Optimal. Leaf size=118 \[ \frac{c^2 (b c-a d)^2 \log \left (c+d x^n\right )}{d^5 n}-\frac{c x^n (b c-a d)^2}{d^4 n}+\frac{x^{2 n} (b c-a d)^2}{2 d^3 n}-\frac{b x^{3 n} (b c-2 a d)}{3 d^2 n}+\frac{b^2 x^{4 n}}{4 d n} \]
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Rubi [A] time = 0.115469, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {446, 88} \[ \frac{c^2 (b c-a d)^2 \log \left (c+d x^n\right )}{d^5 n}-\frac{c x^n (b c-a d)^2}{d^4 n}+\frac{x^{2 n} (b c-a d)^2}{2 d^3 n}-\frac{b x^{3 n} (b c-2 a d)}{3 d^2 n}+\frac{b^2 x^{4 n}}{4 d n} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rubi steps
\begin{align*} \int \frac{x^{-1+3 n} \left (a+b x^n\right )^2}{c+d x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 (a+b x)^2}{c+d x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c (b c-a d)^2}{d^4}+\frac{(-b c+a d)^2 x}{d^3}-\frac{b (b c-2 a d) x^2}{d^2}+\frac{b^2 x^3}{d}+\frac{c^2 (b c-a d)^2}{d^4 (c+d x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{c (b c-a d)^2 x^n}{d^4 n}+\frac{(b c-a d)^2 x^{2 n}}{2 d^3 n}-\frac{b (b c-2 a d) x^{3 n}}{3 d^2 n}+\frac{b^2 x^{4 n}}{4 d n}+\frac{c^2 (b c-a d)^2 \log \left (c+d x^n\right )}{d^5 n}\\ \end{align*}
Mathematica [A] time = 0.146358, size = 103, normalized size = 0.87 \[ \frac{12 c^2 (b c-a d)^2 \log \left (c+d x^n\right )+6 d^2 x^{2 n} (b c-a d)^2-4 b d^3 x^{3 n} (b c-2 a d)-12 c d x^n (b c-a d)^2+3 b^2 d^4 x^{4 n}}{12 d^5 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 236, normalized size = 2. \begin{align*}{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,dn}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{a}^{2}}{2\,dn}}-{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}abc}{{d}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}{b}^{2}{c}^{2}}{2\,{d}^{3}n}}+{\frac{2\,b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}a}{3\,dn}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}c}{3\,{d}^{2}n}}-{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}{a}^{2}}{{d}^{2}n}}+2\,{\frac{{c}^{2}{{\rm e}^{n\ln \left ( x \right ) }}ab}{{d}^{3}n}}-{\frac{{c}^{3}{{\rm e}^{n\ln \left ( x \right ) }}{b}^{2}}{{d}^{4}n}}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){a}^{2}}{{d}^{3}n}}-2\,{\frac{{c}^{3}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) ab}{{d}^{4}n}}+{\frac{{c}^{4}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ){b}^{2}}{{d}^{5}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961493, size = 259, normalized size = 2.19 \begin{align*} \frac{1}{12} \, b^{2}{\left (\frac{12 \, c^{4} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{5} n} + \frac{3 \, d^{3} x^{4 \, n} - 4 \, c d^{2} x^{3 \, n} + 6 \, c^{2} d x^{2 \, n} - 12 \, c^{3} x^{n}}{d^{4} n}\right )} - \frac{1}{3} \, a b{\left (\frac{6 \, c^{3} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{4} n} - \frac{2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac{1}{2} \, a^{2}{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06209, size = 306, normalized size = 2.59 \begin{align*} \frac{3 \, b^{2} d^{4} x^{4 \, n} - 4 \,{\left (b^{2} c d^{3} - 2 \, a b d^{4}\right )} x^{3 \, n} + 6 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2 \, n} - 12 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{n} + 12 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \log \left (d x^{n} + c\right )}{12 \, d^{5} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{2} x^{3 \, n - 1}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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