Optimal. Leaf size=16 \[ \frac {1}{8} \left (a+b x+d x^3\right )^8 \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1588} \begin {gather*} \frac {1}{8} \left (a+b x+d x^3\right )^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx &=\frac {1}{8} \left (a+b x+d x^3\right )^8\\ \end {align*}
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Mathematica [B] time = 0.06, size = 127, normalized size = 7.94 \begin {gather*} \frac {1}{8} x \left (b+d x^2\right ) \left (8 a^7+28 a^6 x \left (b+d x^2\right )+56 a^5 x^2 \left (b+d x^2\right )^2+70 a^4 x^3 \left (b+d x^2\right )^3+56 a^3 x^4 \left (b+d x^2\right )^4+28 a^2 x^5 \left (b+d x^2\right )^5+8 a x^6 \left (b+d x^2\right )^6+x^7 \left (b+d x^2\right )^7\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.94, size = 486, normalized size = 30.38 \begin {gather*} \frac {1}{8} x^{24} d^{8} + x^{22} d^{7} b + x^{21} d^{7} a + \frac {7}{2} x^{20} d^{6} b^{2} + 7 x^{19} d^{6} b a + 7 x^{18} d^{5} b^{3} + \frac {7}{2} x^{18} d^{6} a^{2} + 21 x^{17} d^{5} b^{2} a + \frac {35}{4} x^{16} d^{4} b^{4} + 21 x^{16} d^{5} b a^{2} + 35 x^{15} d^{4} b^{3} a + 7 x^{15} d^{5} a^{3} + 7 x^{14} d^{3} b^{5} + \frac {105}{2} x^{14} d^{4} b^{2} a^{2} + 35 x^{13} d^{3} b^{4} a + 35 x^{13} d^{4} b a^{3} + \frac {7}{2} x^{12} d^{2} b^{6} + 70 x^{12} d^{3} b^{3} a^{2} + \frac {35}{4} x^{12} d^{4} a^{4} + 21 x^{11} d^{2} b^{5} a + 70 x^{11} d^{3} b^{2} a^{3} + x^{10} d b^{7} + \frac {105}{2} x^{10} d^{2} b^{4} a^{2} + 35 x^{10} d^{3} b a^{4} + 7 x^{9} d b^{6} a + 70 x^{9} d^{2} b^{3} a^{3} + 7 x^{9} d^{3} a^{5} + \frac {1}{8} x^{8} b^{8} + 21 x^{8} d b^{5} a^{2} + \frac {105}{2} x^{8} d^{2} b^{2} a^{4} + x^{7} b^{7} a + 35 x^{7} d b^{4} a^{3} + 21 x^{7} d^{2} b a^{5} + \frac {7}{2} x^{6} b^{6} a^{2} + 35 x^{6} d b^{3} a^{4} + \frac {7}{2} x^{6} d^{2} a^{6} + 7 x^{5} b^{5} a^{3} + 21 x^{5} d b^{2} a^{5} + \frac {35}{4} x^{4} b^{4} a^{4} + 7 x^{4} d b a^{6} + 7 x^{3} b^{3} a^{5} + x^{3} d a^{7} + \frac {7}{2} x^{2} b^{2} a^{6} + x b a^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 120, normalized size = 7.50 \begin {gather*} \frac {1}{8} \, {\left (d x^{3} + b x\right )}^{8} + {\left (d x^{3} + b x\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + b x\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + b x\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + b x\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + b x\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + b x\right )}^{2} a^{6} + {\left (d x^{3} + b x\right )} a^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 2185, normalized size = 136.