Optimal. Leaf size=57 \[ \frac {x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1357, 251}
\begin {gather*} \frac {x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac {\left (2 a b+2 b^2 x^n\right )^3 \int \frac {1}{\left (2 a b+2 b^2 x^n\right )^3} \, dx}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}\\ &=\frac {x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.81 \begin {gather*} \frac {x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 \left (\left (a+b x^n\right )^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}\right )^{\frac {3}{2}}}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac {3}{2}}}\, dx \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 = -0.02 \begin {gather*} \int \frac {1}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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