Integrand size = 25, antiderivative size = 116 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c (1+2 n)}+\frac {2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 (1+n) (1+2 n)}+\frac {2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (1+n) (1+2 n)} \]
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Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {386, 197} \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (n+1) (2 n+1)}+\frac {2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c^2 (n+1) (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)} \]
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Rule 197
Rule 386
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c (1+2 n)}+\frac {(2 a n) \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx}{c (1+2 n)} \\ & = \frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c (1+2 n)}+\frac {2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 (1+n) (1+2 n)}+\frac {\left (2 a^2 n^2\right ) \int \left (c+d x^n\right )^{-1-\frac {1}{n}} \, dx}{c^2 (1+n) (1+2 n)} \\ & = \frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c (1+2 n)}+\frac {2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 (1+n) (1+2 n)}+\frac {2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (1+n) (1+2 n)} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-2-\frac {1}{n}} \left (b^2 c^2 (1+n) x^{2 n}+2 a b c x^n \left (c+2 c n+d n x^n\right )+a^2 \left (c^2 \left (1+3 n+2 n^2\right )+2 c d n (1+2 n) x^n+2 d^2 n^2 x^{2 n}\right )\right )}{c^3 (1+n) (1+2 n)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(587\) vs. \(2(116)=232\).
Time = 4.51 (sec) , antiderivative size = 588, normalized size of antiderivative = 5.07
method | result | size |
parallelrisch | \(\frac {2 x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} d^{3} n^{2}+2 x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b c \,d^{2} n +x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{2} d n +6 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c \,d^{2} n^{2}+x \,x^{3 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{2} d +2 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c \,d^{2} n +6 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{2} d n +x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{3} n +6 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{2} d \,n^{2}+2 x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{2} d +x \,x^{2 n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} b^{2} c^{3}+5 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{2} d n +4 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{3} n +2 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{3} n^{2}+x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{2} d +2 x \,x^{n} \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a b \,c^{3}+3 x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{3} n +x \left (c +d \,x^{n}\right )^{-\frac {1+3 n}{n}} a^{2} c^{3}}{\left (1+n \right ) \left (1+2 n \right ) c^{3}}\) | \(588\) |
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none
Time = 0.25 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.99 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\frac {{\left (2 \, a^{2} d^{3} n^{2} + b^{2} c^{2} d + {\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} n\right )} x x^{3 \, n} + {\left (6 \, a^{2} c d^{2} n^{2} + b^{2} c^{3} + 2 \, a b c^{2} d + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (6 \, a^{2} c^{2} d n^{2} + 2 \, a b c^{3} + a^{2} c^{2} d + {\left (4 \, a b c^{3} + 5 \, a^{2} c^{2} d\right )} n\right )} x x^{n} + {\left (2 \, a^{2} c^{3} n^{2} + 3 \, a^{2} c^{3} n + a^{2} c^{3}\right )} x}{{\left (2 \, c^{3} n^{2} + 3 \, c^{3} n + c^{3}\right )} {\left (d x^{n} + c\right )}^{\frac {3 \, n + 1}{n}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (104) = 208\).
Time = 12.92 (sec) , antiderivative size = 1035, normalized size of antiderivative = 8.92 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\text {Too large to display} \]
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\[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 3} \,d x } \]
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Exception generated. \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+3}} \,d x \]
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