{4} f x^{23} + \frac {1}{22} \, b^{4} e x^{22} + \frac {1}{21} \, b^{4} d x^{21} + \frac {1}{20} \, b^{4} c x^{20} + \frac {4}{19} \, a b^{3} f x^{19} + \frac {2}{9} \, a b^{3} e x^{18} + \frac {4}{17} \, a b^{3} d x^{17} + \frac {1}{4} \, a b^{3} c x^{16} + \frac {2}{5} \, a^{2} b^{2} f x^{15} + \frac {3}{7} \, a^{2} b^{2} e x^{14} + \frac {6}{13} \, a^{2} b^{2} d x^{13} + \frac {1}{2} \, a^{2} b^{2} c x^{12} + \frac {4}{11} \, a^{3} b f x^{11} + \frac {2}{5} \, a^{3} b e x^{10} + \frac {4}{9} \, a^{3} b d x^{9} + \frac {1}{2} \, a^{3} b c x^{8} + \frac {1}{7} \, a^{4} f x^{7} + \frac {1}{6} \, a^{4} e x^{6} + \frac {1}{5} \, a^{4} d x^{5} + \frac {1}{4} \, a^{4} c x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 245, normalized size = 1.24 \begin {gather*} \frac {a^{4} c x^{4}}{4} + \frac {a^{4} d x^{5}}{5} + \frac {a^{4} e x^{6}}{6} + \frac {a^{4} f x^{7}}{7} + \frac {a^{3} b c x^{8}}{2} + \frac {4 a^{3} b d x^{9}}{9} + \frac {2 a^{3} b e x^{10}}{5} + \frac {4 a^{3} b f x^{11}}{11} + \frac {a^{2} b^{2} c x^{12}}{2} + \frac {6 a^{2} b^{2} d x^{13}}{13} + \frac {3 a^{2} b^{2} e x^{14}}{7} + \frac {2 a^{2} b^{2} f x^{15}}{5} + \frac {a b^{3} c x^{16}}{4} + \frac {4 a b^{3} d x^{17}}{17} + \frac {2 a b^{3} e x^{18}}{9} + \frac {4 a b^{3} f x^{19}}{19} + \frac {b^{4} c x^{20}}{20} + \frac {b^{4} d x^{21}}{21} + \frac {b^{4} e x^{22}}{22} + \frac {b^{4} f x^{23}}{23} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 206, normalized size = 1.04 \begin {gather*} \frac {1}{23} \, b^{4} f x^{23} + \frac {1}{22} \, b^{4} x^{22} e + \frac {1}{21} \, b^{4} d x^{21} + \frac {1}{20} \, b^{4} c x^{20} + \frac {4}{19} \, a b^{3} f x^{19} + \frac {2}{9} \, a b^{3} x^{18} e + \frac {4}{17} \, a b^{3} d x^{17} + \frac {1}{4} \, a b^{3} c x^{16} + \frac {2}{5} \, a^{2} b^{2} f x^{15} + \frac {3}{7} \, a^{2} b^{2} x^{14} e + \frac {6}{13} \, a^{2} b^{2} d x^{13} + \frac {1}{2} \, a^{2} b^{2} c x^{12} + \frac {4}{11} \, a^{3} b f x^{11} + \frac {2}{5} \, a^{3} b x^{10} e + \frac {4}{9} \, a^{3} b d x^{9} + \frac {1}{2} \, a^{3} b c x^{8} + \frac {1}{7} \, a^{4} f x^{7} + \frac {1}{6} \, a^{4} x^{6} e + \frac {1}{5} \, a^{4} d x^{5} + \frac {1}{4} \, a^{4} c x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 201, normalized size = 1.02 \begin {gather*} \frac {f\,a^4\,x^7}{7}+\frac {e\,a^4\,x^6}{6}+\frac {d\,a^4\,x^5}{5}+\frac {c\,a^4\,x^4}{4}+\frac {4\,f\,a^3\,b\,x^{11}}{11}+\frac {2\,e\,a^3\,b\,x^{10}}{5}+\frac {4\,d\,a^3\,b\,x^9}{9}+\frac {c\,a^3\,b\,x^8}{2}+\frac {2\,f\,a^2\,b^2\,x^{15}}{5}+\frac {3\,e\,a^2\,b^2\,x^{14}}{7}+\frac {6\,d\,a^2\,b^2\,x^{13}}{13}+\frac {c\,a^2\,b^2\,x^{12}}{2}+\frac {4\,f\,a\,b^3\,x^{19}}{19}+\frac {2\,e\,a\,b^3\,x^{18}}{9}+\frac {4\,d\,a\,b^3\,x^{17}}{17}+\frac {c\,a\,b^3\,x^{16}}{4}+\frac {f\,b^4\,x^{23}}{23}+\frac {e\,b^4\,x^{22}}{22}+\frac {d\,b^4\,x^{21}}{21}+\frac {c\,b^4\,x^{20}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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