Optimal. Leaf size=127 \[ -\frac {(b c-a d) x \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c d (1+2 n)}+\frac {(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac {n (b c+2 a d n) x \left (c+d x^n\right )^{-1/n}}{c^3 d (1+n) (1+2 n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {393, 198, 197}
\begin {gather*} \frac {n x \left (c+d x^n\right )^{-1/n} (2 a d n+b c)}{c^3 d (n+1) (2 n+1)}+\frac {x \left (c+d x^n\right )^{-\frac {1}{n}-1} (2 a d n+b c)}{c^2 d (n+1) (2 n+1)}-\frac {x (b c-a d) \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c d (2 n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 393
Rubi steps
\begin {align*} \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx &=-\frac {(b c-a d) x \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c d (1+2 n)}+\frac {(b c+2 a d n) \int \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx}{c d (1+2 n)}\\ &=-\frac {(b c-a d) x \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c d (1+2 n)}+\frac {(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac {(n (b c+2 a d n)) \int \left (c+d x^n\right )^{-1-\frac {1}{n}} \, dx}{c^2 d (1+n) (1+2 n)}\\ &=-\frac {(b c-a d) x \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c d (1+2 n)}+\frac {(b c+2 a d n) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^2 d (1+n) (1+2 n)}+\frac {n (b c+2 a d n) x \left (c+d x^n\right )^{-1/n}}{c^3 d (1+n) (1+2 n)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.13, size = 94, normalized size = 0.74 \begin {gather*} \frac {x \left (c+d x^n\right )^{-1/n} \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (b c \, _2F_1\left (2+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )+(-b c+a d) \, _2F_1\left (3+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )\right )}{c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )^{-3-\frac {1}{n}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.47, size = 173, normalized size = 1.36 \begin {gather*} \frac {{\left (2 \, a d^{3} n^{2} + b c d^{2} n\right )} x x^{3 \, n} + {\left (6 \, a c d^{2} n^{2} + b c^{2} d + {\left (3 \, b c^{2} d + 2 \, a c d^{2}\right )} n\right )} x x^{2 \, n} + {\left (6 \, a c^{2} d n^{2} + b c^{3} + a c^{2} d + {\left (2 \, b c^{3} + 5 \, a c^{2} d\right )} n\right )} x x^{n} + {\left (2 \, a c^{3} n^{2} + 3 \, a c^{3} n + a 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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,x^n}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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