Optimal. Leaf size=193 \[ a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19}+\frac {f \left (a+b x^4\right )^5}{20 b} \]
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Rubi [A]
time = 0.10, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1596, 1671}
\begin {gather*} a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {f \left (a+b x^4\right )^5}{20 b}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19} \end {gather*}
Antiderivative was successfully verified.
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Rule 1596
Rule 1671
Rubi steps
\begin {align*} \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^4 \, dx &=\frac {f \left (a+b x^4\right )^5}{20 b}+\int \left (c+d x+e x^2\right ) \left (a+b x^4\right )^4 \, dx\\ &=\frac {f \left (a+b x^4\right )^5}{20 b}+\int \left (a^4 c+a^4 d x+a^4 e x^2+4 a^3 b c x^4+4 a^3 b d x^5+4 a^3 b e x^6+6 a^2 b^2 c x^8+6 a^2 b^2 d x^9+6 a^2 b^2 e x^{10}+4 a b^3 c x^{12}+4 a b^3 d x^{13}+4 a b^3 e x^{14}+b^4 c x^{16}+b^4 d x^{17}+b^4 e x^{18}\right ) \, dx\\ &=a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19}+\frac {f \left (a+b x^4\right )^5}{20 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 236, normalized size = 1.22 \begin {gather*} a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+\frac {1}{4} a^4 f x^4+\frac {4}{5} a^3 b c x^5+\frac {2}{3} a^3 b d x^6+\frac {4}{7} a^3 b e x^7+\frac {1}{2} a^3 b f x^8+\frac {2}{3} a^2 b^2 c x^9+\frac {3}{5} a^2 b^2 d x^{10}+\frac {6}{11} a^2 b^2 e x^{11}+\frac {1}{2} a^2 b^2 f x^{12}+\frac {4}{13} a b^3 c x^{13}+\frac {2}{7} a b^3 d x^{14}+\frac {4}{15} a b^3 e x^{15}+\frac {1}{4} a b^3 f x^{16}+\frac {1}{17} b^4 c x^{17}+\frac {1}{18} b^4 d x^{18}+\frac {1}{19} b^4 e x^{19}+\frac {1}{20} b^4 f x^{20} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 199, normalized size = 1.03
method | result | size |
gosper | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
default | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
norman | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
risch | \(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+\frac {1}{4} a^{4} f \,x^{4}+\frac {4}{5} a^{3} b c \,x^{5}+\frac {2}{3} a^{3} b d \,x^{6}+\frac {4}{7} a^{3} b e \,x^{7}+\frac {1}{2} a^{3} b f \,x^{8}+\frac {2}{3} a^{2} b^{2} c \,x^{9}+\frac {3}{5} a^{2} b^{2} d \,x^{10}+\frac {6}{11} a^{2} b^{2} e \,x^{11}+\frac {1}{2} f \,a^{2} b^{2} x^{12}+\frac {4}{13} a \,b^{3} c \,x^{13}+\frac {2}{7} a \,b^{3} d \,x^{14}+\frac {4}{15} a \,b^{3} e \,x^{15}+\frac {1}{4} f a \,b^{3} x^{16}+\frac {1}{17} b^{4} c \,x^{17}+\frac {1}{18} b^{4} d \,x^{18}+\frac {1}{19} b^{4} e \,x^{19}+\frac {1}{20} f \,b^{4} x^{20}\) | \(199\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 203, normalized size = 1.05 \begin {gather*} \frac {1}{20} \, b^{4} f x^{20} + \frac {1}{19} \, b^{4} x^{19} e + \frac {1}{18} \, b^{4} d x^{18} + \frac {1}{17} \, b^{4} c x^{17} + \frac {1}{4} \, a b^{3} f x^{16} + \frac {4}{15} \, a b^{3} x^{15} e + \frac {2}{7} \, a b^{3} d x^{14} + \frac {4}{13} \, a b^{3} c x^{13} + \frac {1}{2} \, a^{2} b^{2} f x^{12} + \frac {6}{11} \, a^{2} b^{2} x^{11} e + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b f x^{8} + \frac {4}{7} \, a^{3} b x^{7} e + \frac {2}{3} \, a^{3} b d x^{6} + \frac {4}{5} \, a^{3} b c x^{5} + \frac {1}{4} \, a^{4} f x^{4} + \frac {1}{3} \, a^{4} x^{3} e + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 198, normalized size = 1.03 \begin {gather*} \frac {1}{20} \, b^{4} f x^{20} + \frac {1}{19} \, b^{4} e x^{19} + \frac {1}{18} \, b^{4} d x^{18} + \frac {1}{17} \, b^{4} c x^{17} + \frac {1}{4} \, a b^{3} f x^{16} + \frac {4}{15} \, a b^{3} e x^{15} + \frac {2}{7} \, a b^{3} d x^{14} + \frac {4}{13} \, a b^{3} c x^{13} + \frac {1}{2} \, a^{2} b^{2} f x^{12} + \frac {6}{11} \, a^{2} b^{2} e x^{11} + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b f x^{8} + \frac {4}{7} \, a^{3} b e x^{7} + \frac {2}{3} \, a^{3} b d x^{6} + \frac {4}{5} \, a^{3} b c x^{5} + \frac {1}{4} \, a^{4} f x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 241, normalized size = 1.25 \begin {gather*} a^{4} c x + \frac {a^{4} d x^{2}}{2} + \frac {a^{4} e x^{3}}{3} + \frac {a^{4} f x^{4}}{4} + \frac {4 a^{3} b c x^{5}}{5} + \frac {2 a^{3} b d x^{6}}{3} + \frac {4 a^{3} b e x^{7}}{7} + \frac {a^{3} b f x^{8}}{2} + \frac {2 a^{2} b^{2} c x^{9}}{3} + \frac {3 a^{2} b^{2} d x^{10}}{5} + \frac {6 a^{2} b^{2} e x^{11}}{11} + \frac {a^{2} b^{2} f x^{12}}{2} + \frac {4 a b^{3} c x^{13}}{13} + \frac {2 a b^{3} d x^{14}}{7} + \frac {4 a b^{3} e x^{15}}{15} + \frac {a b^{3} f x^{16}}{4} + \frac {b^{4} c x^{17}}{17} + \frac {b^{4} d x^{18}}{18} + \frac {b^{4} e x^{19}}{19} + \frac {b^{4} f x^{20}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 203, normalized size = 1.05 \begin {gather*} \frac {1}{20} \, b^{4} f x^{20} + \frac {1}{19} \, b^{4} x^{19} e + \frac {1}{18} \, b^{4} d x^{18} + \frac {1}{17} \, b^{4} c x^{17} + \frac {1}{4} \, a b^{3} f x^{16} + \frac {4}{15} \, a b^{3} x^{15} e + \frac {2}{7} \, a b^{3} d x^{14} + \frac {4}{13} \, a b^{3} c x^{13} + \frac {1}{2} \, a^{2} b^{2} f x^{12} + \frac {6}{11} \, a^{2} b^{2} x^{11} e + \frac {3}{5} \, a^{2} b^{2} d x^{10} + \frac {2}{3} \, a^{2} b^{2} c x^{9} + \frac {1}{2} \, a^{3} b f x^{8} + \frac {4}{7} \, a^{3} b x^{7} e + \frac {2}{3} \, a^{3} b d x^{6} + \frac {4}{5} \, a^{3} b c x^{5} + \frac {1}{4} \, a^{4} f x^{4} + \frac {1}{3} \, a^{4} x^{3} e + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.08, size = 198, normalized size = 1.03 \begin {gather*} \frac {f\,a^4\,x^4}{4}+\frac {e\,a^4\,x^3}{3}+\frac {d\,a^4\,x^2}{2}+c\,a^4\,x+\frac {f\,a^3\,b\,x^8}{2}+\frac {4\,e\,a^3\,b\,x^7}{7}+\frac {2\,d\,a^3\,b\,x^6}{3}+\frac {4\,c\,a^3\,b\,x^5}{5}+\frac {f\,a^2\,b^2\,x^{12}}{2}+\frac {6\,e\,a^2\,b^2\,x^{11}}{11}+\frac {3\,d\,a^2\,b^2\,x^{10}}{5}+\frac {2\,c\,a^2\,b^2\,x^9}{3}+\frac {f\,a\,b^3\,x^{16}}{4}+\frac {4\,e\,a\,b^3\,x^{15}}{15}+\frac {2\,d\,a\,b^3\,x^{14}}{7}+\frac {4\,c\,a\,b^3\,x^{13}}{13}+\frac {f\,b^4\,x^{20}}{20}+\frac {e\,b^4\,x^{19}}{19}+\frac {d\,b^4\,x^{18}}{18}+\frac {c\,b^4\,x^{17}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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