Optimal. Leaf size=109 \[ a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11}+\frac {f \left (a+b x^4\right )^3}{12 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, 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^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {f \left (a+b x^4\right )^3}{12 b}+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11} \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 )^2 \, dx &=\frac {f \left (a+b x^4\right )^3}{12 b}+\int \left (c+d x+e x^2\right ) \left (a+b x^4\right )^2 \, dx\\ &=\frac {f \left (a+b x^4\right )^3}{12 b}+\int \left (a^2 c+a^2 d x+a^2 e x^2+2 a b c x^4+2 a b d x^5+2 a b e x^6+b^2 c x^8+b^2 d x^9+b^2 e x^{10}\right ) \, dx\\ &=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11}+\frac {f \left (a+b x^4\right )^3}{12 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 124, normalized size = 1.14 \begin {gather*} a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {1}{4} a^2 f x^4+\frac {2}{5} a b c x^5+\frac {1}{3} a b d x^6+\frac {2}{7} a b e x^7+\frac {1}{4} a b f x^8+\frac {1}{9} b^2 c x^9+\frac {1}{10} b^2 d x^{10}+\frac {1}{11} b^2 e x^{11}+\frac {1}{12} b^2 f x^{12} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 103, normalized size = 0.94
method | result | size |
gosper | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} x^{9} b^{2} c +\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} x^{5} a b c +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
default | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} x^{9} b^{2} c +\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} x^{5} a b c +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
norman | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} x^{9} b^{2} c +\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} x^{5} a b c +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
risch | \(\frac {1}{12} b^{2} f \,x^{12}+\frac {1}{11} b^{2} e \,x^{11}+\frac {1}{10} b^{2} d \,x^{10}+\frac {1}{9} x^{9} b^{2} c +\frac {1}{4} a b f \,x^{8}+\frac {2}{7} a b e \,x^{7}+\frac {1}{3} a b d \,x^{6}+\frac {2}{5} x^{5} a b c +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 105, normalized size = 0.96 \begin {gather*} \frac {1}{12} \, b^{2} f x^{12} + \frac {1}{11} \, b^{2} x^{11} e + \frac {1}{10} \, b^{2} d x^{10} + \frac {1}{9} \, b^{2} c x^{9} + \frac {1}{4} \, a b f x^{8} + \frac {2}{7} \, a b x^{7} e + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} x^{3} e + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 102, normalized size = 0.94 \begin {gather*} \frac {1}{12} \, b^{2} f x^{12} + \frac {1}{11} \, b^{2} e x^{11} + \frac {1}{10} \, b^{2} d x^{10} + \frac {1}{9} \, b^{2} c x^{9} + \frac {1}{4} \, a b f x^{8} + \frac {2}{7} \, a b e x^{7} + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 121, normalized size = 1.11 \begin {gather*} a^{2} c x + \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} + \frac {a^{2} f x^{4}}{4} + \frac {2 a b c x^{5}}{5} + \frac {a b d x^{6}}{3} + \frac {2 a b e x^{7}}{7} + \frac {a b f x^{8}}{4} + \frac {b^{2} c x^{9}}{9} + \frac {b^{2} d x^{10}}{10} + \frac {b^{2} e x^{11}}{11} + \frac {b^{2} f x^{12}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 105, normalized size = 0.96 \begin {gather*} \frac {1}{12} \, b^{2} f x^{12} + \frac {1}{11} \, b^{2} x^{11} e + \frac {1}{10} \, b^{2} d x^{10} + \frac {1}{9} \, b^{2} c x^{9} + \frac {1}{4} \, a b f x^{8} + \frac {2}{7} \, a b x^{7} e + \frac {1}{3} \, a b d x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} x^{3} e + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 102, normalized size = 0.94 \begin {gather*} \frac {f\,a^2\,x^4}{4}+\frac {e\,a^2\,x^3}{3}+\frac {d\,a^2\,x^2}{2}+c\,a^2\,x+\frac {f\,a\,b\,x^8}{4}+\frac {2\,e\,a\,b\,x^7}{7}+\frac {d\,a\,b\,x^6}{3}+\frac {2\,c\,a\,b\,x^5}{5}+\frac {f\,b^2\,x^{12}}{12}+\frac {e\,b^2\,x^{11}}{11}+\frac {d\,b^2\,x^{10}}{10}+\frac {c\,b^2\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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