Optimal. Leaf size=56 \[ a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {1}{7} c^3 d x^7+\frac {e \left (a+c x^2\right )^4}{8 c} \]
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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 200}
\begin {gather*} a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {e \left (a+c x^2\right )^4}{8 c}+\frac {1}{7} c^3 d x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 655
Rubi steps
\begin {align*} \int (d+e x) \left (a+c x^2\right )^3 \, dx &=\frac {e \left (a+c x^2\right )^4}{8 c}+d \int \left (a+c x^2\right )^3 \, dx\\ &=\frac {e \left (a+c x^2\right )^4}{8 c}+d \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {1}{7} c^3 d x^7+\frac {e \left (a+c x^2\right )^4}{8 c}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 85, normalized size = 1.52 \begin {gather*} a^3 d x+\frac {1}{2} a^3 e x^2+a^2 c d x^3+\frac {3}{4} a^2 c e x^4+\frac {3}{5} a c^2 d x^5+\frac {1}{2} a c^2 e x^6+\frac {1}{7} c^3 d x^7+\frac {1}{8} c^3 e x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 74, normalized size = 1.32
method | result | size |
gosper | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
default | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
norman | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
risch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 77, normalized size = 1.38 \begin {gather*} \frac {1}{8} \, c^{3} x^{8} e + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} x^{6} e + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c x^{4} e + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} x^{2} e + a^{3} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.86, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{7} \, c^{3} d x^{7} + \frac {3}{5} \, a c^{2} d x^{5} + a^{2} c d x^{3} + a^{3} d x + \frac {1}{8} \, {\left (c^{3} x^{8} + 4 \, a c^{2} x^{6} + 6 \, a^{2} c x^{4} + 4 \, a^{3} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 85, normalized size = 1.52 \begin {gather*} a^{3} d x + \frac {a^{3} e x^{2}}{2} + a^{2} c d x^{3} + \frac {3 a^{2} c e x^{4}}{4} + \frac {3 a c^{2} d x^{5}}{5} + \frac {a c^{2} e x^{6}}{2} + \frac {c^{3} d x^{7}}{7} + \frac {c^{3} e x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 77, normalized size = 1.38 \begin {gather*} \frac {1}{8} \, c^{3} x^{8} e + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} x^{6} e + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c x^{4} e + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} x^{2} e + a^{3} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 73, normalized size = 1.30 \begin {gather*} \frac {e\,a^3\,x^2}{2}+d\,a^3\,x+\frac {3\,e\,a^2\,c\,x^4}{4}+d\,a^2\,c\,x^3+\frac {e\,a\,c^2\,x^6}{2}+\frac {3\,d\,a\,c^2\,x^5}{5}+\frac {e\,c^3\,x^8}{8}+\frac {d\,c^3\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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