Optimal. Leaf size=151 \[ a^3 c x+\frac {1}{2} a^3 d x^2+\frac {1}{3} a^3 e x^3+\frac {3}{5} a^2 b c x^5+\frac {1}{2} a^2 b d x^6+\frac {3}{7} a^2 b e x^7+\frac {1}{3} a b^2 c x^9+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {1}{13} b^3 c x^{13}+\frac {1}{14} b^3 d x^{14}+\frac {1}{15} b^3 e x^{15}+\frac {f \left (a+b x^4\right )^4}{16 b} \]
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Rubi [A]
time = 0.07, antiderivative size = 151, 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^3 c x+\frac {1}{2} a^3 d x^2+\frac {1}{3} a^3 e x^3+\frac {3}{5} a^2 b c x^5+\frac {1}{2} a^2 b d x^6+\frac {3}{7} a^2 b e x^7+\frac {1}{3} a b^2 c x^9+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {f \left (a+b x^4\right )^4}{16 b}+\frac {1}{13} b^3 c x^{13}+\frac {1}{14} b^3 d x^{14}+\frac {1}{15} b^3 e x^{15} \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 )^3 \, dx &=\frac {f \left (a+b x^4\right )^4}{16 b}+\int \left (c+d x+e x^2\right ) \left (a+b x^4\right )^3 \, dx\\ &=\frac {f \left (a+b x^4\right )^4}{16 b}+\int \left (a^3 c+a^3 d x+a^3 e x^2+3 a^2 b c x^4+3 a^2 b d x^5+3 a^2 b e x^6+3 a b^2 c x^8+3 a b^2 d x^9+3 a b^2 e x^{10}+b^3 c x^{12}+b^3 d x^{13}+b^3 e x^{14}\right ) \, dx\\ &=a^3 c x+\frac {1}{2} a^3 d x^2+\frac {1}{3} a^3 e x^3+\frac {3}{5} a^2 b c x^5+\frac {1}{2} a^2 b d x^6+\frac {3}{7} a^2 b e x^7+\frac {1}{3} a b^2 c x^9+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {1}{13} b^3 c x^{13}+\frac {1}{14} b^3 d x^{14}+\frac {1}{15} b^3 e x^{15}+\frac {f \left (a+b x^4\right )^4}{16 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 180, normalized size = 1.19 \begin {gather*} a^3 c x+\frac {1}{2} a^3 d x^2+\frac {1}{3} a^3 e x^3+\frac {1}{4} a^3 f x^4+\frac {3}{5} a^2 b c x^5+\frac {1}{2} a^2 b d x^6+\frac {3}{7} a^2 b e x^7+\frac {3}{8} a^2 b f x^8+\frac {1}{3} a b^2 c x^9+\frac {3}{10} a b^2 d x^{10}+\frac {3}{11} a b^2 e x^{11}+\frac {1}{4} a b^2 f x^{12}+\frac {1}{13} b^3 c x^{13}+\frac {1}{14} b^3 d x^{14}+\frac {1}{15} b^3 e x^{15}+\frac {1}{16} b^3 f x^{16} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 151, normalized size = 1.00
method | result | size |
gosper | \(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {1}{3} a^{3} e \,x^{3}+\frac {1}{4} a^{3} f \,x^{4}+\frac {3}{5} a^{2} b c \,x^{5}+\frac {1}{2} a^{2} b d \,x^{6}+\frac {3}{7} a^{2} b e \,x^{7}+\frac {3}{8} f \,a^{2} b \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{4} a \,b^{2} f \,x^{12}+\frac {1}{13} b^{3} c \,x^{13}+\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} b^{3} f \,x^{16}\) | \(151\) |
default | \(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {1}{3} a^{3} e \,x^{3}+\frac {1}{4} a^{3} f \,x^{4}+\frac {3}{5} a^{2} b c \,x^{5}+\frac {1}{2} a^{2} b d \,x^{6}+\frac {3}{7} a^{2} b e \,x^{7}+\frac {3}{8} f \,a^{2} b \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{4} a \,b^{2} f \,x^{12}+\frac {1}{13} b^{3} c \,x^{13}+\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} b^{3} f \,x^{16}\) | \(151\) |
norman | \(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {1}{3} a^{3} e \,x^{3}+\frac {1}{4} a^{3} f \,x^{4}+\frac {3}{5} a^{2} b c \,x^{5}+\frac {1}{2} a^{2} b d \,x^{6}+\frac {3}{7} a^{2} b e \,x^{7}+\frac {3}{8} f \,a^{2} b \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{4} a \,b^{2} f \,x^{12}+\frac {1}{13} b^{3} c \,x^{13}+\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} b^{3} f \,x^{16}\) | \(151\) |
