Optimal. Leaf size=400 \[ -\frac {8\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{55 b^{4/3} c^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {12 a x^2 \sqrt {a+b \sqrt {c x^3}}}{55 b \sqrt {c x^3}}+\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}} \]
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Rubi [A] time = 0.25, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {369, 341, 279, 321, 218} \[ -\frac {8\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{55 b^{4/3} c^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {12 a x^2 \sqrt {a+b \sqrt {c x^3}}}{55 b \sqrt {c x^3}}+\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}} \]
Antiderivative was successfully verified.
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Rule 218
Rule 279
Rule 321
Rule 341
Rule 369
Rubi steps
\begin {align*} \int x \sqrt {a+b \sqrt {c x^3}} \, dx &=\operatorname {Subst}\left (\int x \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int x^3 \sqrt {a+b \sqrt {c} x^3} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}}+\operatorname {Subst}\left (\frac {1}{11} (6 a) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}}+\frac {12 a x^2 \sqrt {a+b \sqrt {c x^3}}}{55 b \sqrt {c x^3}}-\operatorname {Subst}\left (\frac {\left (12 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sqrt {c} x^3}} \, dx,x,\sqrt {x}\right )}{55 b \sqrt {c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{11} x^2 \sqrt {a+b \sqrt {c x^3}}+\frac {12 a x^2 \sqrt {a+b \sqrt {c x^3}}}{55 b \sqrt {c x^3}}-\frac {8\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right )|-7-4 \sqrt {3}\right )}{55 b^{4/3} c^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}\\ \end {align*}
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Mathematica [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x \sqrt {a+b \sqrt {c x^3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 6.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sqrt {c x^{3}} b + a} x, x\right ) \]
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 \[ \int \sqrt {\sqrt {c x^{3}} b + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 350, normalized size = 0.88 \[ \frac {\frac {4 b^{3} c^{2} x^{5}}{11}+\frac {12 a^{2} b c \,x^{2}}{55}+\frac {32 \sqrt {c \,x^{3}}\, a \,b^{2} c \,x^{2}}{55}+\frac {4 i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {-\frac {i \left (-2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {\frac {\sqrt {c \,x^{3}}\, b -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right ) x}}\, \sqrt {-\frac {i \left (2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x +\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {c \,x^{3}}\, a^{2} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (-2 \sqrt {c \,x^{3}}\, b +i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} x -\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )}{55}}{\sqrt {c \,x^{3}}\, \sqrt {a +\sqrt {c \,x^{3}}\, b}\, b^{2} c} \]
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 \[ \int \sqrt {\sqrt {c x^{3}} b + a} x\,{d x} \]
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 \[ \int x\,\sqrt {a+b\,\sqrt {c\,x^3}} \,d x \]
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 \[ \int x \sqrt {a + b \sqrt {c x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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