Optimal. Leaf size=341 \[ -\frac{d x \left (c+d x^n\right ) \left (a^2 d^2 \left (6 n^2+5 n+1\right )-2 a b c d \left (5 n^2+4 n+1\right )+b^2 c^2 \left (2 n^2+3 n+1\right )\right )}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left (a^2 b c d^2 \left (16 n^3+26 n^2+15 n+3\right )-a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )-a b^2 c^2 d \left (12 n^3+17 n^2+12 n+3\right )+b^3 c^3 \left (2 n^2+3 n+1\right )\right )}{a b^4 n (n+1) (2 n+1)}-\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}+\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.547905, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {413, 528, 388, 245} \[ -\frac{d x \left (c+d x^n\right ) \left (a^2 d^2 \left (6 n^2+5 n+1\right )-2 a b c d \left (5 n^2+4 n+1\right )+b^2 c^2 \left (2 n^2+3 n+1\right )\right )}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left (a^2 b c d^2 \left (16 n^3+26 n^2+15 n+3\right )-a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )-a b^2 c^2 d \left (12 n^3+17 n^2+12 n+3\right )+b^3 c^3 \left (2 n^2+3 n+1\right )\right )}{a b^4 n (n+1) (2 n+1)}-\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}+\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 413
Rule 528
Rule 388
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (c+d x^n\right )^4}{\left (a+b x^n\right )^2} \, dx &=\frac{(b c-a d) x \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )}+\frac{\int \frac{\left (c+d x^n\right )^2 \left (c (a d-b c (1-n))+d (a d (1+3 n)-b (c+2 c n)) x^n\right )}{a+b x^n} \, dx}{a b n}\\ &=\frac{d (a d (1+3 n)-b (c+2 c n)) x \left (c+d x^n\right )^2}{a b^2 n (1+2 n)}+\frac{(b c-a d) x \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )}+\frac{\int \frac{\left (c+d x^n\right ) \left (c \left (2 a b c d (1+2 n)-a^2 d^2 (1+3 n)-b^2 c^2 \left (1+n-2 n^2\right )\right )-d \left (b^2 c^2 \left (1+3 n+2 n^2\right )-2 a b c d \left (1+4 n+5 n^2\right )+a^2 d^2 \left (1+5 n+6 n^2\right )\right ) x^n\right )}{a+b x^n} \, dx}{a b^2 n (1+2 n)}\\ &=-\frac{d \left (b^2 c^2 \left (1+3 n+2 n^2\right )-2 a b c d \left (1+4 n+5 n^2\right )+a^2 d^2 \left (1+5 n+6 n^2\right )\right ) x \left (c+d x^n\right )}{a b^3 n (1+n) (1+2 n)}+\frac{d (a d (1+3 n)-b (c+2 c n)) x \left (c+d x^n\right )^2}{a b^2 n (1+2 n)}+\frac{(b c-a d) x \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )}+\frac{\int \frac{c \left (3 a b^2 c^2 d \left (1+3 n+2 n^2\right )+a^3 d^3 \left (1+5 n+6 n^2\right )-a^2 b c d^2 \left (3+12 n+13 n^2\right )-b^3 c^3 \left (1+2 n-n^2-2 n^3\right )\right )-d \left (b^3 c^3 \left (1+3 n+2 n^2\right )-a^3 d^3 \left (1+6 n+11 n^2+6 n^3\right )-a b^2 c^2 d \left (3+12 n+17 n^2+12 n^3\right )+a^2 b c d^2 \left (3+15 n+26 n^2+16 n^3\right )\right ) x^n}{a+b x^n} \, dx}{a b^3 n (1+n) (1+2 n)}\\ &=-\frac{d \left (b^3 c^3 \left (1+3 n+2 n^2\right )-a^3 d^3 \left (1+6 n+11 n^2+6 n^3\right )-a b^2 c^2 d \left (3+12 n+17 n^2+12 n^3\right )+a^2 b c d^2 \left (3+15 n+26 n^2+16 n^3\right )\right ) x}{a b^4 n (1+n) (1+2 n)}-\frac{d \left (b^2 c^2 \left (1+3 n+2 n^2\right )-2 a b c d \left (1+4 n+5 n^2\right )+a^2 d^2 \left (1+5 n+6 n^2\right )\right ) x \left (c+d x^n\right )}{a b^3 n (1+n) (1+2 n)}+\frac{d (a d (1+3 