3.549 \(\int \sqrt{3 x+\sqrt{-7+8 x}} \, dx\)

Optimal. Leaf size=109 \[ \frac{\left (-3 (7-8 x)+8 \sqrt{8 x-7}+21\right )^{3/2}}{72 \sqrt{2}}-\frac{\left (3 \sqrt{8 x-7}+4\right ) \sqrt{-3 (7-8 x)+8 \sqrt{8 x-7}+21}}{36 \sqrt{2}}-\frac{47 \sinh ^{-1}\left (\frac{3 \sqrt{8 x-7}+4}{\sqrt{47}}\right )}{36 \sqrt{6}} \]

[Out]

-((4 + 3*Sqrt[-7 + 8*x])*Sqrt[21 - 3*(7 - 8*x) + 8*Sqrt[-7 + 8*x]])/(36*Sqrt[2])
+ (21 - 3*(7 - 8*x) + 8*Sqrt[-7 + 8*x])^(3/2)/(72*Sqrt[2]) - (47*ArcSinh[(4 + 3
*Sqrt[-7 + 8*x])/Sqrt[47]])/(36*Sqrt[6])

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Rubi [A]  time = 0.135887, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{\left (-3 (7-8 x)+8 \sqrt{8 x-7}+21\right )^{3/2}}{72 \sqrt{2}}-\frac{\left (3 \sqrt{8 x-7}+4\right ) \sqrt{-3 (7-8 x)+8 \sqrt{8 x-7}+21}}{36 \sqrt{2}}-\frac{47 \sinh ^{-1}\left (\frac{3 \sqrt{8 x-7}+4}{\sqrt{47}}\right )}{36 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3*x + Sqrt[-7 + 8*x]],x]

[Out]

-((4 + 3*Sqrt[-7 + 8*x])*Sqrt[21 - 3*(7 - 8*x) + 8*Sqrt[-7 + 8*x]])/(36*Sqrt[2])
+ (21 - 3*(7 - 8*x) + 8*Sqrt[-7 + 8*x])^(3/2)/(72*Sqrt[2]) - (47*ArcSinh[(4 + 3
*Sqrt[-7 + 8*x])/Sqrt[47]])/(36*Sqrt[6])

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Rubi in Sympy [A]  time = 3.49236, size = 94, normalized size = 0.86 \[ \frac{\left (48 x + 16 \sqrt{8 x - 7}\right )^{\frac{3}{2}}}{288} - \frac{\sqrt{48 x + 16 \sqrt{8 x - 7}} \left (12 \sqrt{8 x - 7} + 16\right )}{288} - \frac{47 \sqrt{6} \operatorname{atanh}{\left (\frac{\sqrt{6} \left (12 \sqrt{8 x - 7} + 16\right )}{12 \sqrt{48 x + 16 \sqrt{8 x - 7}}} \right )}}{216} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x+(-7+8*x)**(1/2))**(1/2),x)

[Out]

(48*x + 16*sqrt(8*x - 7))**(3/2)/288 - sqrt(48*x + 16*sqrt(8*x - 7))*(12*sqrt(8*
x - 7) + 16)/288 - 47*sqrt(6)*atanh(sqrt(6)*(12*sqrt(8*x - 7) + 16)/(12*sqrt(48*
x + 16*sqrt(8*x - 7))))/216

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Mathematica [A]  time = 0.0899018, size = 65, normalized size = 0.6 \[ \frac{1}{18} \sqrt{3 x+\sqrt{8 x-7}} \left (12 x+\sqrt{8 x-7}-4\right )-\frac{47 \sinh ^{-1}\left (\frac{3 \sqrt{8 x-7}+4}{\sqrt{47}}\right )}{36 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3*x + Sqrt[-7 + 8*x]],x]

[Out]

(Sqrt[3*x + Sqrt[-7 + 8*x]]*(-4 + 12*x + Sqrt[-7 + 8*x]))/18 - (47*ArcSinh[(4 +
3*Sqrt[-7 + 8*x])/Sqrt[47]])/(36*Sqrt[6])

