Optimal. Leaf size=56 \[ \frac {4 \left (a+b \sqrt {c x^3}\right )^{5/2}}{15 b^2 c}-\frac {4 a \left (a+b \sqrt {c x^3}\right )^{3/2}}{9 b^2 c} \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {369, 266, 43} \begin {gather*} \frac {4 \left (a+b \sqrt {c x^3}\right )^{5/2}}{15 b^2 c}-\frac {4 a \left (a+b \sqrt {c x^3}\right )^{3/2}}{9 b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b \sqrt {c x^3}} \, dx &=\operatorname {Subst}\left (\int x^2 \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\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 \sqrt {c} x} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{3} \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b \sqrt {c} x}}{b \sqrt {c}}+\frac {\left (a+b \sqrt {c} x\right )^{3/2}}{b \sqrt {c}}\right ) \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {4 a \left (a+b \sqrt {c x^3}\right )^{3/2}}{9 b^2 c}+\frac {4 \left (a+b \sqrt {c x^3}\right )^{5/2}}{15 b^2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.77 \begin {gather*} \frac {4 \left (a+b \sqrt {c x^3}\right )^{3/2} \left (3 b \sqrt {c x^3}-2 a\right )}{45 b^2 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 50, normalized size = 0.89 \begin {gather*} -\frac {4 \left (5 a \left (a+b \sqrt {c x^3}\right )^{3/2}-3 \left (a+b \sqrt {c x^3}\right )^{5/2}\right )}{45 b^2 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 46, normalized size = 0.82 \begin {gather*} \frac {4 \, {\left (3 \, b^{2} c x^{3} + \sqrt {c x^{3}} a b - 2 \, a^{2}\right )} \sqrt {\sqrt {c x^{3}} b + a}}{45 \, b^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 57, normalized size = 1.02 \begin {gather*} -\frac {4 \, {\left (5 \, {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {3}{2}} a c^{2} - 3 \, {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {5}{2}}\right )} {\left | c \right |}}{45 \, b^{2} c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 65, normalized size = 1.16 \begin {gather*} \frac {4 \sqrt {a +\sqrt {c \,x^{3}}\, b}\, \left (a b c \,x^{3}+3 \sqrt {c \,x^{3}}\, b^{2} c \,x^{3}-2 \sqrt {c \,x^{3}}\, a^{2}\right )}{45 \sqrt {c \,x^{3}}\, b^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 43, normalized size = 0.77 \begin {gather*} \frac {4 \, {\left (\frac {3 \, {\left (\sqrt {c x^{3}} b + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (\sqrt {c x^{3}} b + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )}}{45 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 52, normalized size = 0.93 \begin {gather*} \frac {x^3\,\sqrt {a+b\,\sqrt {c\,x^3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {c\,x^3}}{a}\right )}{3\,\sqrt {\frac {b\,\sqrt {c\,x^3}}{a}+1}} \end {gather*}
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 \begin {gather*} \int x^{2} \sqrt {a + b \sqrt {c x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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