Optimal. Leaf size=64 \[ \frac {b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);1+\frac {b x^2}{a}\right )}{2 a^2 (1+2 p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 64, 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 = {1127, 272, 67}
\begin {gather*} \frac {b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);\frac {b x^2}{a}+1\right )}{2 a^2 (2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 1127
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, 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^3} \, 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 ) \text {Subst}\left (\int \frac {\left (1+\frac {b x}{a}\right )^{2 p}}{x^2} \, dx,x,x^2\right )\\ &=\frac {b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);1+\frac {b x^2}{a}\right )}{2 a^2 (1+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.86 \begin {gather*} \frac {b \left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^p \, _2F_1\left (2,1+2 p;2+2 p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{x^{3}}\, dx\]
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 {\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{x^{3}}\, 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 {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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