Optimal. Leaf size=642 \[ \frac{4 \sqrt{2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}}\right ),-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4 a \sqrt{a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}+\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}} \]
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Rubi [A] time = 0.734826, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {369, 341, 275, 279, 303, 218, 1877} \[ \frac{4 \sqrt{2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4 a \sqrt{a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}+\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 369
Rule 341
Rule 275
Rule 279
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx &=\operatorname{Subst}\left (\int x^2 \sqrt{a+b c^{3/2} x^{9/2}} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int x^5 \sqrt{a+b c^{3/2} x^9} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int x \sqrt{a+b c^{3/2} x^3} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}}+\operatorname{Subst}\left (\frac{1}{7} (2 a) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b c^{3/2} x^3}} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}}+\operatorname{Subst}\left (\frac{(2 a) \operatorname{Subst}\left (\int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c} x}{\sqrt{a+b c^{3/2} x^3}} \, dx,x,x^{3/2}\right )}{7 \sqrt [3]{b} \sqrt{c}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )+\operatorname{Subst}\left (\frac{\left (2 \sqrt{2 \left (2-\sqrt{3}\right )} a^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b c^{3/2} x^3}} \, dx,x,x^{3/2}\right )}{7 \sqrt [3]{b} \sqrt{c}},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}}+\frac{4 a \sqrt{a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}-\frac{2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4 \sqrt{2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}\\ \end{align*}
Mathematica [C] time = 0.0456987, size = 69, normalized size = 0.11 \[ \frac{x^3 \sqrt{a+b \left (c x^3\right )^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )}{3 \sqrt{\frac{b \left (c x^3\right )^{3/2}}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 495, normalized size = 0.8 \begin{align*}{\frac{1}{3\,c} \left ({\frac{4\,c{x}^{3}}{7}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}-{\frac{{\frac{4\,i}{7}}a\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}\sqrt{{ \left ( \sqrt{c{x}^{3}}-{\frac{1}{b}\sqrt [3]{-{b}^{2}a}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}} \left ( \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a} \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}} \right ) +{\frac{1}{b}\sqrt [3]{-{b}^{2}a}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a} \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}} \right ) } \right ){\frac{1}{\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{\sqrt{c x^{3}} b c x^{3} + a} x^{2}, x\right ) \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^{2} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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