Optimal. Leaf size=158 \[ -\frac {3 a^2}{2 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^3}{4 b^4 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1125, 660, 45}
\begin {gather*} -\frac {3 a^2}{2 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^3}{4 b^4 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rule 1125
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^3} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \left (\frac {1}{b^6}-\frac {a^3}{b^6 (a+b x)^3}+\frac {3 a^2}{b^6 (a+b x)^2}-\frac {3 a}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {3 a^2}{2 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^3}{4 b^4 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 81, normalized size = 0.51 \begin {gather*} \frac {-5 a^3-4 a^2 b x^2+4 a b^2 x^4+2 b^3 x^6-6 a \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 b^4 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 103, normalized size = 0.65
method | result | size |
default | \(-\frac {\left (-2 b^{3} x^{6}+6 \ln \left (b \,x^{2}+a \right ) a \,b^{2} x^{4}-4 a \,b^{2} x^{4}+12 \ln \left (b \,x^{2}+a \right ) a^{2} b \,x^{2}+4 a^{2} b \,x^{2}+6 \ln \left (b \,x^{2}+a \right ) a^{3}+5 a^{3}\right ) \left (b \,x^{2}+a \right )}{4 b^{4} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}\) | \(103\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{2}}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {3 a^{2} x^{2}}{2}-\frac {5 a^{3}}{4 b}\right )}{\left (b \,x^{2}+a \right )^{3} b^{3}}-\frac {3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, a \ln \left (b \,x^{2}+a \right )}{2 \left (b \,x^{2}+a \right ) b^{4}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 66, normalized size = 0.42 \begin {gather*} -\frac {6 \, a^{2} b x^{2} + 5 \, a^{3}}{4 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac {x^{2}}{2 \, b^{3}} - \frac {3 \, a \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 91, normalized size = 0.58 \begin {gather*} \frac {2 \, b^{3} x^{6} + 4 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} - 5 \, a^{3} - 6 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \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^{7}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.68, size = 92, normalized size = 0.58 \begin {gather*} \frac {x^{2}}{2 \, b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {3 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {9 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} + 4 \, a^{3}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} \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 = -0.01 \begin {gather*} \int \frac {x^7}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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