n)-b (c+2 c n)) x \left (c+d x^n\right )^2}{a b^2 n (1+2 n)}+\frac{(b c-a d) x \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )}-\frac{\left ((b c-a d)^3 (b c (1-n)-a d (1+3 n))\right ) \int \frac{1}{a+b x^n} \, dx}{a b^4 n}\\ &=-\frac{d \left (b^3 c^3 \left (1+3 n+2 n^2\right )-a^3 d^3 \left (1+6 n+11 n^2+6 n^3\right )-a b^2 c^2 d \left (3+12 n+17 n^2+12 n^3\right )+a^2 b c d^2 \left (3+15 n+26 n^2+16 n^3\right )\right ) x}{a b^4 n (1+n) (1+2 n)}-\frac{d \left (b^2 c^2 \left (1+3 n+2 n^2\right )-2 a b c d \left (1+4 n+5 n^2\right )+a^2 d^2 \left (1+5 n+6 n^2\right )\right ) x \left (c+d x^n\right )}{a b^3 n (1+n) (1+2 n)}+\frac{d (a d (1+3 n)-b (c+2 c n)) x \left (c+d x^n\right )^2}{a b^2 n (1+2 n)}+\frac{(b c-a d) x \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )}-\frac{(b c-a d)^3 (b c (1-n)-a d (1+3 n)) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}\\ \end{align*}
Mathematica [A] time = 5.18974, size = 217, normalized size = 0.64 \[ \frac{x \left (\frac{-6 a^2 b^2 c^2 d^2+4 a^3 b c d^3-a^4 d^4+4 a b^3 c^3 d+b^4 c^4 (n-1)}{a^2 n}+\frac{(b c-a d)^3 (a d (3 n+1)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n}+\frac{(a d-b c)^3 (a d (3 n+1)+b c (n-1))}{a^2 n}+\frac{2 b d^3 x^n (2 b c-a d)}{n+1}+\frac{(b c-a d)^4}{a n \left (a+b x^n\right )}+\frac{b^2 d^4 x^{2 n}}{2 n+1}\right )}{b^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.379, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c+d{x}^{n} \right ) ^{4}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, 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 (a^{4} d^{4}{\left (3 \, n + 1\right )} - 4 \, a^{3} b c d^{3}{\left (2 \, n + 1\right )} + 6 \, a^{2} b^{2} c^{2} d^{2}{\left (n + 1\right )} - b^{4} c^{4}{\left (n - 1\right )} - 4 \, a b^{3} c^{3} d\right )} \int \frac{1}{a b^{5} n x^{n} + a^{2} b^{4} n}\,{d x} + \frac{{\left (n^{2} + n\right )} a b^{3} d^{4} x x^{3 \, n} +{\left (4 \,{\left (2 \, n^{2} + n\right )} a b^{3} c d^{3} -{\left (3 \, n^{2} + n\right )} a^{2} b^{2} d^{4}\right )} x x^{2 \, n} +{\left (6 \,{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a b^{3} c^{2} d^{2} - 4 \,{\left (4 \, n^{3} + 4 \, n^{2} + n\right )} a^{2} b^{2} c d^{3} +{\left (6 \, n^{3} + 5 \, n^{2} + n\right )} a^{3} b d^{4}\right )} x x^{n} +{\left ({\left (2 \, n^{2} + 3 \, n + 1\right )} b^{4} c^{4} - 4 \,{\left (2 \, n^{2} + 3 \, n + 1\right )} a b^{3} c^{3} d + 6 \,{\left (2 \, n^{3} + 5 \, n^{2} + 4 \, n + 1\right )} a^{2} b^{2} c^{2} d^{2} - 4 \,{\left (4 \, n^{3} + 8 \, n^{2} + 5 \, n + 1\right )} a^{3} b c d^{3} +{\left (6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1\right )} a^{4} d^{4}\right )} x}{{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a b^{5} x^{n} +{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a^{2} b^{4}} \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^{4} x^{4 \, n} + 4 \, c d^{3} x^{3 \, n} + 6 \, c^{2} d^{2} x^{2 \, n} + 4 \, c^{3} d x^{n} + c^{4}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \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 (d x^{n} + c\right )}^{4}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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