Optimal. Leaf size=112 \[ \frac{x^{2 n+1} \left (a^2 e^2+4 a b d e+b^2 d^2\right )}{2 n+1}+a^2 d^2 x+\frac{2 a d x^{n+1} (a e+b d)}{n+1}+\frac{2 b e x^{3 n+1} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n+1}}{4 n+1} \]
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Rubi [A] time = 0.0816809, antiderivative size = 112, 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} \[ \frac{x^{2 n+1} \left (a^2 e^2+4 a b d e+b^2 d^2\right )}{2 n+1}+a^2 d^2 x+\frac{2 a d x^{n+1} (a e+b d)}{n+1}+\frac{2 b e x^{3 n+1} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n+1}}{4 n+1} \]
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 )^2 \, dx &=\int \left (a^2 d^2+2 a d (b d+a e) x^n+\left (b^2 d^2+4 a b d e+a^2 e^2\right ) x^{2 n}+2 b e (b d+a e) x^{3 n}+b^2 e^2 x^{4 n}\right ) \, dx\\ &=a^2 d^2 x+\frac{2 a d (b d+a e) x^{1+n}}{1+n}+\frac{\left (b^2 d^2+4 a b d e+a^2 e^2\right ) x^{1+2 n}}{1+2 n}+\frac{2 b e (b d+a e) x^{1+3 n}}{1+3 n}+\frac{b^2 e^2 x^{1+4 n}}{1+4 n}\\ \end{align*}
Mathematica [A] time = 0.157799, size = 105, normalized size = 0.94 \[ x \left (\frac{x^{2 n} \left (a^2 e^2+4 a b d e+b^2 d^2\right )}{2 n+1}+a^2 d^2+\frac{2 a d x^n (a e+b d)}{n+1}+\frac{2 b e x^{3 n} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n}}{4 n+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 117, normalized size = 1. \begin{align*}{a}^{2}{d}^{2}x+{\frac{ \left ({a}^{2}{e}^{2}+4\,abde+{b}^{2}{d}^{2} \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{{b}^{2}{e}^{2}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{1+4\,n}}+2\,{\frac{ad \left ( ae+bd \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+2\,{\frac{be \left ( ae+bd \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,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 = 1.64533, size = 787, normalized size = 7.03 \begin{align*} \frac{{\left (6 \, b^{2} e^{2} n^{3} + 11 \, b^{2} e^{2} n^{2} + 6 \, b^{2} e^{2} n + b^{2} e^{2}\right )} x x^{4 \, n} + 2 \,{\left (b^{2} d e + a b e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} n^{3} + 14 \,{\left (b^{2} d e + a b e^{2}\right )} n^{2} + 7 \,{\left (b^{2} d e + a b e^{2}\right )} n\right )} x x^{3 \, n} +{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2} + 12 \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} n^{3} + 19 \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} n^{2} + 8 \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} n\right )} x x^{2 \, n} + 2 \,{\left (a b d^{2} + a^{2} d e + 24 \,{\left (a b d^{2} + a^{2} d e\right )} n^{3} + 26 \,{\left (a b d^{2} + a^{2} d e\right )} n^{2} + 9 \,{\left (a b d^{2} + a^{2} d e\right )} n\right )} x x^{n} +{\left (24 \, a^{2} d^{2} n^{4} + 50 \, a^{2} d^{2} n^{3} + 35 \, a^{2} d^{2} n^{2} + 10 \, a^{2} d^{2} n + a^{2} d^{2}\right )} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, 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 = 1.13428, size = 728, normalized size = 6.5 \begin{align*} \frac{24 \, a^{2} d^{2} n^{4} x + 12 \, b^{2} d^{2} n^{3} x x^{2 \, n} + 48 \, a b d^{2} n^{3} x x^{n} + 16 \, b^{2} d n^{3} x x^{3 \, n} e + 48 \, a b d n^{3} x x^{2 \, n} e + 48 \, a^{2} d n^{3} x x^{n} e + 50 \, a^{2} d^{2} n^{3} x + 19 \, b^{2} d^{2} n^{2} x x^{2 \, n} + 52 \, a b d^{2} n^{2} x x^{n} + 6 \, b^{2} n^{3} x x^{4 \, n} e^{2} + 16 \, a b n^{3} x x^{3 \, n} e^{2} + 12 \, a^{2} n^{3} x x^{2 \, n} e^{2} + 28 \, b^{2} d n^{2} x x^{3 \, n} e + 76 \, a b d n^{2} x x^{2 \, n} e + 52 \, a^{2} d n^{2} x x^{n} e + 35 \, a^{2} d^{2} n^{2} x + 8 \, b^{2} d^{2} n x x^{2 \, n} + 18 \, a b d^{2} n x x^{n} + 11 \, b^{2} n^{2} x x^{4 \, n} e^{2} + 28 \, a b n^{2} x x^{3 \, n} e^{2} + 19 \, a^{2} n^{2} x x^{2 \, n} e^{2} + 14 \, b^{2} d n x x^{3 \, n} e + 32 \, a b d n x x^{2 \, n} e + 18 \, a^{2} d n x x^{n} e + 10 \, a^{2} d^{2} n x + b^{2} d^{2} x x^{2 \, n} + 2 \, a b d^{2} x x^{n} + 6 \, b^{2} n x x^{4 \, n} e^{2} + 14 \, a b n x x^{3 \, n} e^{2} + 8 \, a^{2} n x x^{2 \, n} e^{2} + 2 \, b^{2} d x x^{3 \, n} e + 4 \, a b d x x^{2 \, n} e + 2 \, a^{2} d x x^{n} e + a^{2} d^{2} x + b^{2} x x^{4 \, n} e^{2} + 2 \, a b x x^{3 \, n} e^{2} + a^{2} x x^{2 \, n} e^{2}}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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