Optimal. Leaf size=313 \[ \frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+9}}{d^9 (m+9) \left (a+b x^2\right )}+\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+11}}{d^{11} (m+11) \left (a+b x^2\right )}+\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]
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Rubi [A] time = 0.120312, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1112, 270} \[ \frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac{10 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+9}}{d^9 (m+9) \left (a+b x^2\right )}+\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+11}}{d^{11} (m+11) \left (a+b x^2\right )}+\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 270
Rubi steps
\begin{align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int (d x)^m \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 (d x)^m+\frac{5 a^4 b^6 (d x)^{2+m}}{d^2}+\frac{10 a^3 b^7 (d x)^{4+m}}{d^4}+\frac{10 a^2 b^8 (d x)^{6+m}}{d^6}+\frac{5 a b^9 (d x)^{8+m}}{d^8}+\frac{b^{10} (d x)^{10+m}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac{a^5 (d x)^{1+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac{5 a^4 b (d x)^{3+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac{10 a^3 b^2 (d x)^{5+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac{10 a^2 b^3 (d x)^{7+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}+\frac{5 a b^4 (d x)^{9+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^9 (9+m) \left (a+b x^2\right )}+\frac{b^5 (d x)^{11+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^{11} (11+m) \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0900514, size = 111, normalized size = 0.35 \[ \frac{x \left (\left (a+b x^2\right )^2\right )^{5/2} (d x)^m \left (\frac{10 a^2 b^3 x^6}{m+7}+\frac{10 a^3 b^2 x^4}{m+5}+\frac{5 a^4 b x^2}{m+3}+\frac{a^5}{m+1}+\frac{5 a b^4 x^8}{m+9}+\frac{b^5 x^{10}}{m+11}\right )}{\left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.166, size = 453, normalized size = 1.5 \begin{align*}{\frac{ \left ({b}^{5}{m}^{5}{x}^{10}+25\,{b}^{5}{m}^{4}{x}^{10}+5\,a{b}^{4}{m}^{5}{x}^{8}+230\,{b}^{5}{m}^{3}{x}^{10}+135\,a{b}^{4}{m}^{4}{x}^{8}+950\,{b}^{5}{m}^{2}{x}^{10}+10\,{a}^{2}{b}^{3}{m}^{5}{x}^{6}+1310\,a{b}^{4}{m}^{3}{x}^{8}+1689\,{b}^{5}m{x}^{10}+290\,{a}^{2}{b}^{3}{m}^{4}{x}^{6}+5610\,a{b}^{4}{m}^{2}{x}^{8}+945\,{b}^{5}{x}^{10}+10\,{a}^{3}{b}^{2}{m}^{5}{x}^{4}+3020\,{a}^{2}{b}^{3}{m}^{3}{x}^{6}+10205\,a{b}^{4}m{x}^{8}+310\,{a}^{3}{b}^{2}{m}^{4}{x}^{4}+13660\,{a}^{2}{b}^{3}{m}^{2}{x}^{6}+5775\,a{b}^{4}{x}^{8}+5\,{a}^{4}b{m}^{5}{x}^{2}+3500\,{a}^{3}{b}^{2}{m}^{3}{x}^{4}+25770\,{a}^{2}{b}^{3}m{x}^{6}+165\,{a}^{4}b{m}^{4}{x}^{2}+17300\,{a}^{3}{b}^{2}{m}^{2}{x}^{4}+14850\,{a}^{2}{b}^{3}{x}^{6}+{a}^{5}{m}^{5}+2030\,{a}^{4}b{m}^{3}{x}^{2}+34890\,{a}^{3}{b}^{2}m{x}^{4}+35\,{a}^{5}{m}^{4}+11310\,{a}^{4}b{m}^{2}{x}^{2}+20790\,{b}^{2}{a}^{3}{x}^{4}+470\,{a}^{5}{m}^{3}+26765\,{a}^{4}bm{x}^{2}+3010\,{a}^{5}{m}^{2}+17325\,{a}^{4}b{x}^{2}+9129\,{a}^{5}m+10395\,{a}^{5} \right ) x \left ( dx \right ) ^{m}}{ \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994014, size = 328, normalized size = 1.05 \begin{align*} \frac{{\left ({\left (m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945\right )} b^{5} d^{m} x^{11} + 5 \,{\left (m^{5} + 27 \, m^{4} + 262 \, m^{3} + 1122 \, m^{2} + 2041 \, m + 1155\right )} a b^{4} d^{m} x^{9} + 10 \,{\left (m^{5} + 29 \, m^{4} + 302 \, m^{3} + 1366 \, m^{2} + 2577 \, m + 1485\right )} a^{2} b^{3} d^{m} x^{7} + 10 \,{\left (m^{5} + 31 \, m^{4} + 350 \, m^{3} + 1730 \, m^{2} + 3489 \, m + 2079\right )} a^{3} b^{2} d^{m} x^{5} + 5 \,{\left (m^{5} + 33 \, m^{4} + 406 \, m^{3} + 2262 \, m^{2} + 5353 \, m + 3465\right )} a^{4} b d^{m} x^{3} +{\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} a^{5} d^{m} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62986, size = 873, normalized size = 2.79 \begin{align*} \frac{{\left ({\left (b^{5} m^{5} + 25 \, b^{5} m^{4} + 230 \, b^{5} m^{3} + 950 \, b^{5} m^{2} + 1689 \, b^{5} m + 945 \, b^{5}\right )} x^{11} + 5 \,{\left (a b^{4} m^{5} + 27 \, a b^{4} m^{4} + 262 \, a b^{4} m^{3} + 1122 \, a b^{4} m^{2} + 2041 \, a b^{4} m + 1155 \, a b^{4}\right )} x^{9} + 10 \,{\left (a^{2} b^{3} m^{5} + 29 \, a^{2} b^{3} m^{4} + 302 \, a^{2} b^{3} m^{3} + 1366 \, a^{2} b^{3} m^{2} + 2577 \, a^{2} b^{3} m + 1485 \, a^{2} b^{3}\right )} x^{7} + 10 \,{\left (a^{3} b^{2} m^{5} + 31 \, a^{3} b^{2} m^{4} + 350 \, a^{3} b^{2} m^{3} + 1730 \, a^{3} b^{2} m^{2} + 3489 \, a^{3} b^{2} m + 2079 \, a^{3} b^{2}\right )} x^{5} + 5 \,{\left (a^{4} b m^{5} + 33 \, a^{4} b m^{4} + 406 \, a^{4} b m^{3} + 2262 \, a^{4} b m^{2} + 5353 \, a^{4} b m + 3465 \, a^{4} b\right )} x^{3} +{\left (a^{5} m^{5} + 35 \, a^{5} m^{4} + 470 \, a^{5} m^{3} + 3010 \, a^{5} m^{2} + 9129 \, a^{5} m + 10395 \, a^{5}\right )} x\right )} \left (d x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29992, size = 1215, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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