Optimal. Leaf size=173 \[ \frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (6 n^2+7 n+2\right )+b^2 c^2 \left (6 n^2+4 n+1\right )\right )}{b^3 (n+1) (2 n+1)}+\frac{x (b c-a d)^3 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^3}-\frac{d x \left (c+d x^n\right ) (a d (2 n+1)-b (4 c n+c))}{b^2 (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2}{b (2 n+1)} \]
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Rubi [A] time = 0.267249, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {416, 528, 388, 245} \[ \frac{d x \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (6 n^2+7 n+2\right )+b^2 c^2 \left (6 n^2+4 n+1\right )\right )}{b^3 (n+1) (2 n+1)}+\frac{x (b c-a d)^3 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^3}-\frac{d x \left (c+d x^n\right ) (a d (2 n+1)-b (4 c n+c))}{b^2 (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2}{b (2 n+1)} \]
Antiderivative was successfully verified.
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Rule 416
Rule 528
Rule 388
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (c+d x^n\right )^3}{a+b x^n} \, dx &=\frac{d x \left (c+d x^n\right )^2}{b (1+2 n)}+\frac{\int \frac{\left (c+d x^n\right ) \left (-c (a d-b (c+2 c n))-d (a d (1+2 n)-b (c+4 c n)) x^n\right )}{a+b x^n} \, dx}{b (1+2 n)}\\ &=-\frac{d (a d (1+2 n)-b (c+4 c n)) x \left (c+d x^n\right )}{b^2 (1+n) (1+2 n)}+\frac{d x \left (c+d x^n\right )^2}{b (1+2 n)}+\frac{\int \frac{c \left (a^2 d^2 (1+2 n)-a b c d (2+5 n)+b^2 c^2 \left (1+3 n+2 n^2\right )\right )+d \left (a^2 d^2 \left (1+3 n+2 n^2\right )+b^2 c^2 \left (1+4 n+6 n^2\right )-a b c d \left (2+7 n+6 n^2\right )\right ) x^n}{a+b x^n} \, dx}{b^2 (1+n) (1+2 n)}\\ &=\frac{d \left (a^2 d^2 \left (1+3 n+2 n^2\right )+b^2 c^2 \left (1+4 n+6 n^2\right )-a b c d \left (2+7 n+6 n^2\right )\right ) x}{b^3 (1+n) (1+2 n)}-\frac{d (a d (1+2 n)-b (c+4 c n)) x \left (c+d x^n\right )}{b^2 (1+n) (1+2 n)}+\frac{d x \left (c+d x^n\right )^2}{b (1+2 n)}+\frac{(b c-a d)^3 \int \frac{1}{a+b x^n} \, dx}{b^3}\\ &=\frac{d \left (a^2 d^2 \left (1+3 n+2 n^2\right )+b^2 c^2 \left (1+4 n+6 n^2\right )-a b c d \left (2+7 n+6 n^2\right )\right ) x}{b^3 (1+n) (1+2 n)}-\frac{d (a d (1+2 n)-b (c+4 c n)) x \left (c+d x^n\right )}{b^2 (1+n) (1+2 n)}+\frac{d x \left (c+d x^n\right )^2}{b (1+2 n)}+\frac{(b c-a d)^3 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b^3}\\ \end{align*}
Mathematica [C] time = 0.982742, size = 104, normalized size = 0.6 \[ \frac{x \left (3 c^2 d x^n \Phi \left (-\frac{b x^n}{a},1,1+\frac{1}{n}\right )+c^3 \Phi \left (-\frac{b x^n}{a},1,\frac{1}{n}\right )+3 c d^2 x^{2 n} \Phi \left (-\frac{b x^n}{a},1,2+\frac{1}{n}\right )+d^3 x^{3 n} \Phi \left (-\frac{b x^n}{a},1,3+\frac{1}{n}\right )\right )}{a n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.382, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c+d{x}^{n} \right ) ^{3}}{a+b{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \int \frac{1}{b^{4} x^{n} + a b^{3}}\,{d x} + \frac{b^{2} d^{3}{\left (n + 1\right )} x x^{2 \, n} +{\left (3 \, b^{2} c d^{2}{\left (2 \, n + 1\right )} - a b d^{3}{\left (2 \, n + 1\right )}\right )} x x^{n} +{\left (3 \,{\left (2 \, n^{2} + 3 \, n + 1\right )} b^{2} c^{2} d - 3 \,{\left (2 \, n^{2} + 3 \, n + 1\right )} a b c d^{2} +{\left (2 \, n^{2} + 3 \, n + 1\right )} a^{2} d^{3}\right )} x}{{\left (2 \, n^{2} + 3 \, n + 1\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}{b x^{n} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.7349, size = 269, normalized size = 1.55 \begin{align*} - \frac{3 c^{2} d x \Phi \left (\frac{a x^{- n} e^{i \pi }}{b}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{b n^{2} \Gamma \left (1 + \frac{1}{n}\right )} + \frac{c^{3} x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac{1}{n}\right )} + \frac{6 c d^{2} x x^{2 n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 2 + \frac{1}{n}\right ) \Gamma \left (2 + \frac{1}{n}\right )}{a n \Gamma \left (3 + \frac{1}{n}\right )} + \frac{3 c d^{2} x x^{2 n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 2 + \frac{1}{n}\right ) \Gamma \left (2 + \frac{1}{n}\right )}{a n^{2} \Gamma \left (3 + \frac{1}{n}\right )} + \frac{3 d^{3} x x^{3 n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 3 + \frac{1}{n}\right ) \Gamma \left (3 + \frac{1}{n}\right )}{a n \Gamma \left (4 + \frac{1}{n}\right )} + \frac{d^{3} x x^{3 n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 3 + \frac{1}{n}\right ) \Gamma \left (3 + \frac{1}{n}\right )}{a n^{2} \Gamma \left (4 + \frac{1}{n}\right )} \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 (d x^{n} + c\right )}^{3}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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