Optimal. Leaf size=158 \[ \frac {d x^{2 n+1} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac {e x^{3 n+1} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3 x+\frac {a d^2 x^{n+1} (3 a e+2 b d)}{n+1}+\frac {b e^2 x^{4 n+1} (2 a e+3 b d)}{4 n+1}+\frac {b^2 e^3 x^{5 n+1}}{5 n+1} \]
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Rubi [A] time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {373} \[ \frac {d x^{2 n+1} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac {e x^{3 n+1} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3 x+\frac {a d^2 x^{n+1} (3 a e+2 b d)}{n+1}+\frac {b e^2 x^{4 n+1} (2 a e+3 b d)}{4 n+1}+\frac {b^2 e^3 x^{5 n+1}}{5 n+1} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx &=\int \left (a^2 d^3+a d^2 (2 b d+3 a e) x^n+d \left (b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^{2 n}+e \left (3 b^2 d^2+6 a b d e+a^2 e^2\right ) x^{3 n}+b e^2 (3 b d+2 a e) x^{4 n}+b^2 e^3 x^{5 n}\right ) \, dx\\ &=a^2 d^3 x+\frac {a d^2 (2 b d+3 a e) x^{1+n}}{1+n}+\frac {d \left (b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^{1+2 n}}{1+2 n}+\frac {e \left (3 b^2 d^2+6 a b d e+a^2 e^2\right ) x^{1+3 n}}{1+3 n}+\frac {b e^2 (3 b d+2 a e) x^{1+4 n}}{1+4 n}+\frac {b^2 e^3 x^{1+5 n}}{1+5 n}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 149, normalized size = 0.94 \[ x \left (\frac {d x^{2 n} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac {e x^{3 n} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3+\frac {a d^2 x^n (3 a e+2 b d)}{n+1}+\frac {b e^2 x^{4 n} (2 a e+3 b d)}{4 n+1}+\frac {b^2 e^3 x^{5 n}}{5 n+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 667, normalized size = 4.22 \[ \frac {{\left (24 \, b^{2} e^{3} n^{4} + 50 \, b^{2} e^{3} n^{3} + 35 \, b^{2} e^{3} n^{2} + 10 \, b^{2} e^{3} n + b^{2} e^{3}\right )} x x^{5 \, n} + {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3} + 30 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{4} + 61 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{3} + 41 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{2} + 11 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n\right )} x x^{4 \, n} + {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3} + 40 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{4} + 78 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{3} + 49 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{2} + 12 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n\right )} x x^{3 \, n} + {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + 60 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{4} + 107 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{3} + 59 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{2} + 13 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n\right )} x x^{2 \, n} + {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e + 120 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{4} + 154 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{3} + 71 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{2} + 14 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n\right )} x x^{n} + {\left (120 \, a^{2} d^{3} n^{5} + 274 \, a^{2} d^{3} n^{4} + 225 \, a^{2} d^{3} n^{3} + 85 \, a^{2} d^{3} n^{2} + 15 \, a^{2} d^{3} n + a^{2} d^{3}\right )} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 947, normalized size = 5.99 \[ \frac {120 \, a^{2} d^{3} n^{5} x + 60 \, b^{2} d^{3} n^{4} x x^{2 \, n} + 240 \, a b d^{3} n^{4} x x^{n} + 120 \, b^{2} d^{2} n^{4} x x^{3 \, n} e + 360 \, a b d^{2} n^{4} x x^{2 \, n} e + 360 \, a^{2} d^{2} n^{4} x x^{n} e + 274 \, a^{2} d^{3} n^{4} x + 107 \, b^{2} d^{3} n^{3} x x^{2 \, n} + 308 \, a b d^{3} n^{3} x x^{n} + 90 \, b^{2} d n^{4} x x^{4 \, n} e^{2} + 240 \, a b d n^{4} x x^{3 \, n} e^{2} + 180 \, a^{2} d n^{4} x x^{2 \, n} e^{2} + 234 \, b^{2} d^{2} n^{3} x x^{3 \, n} e + 642 \, a b d^{2} n^{3} x x^{2 \, n} e + 462 \, a^{2} d^{2} n^{3} x x^{n} e + 225 \, a^{2} d^{3} n^{3} x + 59 \, b^{2} d^{3} n^{2} x x^{2 \, n} + 142 \, a b d^{3} n^{2} x x^{n} + 24 \, b^{2} n^{4} x x^{5 \, n} e^{3} + 60 \, a b n^{4} x x^{4 \, n} e^{3} + 40 \, a^{2} n^{4} x x^{3 \, n} e^{3} + 183 \, b^{2} d n^{3} x x^{4 \, n} e^{2} + 468 \, a b d n^{3} x x^{3 \, n} e^{2} + 321 \, a^{2} d n^{3} x x^{2 \, n} e^{2} + 147 \, b^{2} d^{2} n^{2} x x^{3 \, n} e + 354 \, a b d^{2} n^{2} x x^{2 \, n} e + 213 \, a^{2} d^{2} n^{2} x x^{n} e + 85 \, a^{2} d^{3} n^{2} x + 13 \, b^{2} d^{3} n x x^{2 \, n} + 28 \, a b d^{3} n x x^{n} + 50 \, b^{2} n^{3} x x^{5 \, n} e^{3} + 122 \, a b n^{3} x x^{4 \, n} e^{3} + 78 \, a^{2} n^{3} x x^{3 \, n} e^{3} + 123 \, b^{2} d n^{2} x x^{4 \, n} e^{2} + 294 \, a b d n^{2} x x^{3 \, n} e^{2} + 177 \, a^{2} d n^{2} x x^{2 \, n} e^{2} + 36 \, b^{2} d^{2} n x x^{3 \, n} e + 78 \, a b d^{2} n x x^{2 \, n} e + 42 \, a^{2} d^{2} n x x^{n} e + 15 \, a^{2} d^{3} n x + b^{2} d^{3} x x^{2 \, n} + 2 \, a b d^{3} x x^{n} + 35 \, b^{2} n^{2} x x^{5 \, n} e^{3} + 82 \, a b n^{2} x x^{4 \, n} e^{3} + 49 \, a^{2} n^{2} x x^{3 \, n} e^{3} + 33 \, b^{2} d n x x^{4 \, n} e^{2} + 72 \, a b d n x x^{3 \, n} e^{2} + 39 \, a^{2} d n x x^{2 \, n} e^{2} + 3 \, b^{2} d^{2} x x^{3 \, n} e + 6 \, a b d^{2} x x^{2 \, n} e + 3 \, a^{2} d^{2} x x^{n} e + a^{2} d^{3} x + 10 \, b^{2} n x x^{5 \, n} e^{3} + 22 \, a b n x x^{4 \, n} e^{3} + 12 \, a^{2} n x x^{3 \, n} e^{3} + 3 \, b^{2} d x x^{4 \, n} e^{2} + 6 \, a b d x x^{3 \, n} e^{2} + 3 \, a^{2} d x x^{2 \, n} e^{2} + b^{2} x x^{5 \, n} e^{3} + 2 \, a b x x^{4 \, n} e^{3} + a^{2} x x^{3 \, n} e^{3}}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 164, normalized size = 1.04 \[ \frac {b^{2} e^{3} x \,{\mathrm e}^{5 n \ln \relax (x )}}{5 n +1}+a^{2} d^{3} x +\frac {\left (3 a e +2 b d \right ) a \,d^{2} x \,{\mathrm e}^{n \ln \relax (x )}}{n +1}+\frac {\left (2 a e +3 b d \right ) b \,e^{2} x \,{\mathrm e}^{4 n \ln \relax (x )}}{4 n +1}+\frac {\left (3 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) d x \,{\mathrm e}^{2 n \ln \relax (x )}}{2 n +1}+\frac {\left (a^{2} e^{2}+6 a b d e +3 b^{2} d^{2}\right ) e x \,{\mathrm e}^{3 n \ln \relax (x )}}{3 n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 242, normalized size = 1.53 \[ a^{2} d^{3} x + \frac {b^{2} e^{3} x^{5 \, n + 1}}{5 \, n + 1} + \frac {3 \, b^{2} d e^{2} x^{4 \, n + 1}}{4 \, n + 1} + \frac {2 \, a b e^{3} x^{4 \, n + 1}}{4 \, n + 1} + \frac {3 \, b^{2} d^{2} e x^{3 \, n + 1}}{3 \, n + 1} + \frac {6 \, a b d e^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {a^{2} e^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {b^{2} d^{3} x^{2 \, n + 1}}{2 \, n + 1} + \frac {6 \, a b d^{2} e x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} d e^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b d^{3} x^{n + 1}}{n + 1} + \frac {3 \, a^{2} d^{2} e x^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 157, normalized size = 0.99 \[ a^2\,d^3\,x+\frac {x\,x^{2\,n}\,\left (3\,a^2\,d\,e^2+6\,a\,b\,d^2\,e+b^2\,d^3\right )}{2\,n+1}+\frac {x\,x^{3\,n}\,\left (a^2\,e^3+6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )}{3\,n+1}+\frac {b^2\,e^3\,x\,x^{5\,n}}{5\,n+1}+\frac {a\,d^2\,x\,x^n\,\left (3\,a\,e+2\,b\,d\right )}{n+1}+\frac {b\,e^2\,x\,x^{4\,n}\,\left (2\,a\,e+3\,b\,d\right )}{4\,n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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