Optimal. Leaf size=58 \[ \frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {378, 191} \[ \frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 191
Rule 378
Rubi steps
\begin {align*} \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx &=\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c (1+n)}+\frac {(a n) \int \left (c+d x^n\right )^{-1-\frac {1}{n}} \, dx}{c (1+n)}\\ &=\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c (1+n)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (1+n)}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 82, normalized size = 1.41 \[ \frac {x \left (c+d x^n\right )^{-\frac {n+1}{n}} \left (a (n+1) \left (c+d x^n\right ) \left (\frac {d x^n}{c}+1\right )^{\frac {1}{n}} \, _2F_1\left (2+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )+b c x^n\right )}{c^2 (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 85, normalized size = 1.47 \[ \frac {{\left (a d^{2} n + b c d\right )} x x^{2 \, n} + {\left (2 \, a c d n + b c^{2} + a c d\right )} x x^{n} + {\left (a c^{2} n + a c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )} {\left (d x^{n} + c\right )}^{\frac {2 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right ) \left (d \,x^{n}+c \right )^{-\frac {1}{n}-2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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