Optimal. Leaf size=306 \[ \frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {c x^2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {2}{5} x \sqrt {a+b \left (c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {254, 195, 218} \[ \frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} \sqrt {c x^2}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {c x^2}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {c x^2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}+\frac {2}{5} x \sqrt {a+b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 218
Rule 254
Rubi steps
\begin {align*} \int \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {x \operatorname {Subst}\left (\int \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{\sqrt {c x^2}}\\ &=\frac {2}{5} x \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {(3 a x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\sqrt {c x^2}\right )}{5 \sqrt {c x^2}}\\ &=\frac {2}{5} x \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a x \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^2-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}}\right )|-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {c x^2} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^2}\right )^2}} \sqrt {a+b \left (c x^2\right )^{3/2}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 64, normalized size = 0.21 \[ \frac {x \sqrt {a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac {1}{2},\frac {1}{3};\frac {4}{3};-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{\sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sqrt {c x^{2}} b c x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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