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{8} \, {\left (d x^{3} + b x + a\right )}^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 438, normalized size = 27.38 \begin {gather*} x^{12}\,\left (\frac {35\,a^4\,d^4}{4}+70\,a^2\,b^3\,d^3+\frac {7\,b^6\,d^2}{2}\right )+x^4\,\left (7\,d\,a^6\,b+\frac {35\,a^4\,b^4}{4}\right )+x^{18}\,\left (\frac {7\,a^2\,d^6}{2}+7\,b^3\,d^5\right )+x^6\,\left (\frac {7\,a^6\,d^2}{2}+35\,a^4\,b^3\,d+\frac {7\,a^2\,b^6}{2}\right )+x^8\,\left (\frac {105\,a^4\,b^2\,d^2}{2}+21\,a^2\,b^5\,d+\frac {b^8}{8}\right )+\frac {d^8\,x^{24}}{8}+x^3\,\left (d\,a^7+7\,a^5\,b^3\right )+a\,d^7\,x^{21}+b\,d^7\,x^{22}+\frac {7\,a^6\,b^2\,x^2}{2}+\frac {7\,b^2\,d^6\,x^{20}}{2}+a^7\,b\,x+21\,a\,b^2\,d^5\,x^{17}+a\,b\,x^7\,\left (21\,a^4\,d^2+35\,a^2\,b^3\,d+b^6\right )+7\,a\,d\,x^9\,\left (a^4\,d^2+10\,a^2\,b^3\,d+b^6\right )+7\,a^3\,b^2\,x^5\,\left (3\,d\,a^2+b^3\right )+7\,a\,d^4\,x^{15}\,\left (d\,a^2+5\,b^3\right )+\frac {7\,b\,d^4\,x^{16}\,\left (12\,d\,a^2+5\,b^3\right )}{4}+\frac {b\,d\,x^{10}\,\left (70\,a^4\,d^2+105\,a^2\,b^3\,d+2\,b^6\right )}{2}+7\,a\,b\,d^6\,x^{19}+\frac {7\,b^2\,d^3\,x^{14}\,\left (15\,d\,a^2+2\,b^3\right )}{2}+7\,a\,b^2\,d^2\,x^{11}\,\left (10\,d\,a^2+3\,b^3\right )+35\,a\,b\,d^3\,x^{13}\,\left (d\,a^2+b^3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 483, normalized size = 30.19 \begin {gather*} a^{7} b x + \frac {7 a^{6} b^{2} x^{2}}{2} + 21 a b^{2} d^{5} x^{17} + 7 a b d^{6} x^{19} + a d^{7} x^{21} + \frac {7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac {d^{8} x^{24}}{8} + x^{18} \left (\frac {7 a^{2} d^{6}}{2} + 7 b^{3} d^{5}\right ) + x^{16} \left (21 a^{2} b d^{5} + \frac {35 b^{4} d^{4}}{4}\right ) + x^{15} \left (7 a^{3} d^{5} + 35 a b^{3} d^{4}\right ) + x^{14} \left (\frac {105 a^{2} b^{2} d^{4}}{2} + 7 b^{5} d^{3}\right ) + x^{13} \left (35 a^{3} b d^{4} + 35 a b^{4} d^{3}\right ) + x^{12} \left (\frac {35 a^{4} d^{4}}{4} + 70 a^{2} b^{3} d^{3} + \frac {7 b^{6} d^{2}}{2}\right ) + x^{11} \left (70 a^{3} b^{2} d^{3} + 21 a b^{5} d^{2}\right ) + x^{10} \left (35 a^{4} b d^{3} + \frac {105 a^{2} b^{4} d^{2}}{2} + b^{7} d\right ) + x^{9} \left (7 a^{5} d^{3} + 70 a^{3} b^{3} d^{2} + 7 a b^{6} d\right ) + x^{8} \left (\frac {105 a^{4} b^{2} d^{2}}{2} + 21 a^{2} b^{5} d + \frac {b^{8}}{8}\right ) + x^{7} \left (21 a^{5} b d^{2} + 35 a^{3} b^{4} d + a b^{7}\right ) + x^{6} \left (\frac {7 a^{6} d^{2}}{2} + 35 a^{4} b^{3} d + \frac {7 a^{2} b^{6}}{2}\right ) + x^{5} \left (21 a^{5} b^{2} d + 7 a^{3} b^{5}\right ) + x^{4} \left (7 a^{6} b d + \frac {35 a^{4} b^{4}}{4}\right ) + x^{3} \left (a^{7} d + 7 a^{5} b^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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