Optimal. Leaf size=109 \[ \frac {\left (-3 (7-8 x)+8 \sqrt {8 x-7}+21\right )^{3/2}}{72 \sqrt {2}}-\frac {\left (3 \sqrt {8 x-7}+4\right ) \sqrt {-3 (7-8 x)+8 \sqrt {8 x-7}+21}}{36 \sqrt {2}}-\frac {47 \sinh ^{-1}\left (\frac {3 \sqrt {8 x-7}+4}{\sqrt {47}}\right )}{36 \sqrt {6}} \]
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Rubi [A] time = 0.07, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {640, 612, 619, 215} \[ \frac {\left (-3 (7-8 x)+8 \sqrt {8 x-7}+21\right )^{3/2}}{72 \sqrt {2}}-\frac {\left (3 \sqrt {8 x-7}+4\right ) \sqrt {-3 (7-8 x)+8 \sqrt {8 x-7}+21}}{36 \sqrt {2}}-\frac {47 \sinh ^{-1}\left (\frac {3 \sqrt {8 x-7}+4}{\sqrt {47}}\right )}{36 \sqrt {6}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x \sqrt {\frac {21}{8}+x+\frac {3 x^2}{8}} \, dx,x,\sqrt {-7+8 x}\right )\\ &=\frac {\left (21-3 (7-8 x)+8 \sqrt {-7+8 x}\right )^{3/2}}{72 \sqrt {2}}-\frac {1}{3} \operatorname {Subst}\left (\int \sqrt {\frac {21}{8}+x+\frac {3 x^2}{8}} \, dx,x,\sqrt {-7+8 x}\right )\\ &=-\frac {\left (4+3 \sqrt {-7+8 x}\right ) \sqrt {21-3 (7-8 x)+8 \sqrt {-7+8 x}}}{36 \sqrt {2}}+\frac {\left (21-3 (7-8 x)+8 \sqrt {-7+8 x}\right )^{3/2}}{72 \sqrt {2}}-\frac {47}{144} \operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {21}{8}+x+\frac {3 x^2}{8}}} \, dx,x,\sqrt {-7+8 x}\right )\\ &=-\frac {\left (4+3 \sqrt {-7+8 x}\right ) \sqrt {21-3 (7-8 x)+8 \sqrt {-7+8 x}}}{36 \sqrt {2}}+\frac {\left (21-3 (7-8 x)+8 \sqrt {-7+8 x}\right )^{3/2}}{72 \sqrt {2}}-\frac {1}{9} \sqrt {\frac {47}{6}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {16 x^2}{47}}} \, dx,x,1+\frac {3}{4} \sqrt {-7+8 x}\right )\\ &=-\frac {\left (4+3 \sqrt {-7+8 x}\right ) \sqrt {21-3 (7-8 x)+8 \sqrt {-7+8 x}}}{36 \sqrt {2}}+\frac {\left (21-3 (7-8 x)+8 \sqrt {-7+8 x}\right )^{3/2}}{72 \sqrt {2}}-\frac {47 \sinh ^{-1}\left (\frac {4+3 \sqrt {-7+8 x}}{\sqrt {47}}\right )}{36 \sqrt {6}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 65, normalized size = 0.60 \[ \frac {1}{216} \left (12 \sqrt {3 x+\sqrt {8 x-7}} \left (12 x+\sqrt {8 x-7}-4\right )-47 \sqrt {6} \sinh ^{-1}\left (\frac {3 \sqrt {8 x-7}+4}{\sqrt {47}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.79, size = 101, normalized size = 0.93 \[ \frac {1}{18} \, {\left (12 \, x + \sqrt {8 \, x - 7} - 4\right )} \sqrt {3 \, x + \sqrt {8 \, x - 7}} + \frac {47}{864} \, \sqrt {6} \log \left (-41472 \, x^{2} - 192 \, {\left (144 \, x - 47\right )} \sqrt {8 \, x - 7} + 8 \, {\left (3 \, \sqrt {6} {\left (144 \, x + 17\right )} \sqrt {8 \, x - 7} + 4 \, \sqrt {6} {\left (432 \, x - 299\right )}\right )} \sqrt {3 \, x + \sqrt {8 \, x - 7}} - 9792 \, x + 30047\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 88, normalized size = 0.81 \[ \frac {1}{216} \, \sqrt {2} {\left (3 \, \sqrt {2} {\left (\sqrt {8 \, x - 7} {\left (3 \, \sqrt {8 \, x - 7} + 2\right )} + 13\right )} \sqrt {3 \, x + \sqrt {8 \, x - 7}} + 47 \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} \sqrt {8 \, x - 7} - 2 \, \sqrt {2} \sqrt {3 \, x + \sqrt {8 \, x - 7}}\right )} - 4\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.61 \[ -\frac {47 \sqrt {6}\, \arcsinh \left (\frac {3 \sqrt {47}\, \left (\sqrt {8 x -7}+\frac {4}{3}\right )}{47}\right )}{216}+\frac {\left (48 x +16 \sqrt {8 x -7}\right )^{\frac {3}{2}}}{288}-\frac {\left (12 \sqrt {8 x -7}+16\right ) \sqrt {48 x +16 \sqrt {8 x -7}}}{288} \]
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 \[ \int \sqrt {3 \, x + \sqrt {8 \, x - 7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {3\,x+\sqrt {8\,x-7}} \,d x \]
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 \[ \int \sqrt {3 x + \sqrt {8 x - 7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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