Optimal. Leaf size=73 \[ -\frac {(b c-a d) x}{c d n \left (c+d x^n\right )}+\frac {(b c-a d (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d n} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {393, 251}
\begin {gather*} \frac {x (b c-a d (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d n}-\frac {x (b c-a d)}{c d n \left (c+d x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 393
Rubi steps
\begin {align*} \int \frac {a+b x^n}{\left (c+d x^n\right )^2} \, dx &=-\frac {(b c-a d) x}{c d n \left (c+d x^n\right )}+\frac {(b c-a d (1-n)) \int \frac {1}{c+d x^n} \, dx}{c d n}\\ &=-\frac {(b c-a d) x}{c d n \left (c+d x^n\right )}+\frac {(b c-a d (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2 d n}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 56, normalized size = 0.77 \begin {gather*} \frac {x \left (\frac {b}{c+d x^n}-\frac {(b c+a d (-1+n)) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c^2}\right )}{d-d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \,x^{n}}{\left (c +d \,x^{n}\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.49, size = 592, normalized size = 8.11 \begin {gather*} a \left (\frac {n x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {d n x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {d x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )}\right ) + b \left (\frac {n^{2} x x^{n} \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {n x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} + \frac {n x x^{n} \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {x x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {d n x x^{2 n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )} - \frac {d x x^{2 n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, 1 + \frac {1}{n}\right ) \Gamma \left (1 + \frac {1}{n}\right )}{c^{2} \left (c n^{3} \Gamma \left (2 + \frac {1}{n}\right ) + d n^{3} x^{n} \Gamma \left (2 + \frac {1}{n}\right )\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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