Optimal. Leaf size=162 \[ \frac {x (a e-b c (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 b n}-\frac {x^{\frac {n+2}{2}} (b d (2-n)-a f (n+2)) \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right );\frac {1}{2} \left (3+\frac {2}{n}\right );-\frac {b x^n}{a}\right )}{a^2 b n (n+2)}+\frac {x \left (x^{n/2} (b d-a f)-a e+b c\right )}{a b n \left (a+b x^n\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1892, 1418, 245, 364} \[ \frac {x (a e-b c (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 b n}-\frac {x^{\frac {n+2}{2}} (b d (2-n)-a f (n+2)) \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right );\frac {1}{2} \left (3+\frac {2}{n}\right );-\frac {b x^n}{a}\right )}{a^2 b n (n+2)}+\frac {x \left (x^{n/2} (b d-a f)-a e+b c\right )}{a b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 245
Rule 364
Rule 1418
Rule 1892
Rubi steps
\begin {align*} \int \frac {c+d x^{n/2}+e x^n+f x^{3 n/2}}{\left (a+b x^n\right )^2} \, dx &=\frac {x \left (b c-a e+(b d-a f) x^{n/2}\right )}{a b n \left (a+b x^n\right )}+\frac {\int \frac {2 (a e-b c (1-n))-(b d (2-n)-a f (2+n)) x^{n/2}}{a+b x^n} \, dx}{2 a b n}\\ &=\frac {x \left (b c-a e+(b d-a f) x^{n/2}\right )}{a b n \left (a+b x^n\right )}+\frac {(a e-b c (1-n)) \int \frac {1}{a+b x^n} \, dx}{a b n}-\frac {(b d (2-n)-a f (2+n)) \int \frac {x^{n/2}}{a+b x^n} \, dx}{2 a b n}\\ &=\frac {x \left (b c-a e+(b d-a f) x^{n/2}\right )}{a b n \left (a+b x^n\right )}-\frac {(b d (2-n)-a f (2+n)) x^{\frac {2+n}{2}} \, _2F_1\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right );\frac {1}{2} \left (3+\frac {2}{n}\right );-\frac {b x^n}{a}\right )}{a^2 b n (2+n)}+\frac {(a e-b c (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 b n}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 147, normalized size = 0.91 \[ \frac {x \left ((b c-a e) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )+\frac {2 x^{n/2} (b d-a f) \, _2F_1\left (2,\frac {1}{2}+\frac {1}{n};\frac {3}{2}+\frac {1}{n};-\frac {b x^n}{a}\right )}{n+2}+a e \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )+\frac {2 a f x^{n/2} \, _2F_1\left (1,\frac {1}{2}+\frac {1}{n};\frac {3}{2}+\frac {1}{n};-\frac {b x^n}{a}\right )}{n+2}\right )}{a^2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f x^{\frac {3}{2} \, n} + d x^{\frac {1}{2} \, n} + e x^{n} + c}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
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 \[ \int \frac {f x^{\frac {3}{2} \, n} + d x^{\frac {1}{2} \, n} + e x^{n} + c}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{\frac {n}{2}}+e \,x^{n}+f \,x^{\frac {3 n}{2}}+c}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
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 \[ \frac {{\left (b d - a f\right )} x x^{\frac {1}{2} \, n} + {\left (b c - a e\right )} x}{a b^{2} n x^{n} + a^{2} b n} + \int \frac {2 \, b c {\left (n - 1\right )} + 2 \, a e + {\left (a f {\left (n + 2\right )} + b d {\left (n - 2\right )}\right )} x^{\frac {1}{2} \, n}}{2 \, {\left (a b^{2} n x^{n} + a^{2} b n\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c+e\,x^n+d\,x^{n/2}+f\,x^{\frac {3\,n}{2}}}{{\left (a+b\,x^n\right )}^2} \,d x \]
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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