Optimal. Leaf size=63 \[ -\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac {b x^2}{a}+1\right )}{2 a (2 p+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1113, 266, 65} \[ -\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac {b x^2}{a}+1\right )}{2 a (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 1113
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x} \, dx &=\left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^{2 p}}{x} \, dx\\ &=\frac {1}{2} \left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x}{a}\right )^{2 p}}{x} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,1+2 p;2 (1+p);1+\frac {b x^2}{a}\right )}{2 a (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 0.86 \[ -\frac {\left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^p \, _2F_1\left (1,2 p+1;2 p+2;\frac {b x^2}{a}+1\right )}{2 a (2 p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{x}\, dx \]
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 \[ \int \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p}{x} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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