Optimal. Leaf size=15 \[ \frac {1}{8} \left (b x+d x^3\right )^8 \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1588} \[ \frac {1}{8} \left (b x+d x^3\right )^8 \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \left (b+3 d x^2\right ) \left (b x+d x^3\right )^7 \, dx &=\frac {1}{8} \left (b x+d x^3\right )^8\\ \end {align*}
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Mathematica [B] time = 0.00, size = 98, normalized size = 6.53 \[ \frac {b^8 x^8}{8}+b^7 d x^{10}+\frac {7}{2} b^6 d^2 x^{12}+7 b^5 d^3 x^{14}+\frac {35}{4} b^4 d^4 x^{16}+7 b^3 d^5 x^{18}+\frac {7}{2} b^2 d^6 x^{20}+b d^7 x^{22}+\frac {d^8 x^{24}}{8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 88, normalized size = 5.87 \[ \frac {1}{8} x^{24} d^{8} + x^{22} d^{7} b + \frac {7}{2} x^{20} d^{6} b^{2} + 7 x^{18} d^{5} b^{3} + \frac {35}{4} x^{16} d^{4} b^{4} + 7 x^{14} d^{3} b^{5} + \frac {7}{2} x^{12} d^{2} b^{6} + x^{10} d b^{7} + \frac {1}{8} x^{8} b^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 13, normalized size = 0.87 \[ \frac {1}{8} \, {\left (d x^{3} + b x\right )}^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 89, normalized size = 5.93 \[ \frac {1}{8} d^{8} x^{24}+b \,d^{7} x^{22}+\frac {7}{2} b^{2} d^{6} x^{20}+7 b^{3} d^{5} x^{18}+\frac {35}{4} b^{4} d^{4} x^{16}+7 b^{5} d^{3} x^{14}+\frac {7}{2} b^{6} d^{2} x^{12}+b^{7} d \,x^{10}+\frac {1}{8} b^{8} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 13, normalized size = 0.87 \[ \frac {1}{8} \, {\left (d x^{3} + b x\right )}^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 88, normalized size = 5.87 \[ \frac {b^8\,x^8}{8}+b^7\,d\,x^{10}+\frac {7\,b^6\,d^2\,x^{12}}{2}+7\,b^5\,d^3\,x^{14}+\frac {35\,b^4\,d^4\,x^{16}}{4}+7\,b^3\,d^5\,x^{18}+\frac {7\,b^2\,d^6\,x^{20}}{2}+b\,d^7\,x^{22}+\frac {d^8\,x^{24}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 97, normalized size = 6.47 \[ \frac {b^{8} x^{8}}{8} + b^{7} d x^{10} + \frac {7 b^{6} d^{2} x^{12}}{2} + 7 b^{5} d^{3} x^{14} + \frac {35 b^{4} d^{4} x^{16}}{4} + 7 b^{3} d^{5} x^{18} + \frac {7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac {d^{8} x^{24}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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