Optimal. Leaf size=174 \[ \frac{20 a^2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac{20 a^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac{2 a^4 \left (a+b \sqrt{c x^2}\right )^{5/2}}{b^6 c^3}-\frac{2 a^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac{10 a \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^6 c^3} \]
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Rubi [A] time = 0.080875, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {368, 43} \[ \frac{20 a^2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac{20 a^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac{2 a^4 \left (a+b \sqrt{c x^2}\right )^{5/2}}{b^6 c^3}-\frac{2 a^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac{10 a \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^6 c^3} \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int x^5 \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{c^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^5 \sqrt{a+b x}}{b^5}+\frac{5 a^4 (a+b x)^{3/2}}{b^5}-\frac{10 a^3 (a+b x)^{5/2}}{b^5}+\frac{10 a^2 (a+b x)^{7/2}}{b^5}-\frac{5 a (a+b x)^{9/2}}{b^5}+\frac{(a+b x)^{11/2}}{b^5}\right ) \, dx,x,\sqrt{c x^2}\right )}{c^3}\\ &=-\frac{2 a^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac{2 a^4 \left (a+b \sqrt{c x^2}\right )^{5/2}}{b^6 c^3}-\frac{20 a^3 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac{20 a^2 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac{10 a \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^6 c^3}+\frac{2 \left (a+b \sqrt{c x^2}\right )^{13/2}}{13 b^6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0560799, size = 103, normalized size = 0.59 \[ \frac{2 \left (a+b \sqrt{c x^2}\right )^{3/2} \left (-480 a^3 b^2 c x^2+560 a^2 b^3 \left (c x^2\right )^{3/2}+384 a^4 b \sqrt{c x^2}-256 a^5-630 a b^4 c^2 x^4+693 b^5 \left (c x^2\right )^{5/2}\right )}{9009 b^6 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 92, normalized size = 0.5 \begin{align*}{\frac{2}{9009\,{c}^{3}{b}^{6}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( 693\, \left ( c{x}^{2} \right ) ^{5/2}{b}^{5}-630\,{c}^{2}{x}^{4}a{b}^{4}+560\, \left ( c{x}^{2} \right ) ^{3/2}{a}^{2}{b}^{3}-480\,c{x}^{2}{a}^{3}{b}^{2}+384\,\sqrt{c{x}^{2}}{a}^{4}b-256\,{a}^{5} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952691, size = 171, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (\frac{693 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{13}{2}}}{b^{6}} - \frac{4095 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{11}{2}} a}{b^{6}} + \frac{10010 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{9}{2}} a^{2}}{b^{6}} - \frac{12870 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{7}{2}} a^{3}}{b^{6}} + \frac{9009 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{5}{2}} a^{4}}{b^{6}} - \frac{3003 \,{\left (\sqrt{c x^{2}} b + a\right )}^{\frac{3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25027, size = 235, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (693 \, b^{6} c^{3} x^{6} - 70 \, a^{2} b^{4} c^{2} x^{4} - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} +{\left (63 \, a b^{5} c^{2} x^{4} + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{9009 \, b^{6} c^{3}} \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 x^{5} \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18346, size = 151, normalized size = 0.87 \begin{align*} \frac{2 \,{\left (693 \,{\left (b \sqrt{c} x + a\right )}^{\frac{13}{2}} \sqrt{c} - 4095 \,{\left (b \sqrt{c} x + a\right )}^{\frac{11}{2}} a \sqrt{c} + 10010 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} a^{2} \sqrt{c} - 12870 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a^{3} \sqrt{c} + 9009 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{4} \sqrt{c} - 3003 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{5} \sqrt{c}\right )}}{9009 \, b^{6} c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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