Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.15, antiderivative size = 255, 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 = {1111, 646, 43} \[ -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{25}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^{13}} \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (\frac {a^5 b^5}{x^{13}}+\frac {5 a^4 b^6}{x^{12}}+\frac {10 a^3 b^7}{x^{11}}+\frac {10 a^2 b^8}{x^{10}}+\frac {5 a b^9}{x^9}+\frac {b^{10}}{x^8}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{24 x^{24} \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{22 x^{22} \left (a+b x^2\right )}-\frac {a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^{20} \left (a+b x^2\right )}-\frac {5 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 x^{18} \left (a+b x^2\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 x^{16} \left (a+b x^2\right )}-\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (462 a^5+2520 a^4 b x^2+5544 a^3 b^2 x^4+6160 a^2 b^3 x^6+3465 a b^4 x^8+792 b^5 x^{10}\right )}{11088 x^{24} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 59, normalized size = 0.23 \[ -\frac {792 \, b^{5} x^{10} + 3465 \, a b^{4} x^{8} + 6160 \, a^{2} b^{3} x^{6} + 5544 \, a^{3} b^{2} x^{4} + 2520 \, a^{4} b x^{2} + 462 \, a^{5}}{11088 \, x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 107, normalized size = 0.42 \[ -\frac {792 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 3465 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 6160 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 5544 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 2520 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 462 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{11088 \, x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.31 \[ -\frac {\left (792 b^{5} x^{10}+3465 a \,b^{4} x^{8}+6160 a^{2} b^{3} x^{6}+5544 a^{3} b^{2} x^{4}+2520 a^{4} b \,x^{2}+462 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}{11088 \left (b \,x^{2}+a \right )^{5} x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 57, normalized size = 0.22 \[ -\frac {b^{5}}{14 \, x^{14}} - \frac {5 \, a b^{4}}{16 \, x^{16}} - \frac {5 \, a^{2} b^{3}}{9 \, x^{18}} - \frac {a^{3} b^{2}}{2 \, x^{20}} - \frac {5 \, a^{4} b}{22 \, x^{22}} - \frac {a^{5}}{24 \, x^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 231, normalized size = 0.91 \[ -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{24\,x^{24}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{16\,x^{16}\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{22\,x^{22}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{9\,x^{18}\,\left (b\,x^2+a\right )}-\frac {a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2\,x^{20}\,\left (b\,x^2+a\right )} \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{25}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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