Optimal. Leaf size=250 \[ \frac {b^5 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {5 a b^4 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {10 a^2 b^3 \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^2 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 250, 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 = {1112, 266, 43} \[ -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^2 \left (a+b x^2\right )}+\frac {5 a b^4 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {b^5 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {10 a^2 b^3 \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^7} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^5}{x^7} \, dx}{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 \frac {\left (a b+b^2 x\right )^5}{x^4} \, 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 (5 a b^9+\frac {a^5 b^5}{x^4}+\frac {5 a^4 b^6}{x^3}+\frac {10 a^3 b^7}{x^2}+\frac {10 a^2 b^8}{x}+b^{10} x\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}}{6 x^6 \left (a+b x^2\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^2 \left (a+b x^2\right )}+\frac {5 a b^4 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {b^5 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 85, normalized size = 0.34 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (-2 a^5-15 a^4 b x^2-60 a^3 b^2 x^4+120 a^2 b^3 x^6 \log (x)+30 a b^4 x^8+3 b^5 x^{10}\right )}{12 x^6 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 61, normalized size = 0.24 \[ \frac {3 \, b^{5} x^{10} + 30 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} \log \relax (x) - 60 \, a^{3} b^{2} x^{4} - 15 \, a^{4} b x^{2} - 2 \, a^{5}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 128, normalized size = 0.51 \[ \frac {1}{4} \, b^{5} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{2} \, a b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {110 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 60 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 15 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 82, normalized size = 0.33 \[ \frac {\left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} \left (3 b^{5} x^{10}+30 a \,b^{4} x^{8}+120 a^{2} b^{3} x^{6} \ln \relax (x )-60 a^{3} b^{2} x^{4}-15 a^{4} b \,x^{2}-2 a^{5}\right )}{12 \left (b \,x^{2}+a \right )^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 56, normalized size = 0.22 \[ \frac {1}{4} \, b^{5} x^{4} + \frac {5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} \log \relax (x) - \frac {5 \, a^{3} b^{2}}{x^{2}} - \frac {5 \, a^{4} b}{4 \, x^{4}} - \frac {a^{5}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}}{x^7} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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