Optimal. Leaf size=64 \[ \frac {x^2 \left (a+b x^n\right ) \, _2F_1\left (3,\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )}{2 a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 371}
\begin {gather*} \frac {x^2 \left (a+b x^n\right ) \, _2F_1\left (3,\frac {2}{n};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 1369
Rubi steps
\begin {align*} \int \frac {x}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {x}{\left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {x^2 \left (a+b x^n\right ) \, _2F_1\left (3,\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )}{2 a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 55, normalized size = 0.86 \begin {gather*} \frac {x^2 \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac {2}{n};1+\frac {2}{n};-\frac {b x^n}{a}\right )}{2 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 \[\int \frac {x}{\left (a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}\right )^{\frac {3}{2}}}\, dx\]
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 {x}{\left (\left (a + b x^{n}\right )^{2}\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 {x}{{\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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