Optimal. Leaf size=116 \[ -\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac{64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac{8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}} \]
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Rubi [A] time = 0.0422674, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac{64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac{8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}} \]
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^{16}} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}}-\frac{(8 b) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{14}} \, dx}{15 a}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac{8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}+\frac{\left (16 b^2\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{65 a^2}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac{8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}-\frac{\left (64 b^3\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{715 a^3}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac{8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac{64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}+\frac{\left (128 b^4\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{6435 a^4}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac{8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac{16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac{64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac{128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7}\\ \end{align*}
Mathematica [A] time = 0.0148535, size = 64, normalized size = 0.55 \[ -\frac{\left (a+b x^2\right )^{7/2} \left (1008 a^2 b^2 x^4-1848 a^3 b x^2+3003 a^4-448 a b^3 x^6+128 b^4 x^8\right )}{45045 a^5 x^{15}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 61, normalized size = 0.5 \begin{align*} -{\frac{128\,{b}^{4}{x}^{8}-448\,{b}^{3}{x}^{6}a+1008\,{b}^{2}{x}^{4}{a}^{2}-1848\,b{x}^{2}{a}^{3}+3003\,{a}^{4}}{45045\,{x}^{15}{a}^{5}} \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 = 2.05689, size = 225, normalized size = 1.94 \begin{align*} -\frac{{\left (128 \, b^{7} x^{14} - 64 \, a b^{6} x^{12} + 48 \, a^{2} b^{5} x^{10} - 40 \, a^{3} b^{4} x^{8} + 35 \, a^{4} b^{3} x^{6} + 4473 \, a^{5} b^{2} x^{4} + 7161 \, a^{6} b x^{2} + 3003 \, a^{7}\right )} \sqrt{b x^{2} + a}}{45045 \, a^{5} x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.81163, size = 1012, normalized size = 8.72 \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 = 2.68954, size = 405, normalized size = 3.49 \begin{align*} \frac{256 \,{\left (18018 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} b^{\frac{15}{2}} + 60060 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a b^{\frac{15}{2}} + 115830 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{2} b^{\frac{15}{2}} + 109395 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{3} b^{\frac{15}{2}} + 65065 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{4} b^{\frac{15}{2}} + 15015 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{5} b^{\frac{15}{2}} + 1365 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{6} b^{\frac{15}{2}} - 455 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{7} b^{\frac{15}{2}} + 105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{8} b^{\frac{15}{2}} - 15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{9} b^{\frac{15}{2}} + a^{10} b^{\frac{15}{2}}\right )}}{45045 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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