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Maple [A]  time = 0.011, size = 67, normalized size = 0.6 \[{\frac{1}{288} \left ( 48\,x+16\,\sqrt{-7+8\,x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{288} \left ( 12\,\sqrt{-7+8\,x}+16 \right ) \sqrt{48\,x+16\,\sqrt{-7+8\,x}}}-{\frac{47\,\sqrt{6}}{216}{\it Arcsinh} \left ({\frac{3\,\sqrt{47}}{47} \left ( \sqrt{-7+8\,x}+{\frac{4}{3}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x+(-7+8*x)^(1/2))^(1/2),x)

[Out]

1/288*(48*x+16*(-7+8*x)^(1/2))^(3/2)-1/288*(12*(-7+8*x)^(1/2)+16)*(48*x+16*(-7+8
*x)^(1/2))^(1/2)-47/216*6^(1/2)*arcsinh(3/47*47^(1/2)*((-7+8*x)^(1/2)+4/3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{3 \, x + \sqrt{8 \, x - 7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x + sqrt(8*x - 7)),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x + sqrt(8*x - 7)), x)

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Fricas [A]  time = 1.32728, size = 153, normalized size = 1.4 \[ \frac{1}{864} \, \sqrt{6}{\left (8 \,{\left (4 \, \sqrt{6}{\left (3 \, x - 1\right )} + \sqrt{6} \sqrt{8 \, x - 7}\right )} \sqrt{3 \, x + \sqrt{8 \, x - 7}} + 47 \, \log \left (-192 \, \sqrt{6}{\left (144 \, x - 47\right )} \sqrt{8 \, x - 7} - \sqrt{6}{\left (41472 \, x^{2} + 9792 \, x - 30047\right )} + 48 \,{\left (3 \,{\left (144 \, x + 17\right )} \sqrt{8 \, x - 7} + 1728 \, x - 1196\right )} \sqrt{3 \, x + \sqrt{8 \, x - 7}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x + sqrt(8*x - 7)),x, algorithm="fricas")

[Out]

1/864*sqrt(6)*(8*(4*sqrt(6)*(3*x - 1) + sqrt(6)*sqrt(8*x - 7))*sqrt(3*x + sqrt(8
*x - 7)) + 47*log(-192*sqrt(6)*(144*x - 47)*sqrt(8*x - 7) - sqrt(6)*(41472*x^2 +
9792*x - 30047) + 48*(3*(144*x + 17)*sqrt(8*x - 7) + 1728*x - 1196)*sqrt(3*x +
sqrt(8*x - 7))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{3 x + \sqrt{8 x - 7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x+(-7+8*x)**(1/2))**(1/2),x)

[Out]

Integral(sqrt(3*x + sqrt(8*x - 7)), x)

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GIAC/XCAS [A]  time = 0.298221, size = 174, normalized size = 1.6 \[ \frac{1}{72} \, \sqrt{2}{\left ({\left (3 \, \sqrt{2} \sqrt{8 \, x - 7} + 2 \, \sqrt{2}\right )} \sqrt{8 \, x - 7} + 13 \, \sqrt{2}\right )} \sqrt{3 \, x + \sqrt{8 \, x - 7}} + \frac{47}{216} \, \sqrt{3} \sqrt{2}{\rm ln}\left (-\sqrt{3}{\left (\sqrt{3} \sqrt{8 \, x - 7} - 2 \, \sqrt{2} \sqrt{3 \, x + \sqrt{8 \, x - 7}}\right )} - 4\right ) - \frac{1}{432} \, \sqrt{3}{\left (13 \, \sqrt{21} \sqrt{3} \sqrt{2} + 94 \, \sqrt{2}{\rm ln}\left (\sqrt{21} \sqrt{3} - 4\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x + sqrt(8*x - 7)),x, algorithm="giac")

[Out]

1/72*sqrt(2)*((3*sqrt(2)*sqrt(8*x - 7) + 2*sqrt(2))*sqrt(8*x - 7) + 13*sqrt(2))*
sqrt(3*x + sqrt(8*x - 7)) + 47/216*sqrt(3)*sqrt(2)*ln(-sqrt(3)*(sqrt(3)*sqrt(8*x
- 7) - 2*sqrt(2)*sqrt(3*x + sqrt(8*x - 7))) - 4) - 1/432*sqrt(3)*(13*sqrt(21)*s
qrt(3)*sqrt(2) + 94*sqrt(2)*ln(sqrt(21)*sqrt(3) - 4))