risch | \(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {1}{3} a^{3} e \,x^{3}+\frac {1}{4} a^{3} f \,x^{4}+\frac {3}{5} a^{2} b c \,x^{5}+\frac {1}{2} a^{2} b d \,x^{6}+\frac {3}{7} a^{2} b e \,x^{7}+\frac {3}{8} f \,a^{2} b \,x^{8}+\frac {1}{3} a \,b^{2} c \,x^{9}+\frac {3}{10} a \,b^{2} d \,x^{10}+\frac {3}{11} a \,b^{2} e \,x^{11}+\frac {1}{4} a \,b^{2} f \,x^{12}+\frac {1}{13} b^{3} c \,x^{13}+\frac {1}{14} b^{3} d \,x^{14}+\frac {1}{15} b^{3} e \,x^{15}+\frac {1}{16} b^{3} f \,x^{16}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 154, normalized size = 1.02 \begin {gather*} \frac {1}{16} \, b^{3} f x^{16} + \frac {1}{15} \, b^{3} x^{15} e + \frac {1}{14} \, b^{3} d x^{14} + \frac {1}{13} \, b^{3} c x^{13} + \frac {1}{4} \, a b^{2} f x^{12} + \frac {3}{11} \, a b^{2} x^{11} e + \frac {3}{10} \, a b^{2} d x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {3}{8} \, a^{2} b f x^{8} + \frac {3}{7} \, a^{2} b x^{7} e + \frac {1}{2} \, a^{2} b d x^{6} + \frac {3}{5} \, a^{2} b c x^{5} + \frac {1}{4} \, a^{3} f x^{4} + \frac {1}{3} \, a^{3} x^{3} e + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 150, normalized size = 0.99 \begin {gather*} \frac {1}{16} \, b^{3} f x^{16} + \frac {1}{15} \, b^{3} e x^{15} + \frac {1}{14} \, b^{3} d x^{14} + \frac {1}{13} \, b^{3} c x^{13} + \frac {1}{4} \, a b^{2} f x^{12} + \frac {3}{11} \, a b^{2} e x^{11} + \frac {3}{10} \, a b^{2} d x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {3}{8} \, a^{2} b f x^{8} + \frac {3}{7} \, a^{2} b e x^{7} + \frac {1}{2} \, a^{2} b d x^{6} + \frac {3}{5} \, a^{2} b c x^{5} + \frac {1}{4} \, a^{3} f x^{4} + \frac {1}{3} \, a^{3} e x^{3} + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 180, normalized size = 1.19 \begin {gather*} a^{3} c x + \frac {a^{3} d x^{2}}{2} + \frac {a^{3} e x^{3}}{3} + \frac {a^{3} f x^{4}}{4} + \frac {3 a^{2} b c x^{5}}{5} + \frac {a^{2} b d x^{6}}{2} + \frac {3 a^{2} b e x^{7}}{7} + \frac {3 a^{2} b f x^{8}}{8} + \frac {a b^{2} c x^{9}}{3} + \frac {3 a b^{2} d x^{10}}{10} + \frac {3 a b^{2} e x^{11}}{11} + \frac {a b^{2} f x^{12}}{4} + \frac {b^{3} c x^{13}}{13} + \frac {b^{3} d x^{14}}{14} + \frac {b^{3} e x^{15}}{15} + \frac {b^{3} f x^{16}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 154, normalized size = 1.02 \begin {gather*} \frac {1}{16} \, b^{3} f x^{16} + \frac {1}{15} \, b^{3} x^{15} e + \frac {1}{14} \, b^{3} d x^{14} + \frac {1}{13} \, b^{3} c x^{13} + \frac {1}{4} \, a b^{2} f x^{12} + \frac {3}{11} \, a b^{2} x^{11} e + \frac {3}{10} \, a b^{2} d x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {3}{8} \, a^{2} b f x^{8} + \frac {3}{7} \, a^{2} b x^{7} e + \frac {1}{2} \, a^{2} b d x^{6} + \frac {3}{5} \, a^{2} b c x^{5} + \frac {1}{4} \, a^{3} f x^{4} + \frac {1}{3} \, a^{3} x^{3} e + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 150, normalized size = 0.99 \begin {gather*} \frac {f\,a^3\,x^4}{4}+\frac {e\,a^3\,x^3}{3}+\frac {d\,a^3\,x^2}{2}+c\,a^3\,x+\frac {3\,f\,a^2\,b\,x^8}{8}+\frac {3\,e\,a^2\,b\,x^7}{7}+\frac {d\,a^2\,b\,x^6}{2}+\frac {3\,c\,a^2\,b\,x^5}{5}+\frac {f\,a\,b^2\,x^{12}}{4}+\frac {3\,e\,a\,b^2\,x^{11}}{11}+\frac {3\,d\,a\,b^2\,x^{10}}{10}+\frac {c\,a\,b^2\,x^9}{3}+\frac {f\,b^3\,x^{16}}{16}+\frac {e\,b^3\,x^{15}}{15}+\frac {d\,b^3\,x^{14}}{14}+\frac {c\,b^3\,x^{13}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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