Optimal. Leaf size=143 \[ \frac{3 a^2 b c x^8 \sqrt{c \left (a+b x^2\right )^2}}{8 \left (a+b x^2\right )}+\frac{a^3 c x^6 \sqrt{c \left (a+b x^2\right )^2}}{6 \left (a+b x^2\right )}+\frac{b^3 c x^{12} \sqrt{c \left (a+b x^2\right )^2}}{12 \left (a+b x^2\right )}+\frac{3 a b^2 c x^{10} \sqrt{c \left (a+b x^2\right )^2}}{10 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.161853, antiderivative size = 134, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1989, 1111, 645} \[ \frac{c \left (a+b x^2\right )^5 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{12 b^3}-\frac{a c \left (a+b x^2\right )^4 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{5 b^3}+\frac{a^2 c \left (a+b x^2\right )^3 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 1989
Rule 1111
Rule 645
Rubi steps
\begin{align*} \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\int x^5 \left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \left (a^2 c+2 a b c x+b^2 c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a^2 c+2 a b c x^2+b^2 c x^4} \operatorname{Subst}\left (\int \left (\frac{a^2 \left (a b c+b^2 c x\right )^3}{b^2}-\frac{2 a \left (a b c+b^2 c x\right )^4}{b^3 c}+\frac{\left (a b c+b^2 c x\right )^5}{b^4 c^2}\right ) \, dx,x,x^2\right )}{2 b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac{a^2 c \left (a+b x^2\right )^3 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{8 b^3}-\frac{a c \left (a+b x^2\right )^4 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{5 b^3}+\frac{c \left (a+b x^2\right )^5 \sqrt{a^2 c+2 a b c x^2+b^2 c x^4}}{12 b^3}\\ \end{align*}
Mathematica [A] time = 0.0215343, size = 63, normalized size = 0.44 \[ \frac{x^6 \left (45 a^2 b x^2+20 a^3+36 a b^2 x^4+10 b^3 x^6\right ) \left (c \left (a+b x^2\right )^2\right )^{3/2}}{120 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 60, normalized size = 0.4 \begin{align*}{\frac{{x}^{6} \left ( 10\,{b}^{3}{x}^{6}+36\,a{b}^{2}{x}^{4}+45\,{a}^{2}b{x}^{2}+20\,{a}^{3} \right ) }{120\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 1.41231, size = 166, normalized size = 1.16 \begin{align*} \frac{{\left (10 \, b^{3} c x^{12} + 36 \, a b^{2} c x^{10} + 45 \, a^{2} b c x^{8} + 20 \, a^{3} c x^{6}\right )} \sqrt{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{120 \,{\left (b x^{2} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21836, size = 97, normalized size = 0.68 \begin{align*} \frac{1}{120} \,{\left (10 \, b^{3} x^{12} \mathrm{sgn}\left (b x^{2} + a\right ) + 36 \, a b^{2} x^{10} \mathrm{sgn}\left (b x^{2} + a\right ) + 45 \, a^{2} b x^{8} \mathrm{sgn}\left (b x^{2} + a\right ) + 20 \, a^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right )\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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