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 = 0.12993, antiderivative size = 158, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.158896, 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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Maple [A] time = 0.055, size = 164, normalized size = 1. \begin{align*}{a}^{2}{d}^{3}x+{\frac{{b}^{2}{e}^{3}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{1+5\,n}}+{\frac{d \left ( 3\,{a}^{2}{e}^{2}+6\,abde+{b}^{2}{d}^{2} \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{e \left ({a}^{2}{e}^{2}+6\,abde+3\,{b}^{2}{d}^{2} \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+{\frac{a{d}^{2} \left ( 3\,ae+2\,bd \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{b{e}^{2} \left ( 2\,ae+3\,bd \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{1+4\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70087, size = 1434, normalized size = 9.08 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.64841, size = 1278, normalized size = 8.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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