Optimal. Leaf size=140 \[ \frac{256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac{32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}} \]
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Rubi [A] time = 0.0557573, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac{32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^{18}} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}}-\frac{(10 b) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx}{17 a}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}+\frac{\left (16 b^2\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{14}} \, dx}{51 a^2}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}-\frac{\left (32 b^3\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{221 a^3}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac{32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}+\frac{\left (128 b^4\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{2431 a^4}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac{32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}-\frac{\left (256 b^5\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{21879 a^5}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac{32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac{256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}\\ \end{align*}
Mathematica [A] time = 0.0177302, size = 75, normalized size = 0.54 \[ \frac{\left (a+b x^2\right )^{7/2} \left (2016 a^2 b^3 x^6-3696 a^3 b^2 x^4+6006 a^4 b x^2-9009 a^5-896 a b^4 x^8+256 b^5 x^{10}\right )}{153153 a^6 x^{17}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 72, normalized size = 0.5 \begin{align*} -{\frac{-256\,{b}^{5}{x}^{10}+896\,a{b}^{4}{x}^{8}-2016\,{a}^{2}{b}^{3}{x}^{6}+3696\,{a}^{3}{b}^{2}{x}^{4}-6006\,{a}^{4}b{x}^{2}+9009\,{a}^{5}}{153153\,{x}^{17}{a}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52008, size = 254, normalized size = 1.81 \begin{align*} \frac{{\left (256 \, b^{8} x^{16} - 128 \, a b^{7} x^{14} + 96 \, a^{2} b^{6} x^{12} - 80 \, a^{3} b^{5} x^{10} + 70 \, a^{4} b^{4} x^{8} - 63 \, a^{5} b^{3} x^{6} - 12705 \, a^{6} b^{2} x^{4} - 21021 \, a^{7} b x^{2} - 9009 \, a^{8}\right )} \sqrt{b x^{2} + a}}{153153 \, a^{6} x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.26149, size = 1346, normalized size = 9.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.46902, size = 443, normalized size = 3.16 \begin{align*} \frac{512 \,{\left (102102 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{22} b^{\frac{17}{2}} + 364650 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} a b^{\frac{17}{2}} + 692835 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a^{2} b^{\frac{17}{2}} + 668525 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{3} b^{\frac{17}{2}} + 384098 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{4} b^{\frac{17}{2}} + 89726 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{5} b^{\frac{17}{2}} + 6188 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{6} b^{\frac{17}{2}} - 2380 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{7} b^{\frac{17}{2}} + 680 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{8} b^{\frac{17}{2}} - 136 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{9} b^{\frac{17}{2}} + 17 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{10} b^{\frac{17}{2}} - a^{11} b^{\frac{17}{2}}\right )}}{153153 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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