Optimal. Leaf size=167 \[ -\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 167, 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} \begin {gather*} -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \end {gather*}
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 )^{3/2}}{x^{15}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^8} \, 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 )^3}{x^8} \, dx,x,x^2\right )}{2 b^2 \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^3 b^3}{x^8}+\frac {3 a^2 b^4}{x^7}+\frac {3 a b^5}{x^6}+\frac {b^6}{x^5}\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 x^{14} \left (a+b x^2\right )}-\frac {a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 x^{12} \left (a+b x^2\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.37 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (20 a^3+70 a^2 b x^2+84 a b^2 x^4+35 b^3 x^6\right )}{280 x^{14} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.17, size = 444, normalized size = 2.66 \begin {gather*} \frac {8 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-20 a^9 b-190 a^8 b^2 x^2-804 a^7 b^3 x^4-1989 a^6 b^4 x^6-3170 a^5 b^5 x^8-3375 a^4 b^6 x^{10}-2400 a^3 b^7 x^{12}-1099 a^2 b^8 x^{14}-294 a b^9 x^{16}-35 b^{10} x^{18}\right )+8 \sqrt {b^2} b^6 \left (20 a^{10}+210 a^9 b x^2+994 a^8 b^2 x^4+2793 a^7 b^3 x^6+5159 a^6 b^4 x^8+6545 a^5 b^5 x^{10}+5775 a^4 b^6 x^{12}+3499 a^3 b^7 x^{14}+1393 a^2 b^8 x^{16}+329 a b^9 x^{18}+35 b^{10} x^{20}\right )}{35 \sqrt {b^2} x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4} \left (-64 a^6 b^6-384 a^5 b^7 x^2-960 a^4 b^8 x^4-1280 a^3 b^9 x^6-960 a^2 b^{10} x^8-384 a b^{11} x^{10}-64 b^{12} x^{12}\right )+35 x^{14} \left (64 a^7 b^7+448 a^6 b^8 x^2+1344 a^5 b^9 x^4+2240 a^4 b^{10} x^6+2240 a^3 b^{11} x^8+1344 a^2 b^{12} x^{10}+448 a b^{13} x^{12}+64 b^{14} x^{14}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 37, normalized size = 0.22 \begin {gather*} -\frac {35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 69, normalized size = 0.41 \begin {gather*} -\frac {35 \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 84 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 70 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 20 \, a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{280 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.35 \begin {gather*} -\frac {\left (35 b^{3} x^{6}+84 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+20 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{280 \left (b \,x^{2}+a \right )^{3} x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 35, normalized size = 0.21 \begin {gather*} -\frac {b^{3}}{8 \, x^{8}} - \frac {3 \, a b^{2}}{10 \, x^{10}} - \frac {a^{2} b}{4 \, x^{12}} - \frac {a^{3}}{14 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.21, size = 151, normalized size = 0.90 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{8\,x^8\,\left (b\,x^2+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{10\,x^{10}\,\left (b\,x^2+a\right )}-\frac {a^2\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^{12}\,\left (b\,x^2+a\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 {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{15}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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