Optimal. Leaf size=327 \[ -\frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left (6 n^2+5 n+1\right ) (b c-a d)}-\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{3 a n (b c-a d)} \]
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Rubi [A] time = 0.184194, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {382, 378, 191} \[ -\frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left (6 n^2+5 n+1\right ) (b c-a d)}-\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{3 a n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 382
Rule 378
Rule 191
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac{1}{n}} \, dx &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac{1}{n}} \, dx}{3 a}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{\left (n \left (3+\frac{b c}{b c n-a d n}\right )\right ) \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac{1}{n}} \, dx}{c (1+3 n)}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{n \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac{\left (2 a n^2 \left (3+\frac{b c}{b c n-a d n}\right )\right ) \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac{1}{n}} \, dx}{c^2 \left (1+5 n+6 n^2\right )}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{n \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac{2 a n^2 \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac{1}{n}}}{c^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac{\left (2 a^2 n^3 \left (3+\frac{b c}{b c n-a d n}\right )\right ) \int \left (c+d x^n\right )^{-1-\frac{1}{n}} \, dx}{c^3 (1+n) \left (1+5 n+6 n^2\right )}\\ &=-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a (b c-a d) n}+\frac{\left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac{1}{n}}}{3 a c (1+3 n)}+\frac{n \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac{1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac{2 a n^2 \left (3+\frac{b c}{b c n-a d n}\right ) x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac{1}{n}}}{c^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac{2 a^2 n^3 \left (3+\frac{b c}{b c n-a d n}\right ) x \left (c+d x^n\right )^{-1/n}}{c^4 (1+n) \left (1+5 n+6 n^2\right )}\\ \end{align*}
Mathematica [C] time = 0.198272, size = 153, normalized size = 0.47 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \left ((n+1) \left (a^2 (2 n+1) \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b^2 x^{2 n} \, _2F_1\left (2+\frac{1}{n},4+\frac{1}{n};3+\frac{1}{n};-\frac{d x^n}{c}\right )\right )+2 a b (2 n+1) x^n \, _2F_1\left (1+\frac{1}{n},4+\frac{1}{n};2+\frac{1}{n};-\frac{d x^n}{c}\right )\right )}{c^4 (n+1) (2 n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.586, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{-4-{n}^{-1}}\, 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*} \int{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65607, size = 819, normalized size = 2.5 \begin{align*} \frac{{\left (6 \, a^{2} d^{4} n^{3} + b^{2} c^{2} d^{2} n +{\left (b^{2} c^{2} d^{2} + 4 \, a b c d^{3}\right )} n^{2}\right )} x x^{4 \, n} +{\left (24 \, a^{2} c d^{3} n^{3} + b^{2} c^{3} d + 2 \,{\left (2 \, b^{2} c^{3} d + 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} n^{2} +{\left (5 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} +{\left (36 \, a^{2} c^{2} d^{2} n^{3} + b^{2} c^{4} + 2 \, a b c^{3} d + 3 \,{\left (b^{2} c^{4} + 8 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} n^{2} +{\left (4 \, b^{2} c^{4} + 14 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} +{\left (24 \, a^{2} c^{3} d n^{3} + 2 \, a b c^{4} + a^{2} c^{3} d + 2 \,{\left (6 \, a b c^{4} + 13 \, a^{2} c^{3} d\right )} n^{2} +{\left (10 \, a b c^{4} + 9 \, a^{2} c^{3} d\right )} n\right )} x x^{n} +{\left (6 \, a^{2} c^{4} n^{3} + 11 \, a^{2} c^{4} n^{2} + 6 \, a^{2} c^{4} n + a^{2} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )}{\left (d x^{n} + c\right )}^{\frac{4 \, n + 1}{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(-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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