Optimal. Leaf size=41 \[ \frac {x^8}{8 a \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 37} \[ \frac {x^8}{8 a \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right )\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {x^8}{8 a \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 1.49 \[ \frac {-a^3-4 a^2 b x^2-6 a b^2 x^4-4 b^3 x^6}{8 b^4 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 80, normalized size = 1.95 \[ -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 54, normalized size = 1.32 \[ -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, {\left (b x^{2} + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.32 \[ -\frac {\left (b \,x^{2}+a \right ) \left (4 b^{3} x^{6}+6 a \,b^{2} x^{4}+4 a^{2} b \,x^{2}+a^{3}\right )}{8 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 80, normalized size = 1.95 \[ -\frac {4 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} + a^{3}}{8 \, {\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.29, size = 144, normalized size = 3.51 \[ \frac {a^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,b^4\,{\left (b\,x^2+a\right )}^5}-\frac {a^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,b^4\,{\left (b\,x^2+a\right )}^4}-\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,b^4\,{\left (b\,x^2+a\right )}^2}+\frac {3\,a\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,b^4\,{\left (b\,x^2+a\right )}^3} \]
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 \[ \int \frac {x^{7}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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