Integrand size = 17, antiderivative size = 109 \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=-\frac {\left (4+3 \sqrt {-7+8 x}\right ) \sqrt {21-3 (7-8 x)+8 \sqrt {-7+8 x}}}{36 \sqrt {2}}+\frac {\left (21-3 (7-8 x)+8 \sqrt {-7+8 x}\right )^{3/2}}{72 \sqrt {2}}-\frac {47 \text {arcsinh}\left (\frac {4+3 \sqrt {-7+8 x}}{\sqrt {47}}\right )}{36 \sqrt {6}} \] Output:
-1/36*(4+3*(-7+8*x)^(1/2))*(6*x+2*(-7+8*x)^(1/2))^(1/2)*2^(1/2)+1/144*(24* x+8*(-7+8*x)^(1/2))^(3/2)*2^(1/2)-47/216*arcsinh(1/47*(4+3*(-7+8*x)^(1/2)) *47^(1/2))*6^(1/2)
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\frac {1}{18} \sqrt {3 x+\sqrt {-7+8 x}} \left (-4+12 x+\sqrt {-7+8 x}\right )+\frac {47 \log \left (-4-3 \sqrt {-7+8 x}+2 \sqrt {6} \sqrt {3 x+\sqrt {-7+8 x}}\right )}{36 \sqrt {6}} \] Input:
Integrate[Sqrt[3*x + Sqrt[-7 + 8*x]],x]
Output:
(Sqrt[3*x + Sqrt[-7 + 8*x]]*(-4 + 12*x + Sqrt[-7 + 8*x]))/18 + (47*Log[-4 - 3*Sqrt[-7 + 8*x] + 2*Sqrt[6]*Sqrt[3*x + Sqrt[-7 + 8*x]]])/(36*Sqrt[6])
Time = 0.45 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {7267, 27, 1160, 1087, 1090, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {3 x+\sqrt {8 x-7}} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt {8 x-7} \sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21}}{2 \sqrt {2}}d\sqrt {8 x-7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \sqrt {8 x-7} \sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21}d\sqrt {8 x-7}}{8 \sqrt {2}}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {\frac {1}{9} \left (3 (8 x-7)+8 \sqrt {8 x-7}+21\right )^{3/2}-\frac {4}{3} \int \sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21}d\sqrt {8 x-7}}{8 \sqrt {2}}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {\frac {1}{9} \left (3 (8 x-7)+8 \sqrt {8 x-7}+21\right )^{3/2}-\frac {4}{3} \left (\frac {47}{6} \int \frac {1}{\sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21}}d\sqrt {8 x-7}+\frac {1}{6} \sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21} \left (3 \sqrt {8 x-7}+4\right )\right )}{8 \sqrt {2}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {\frac {1}{9} \left (3 (8 x-7)+8 \sqrt {8 x-7}+21\right )^{3/2}-\frac {4}{3} \left (\frac {1}{12} \sqrt {\frac {47}{3}} \int \frac {1}{\sqrt {\frac {1}{188} (8 x-7)+1}}d\left (6 \sqrt {8 x-7}+8\right )+\frac {1}{6} \sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21} \left (3 \sqrt {8 x-7}+4\right )\right )}{8 \sqrt {2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\frac {1}{9} \left (3 (8 x-7)+8 \sqrt {8 x-7}+21\right )^{3/2}-\frac {4}{3} \left (\frac {47 \text {arcsinh}\left (\frac {6 \sqrt {8 x-7}+8}{2 \sqrt {47}}\right )}{6 \sqrt {3}}+\frac {1}{6} \sqrt {3 (8 x-7)+8 \sqrt {8 x-7}+21} \left (3 \sqrt {8 x-7}+4\right )\right )}{8 \sqrt {2}}\) |
Input:
Int[Sqrt[3*x + Sqrt[-7 + 8*x]],x]
Output:
((21 + 8*Sqrt[-7 + 8*x] + 3*(-7 + 8*x))^(3/2)/9 - (4*(((4 + 3*Sqrt[-7 + 8* x])*Sqrt[21 + 8*Sqrt[-7 + 8*x] + 3*(-7 + 8*x)])/6 + (47*ArcSinh[(8 + 6*Sqr t[-7 + 8*x])/(2*Sqrt[47])])/(6*Sqrt[3])))/3)/(8*Sqrt[2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(\frac {\left (48 x +16 \sqrt {-7+8 x}\right )^{\frac {3}{2}}}{288}-\frac {\left (12 \sqrt {-7+8 x}+16\right ) \sqrt {48 x +16 \sqrt {-7+8 x}}}{288}-\frac {47 \sqrt {6}\, \operatorname {arcsinh}\left (\frac {3 \sqrt {47}\, \left (\sqrt {-7+8 x}+\frac {4}{3}\right )}{47}\right )}{216}\) | \(67\) |
default | \(\frac {\left (48 x +16 \sqrt {-7+8 x}\right )^{\frac {3}{2}}}{288}-\frac {\left (12 \sqrt {-7+8 x}+16\right ) \sqrt {48 x +16 \sqrt {-7+8 x}}}{288}-\frac {47 \sqrt {6}\, \operatorname {arcsinh}\left (\frac {3 \sqrt {47}\, \left (\sqrt {-7+8 x}+\frac {4}{3}\right )}{47}\right )}{216}\) | \(67\) |
Input:
int((3*x+(-7+8*x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
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))
Time = 0.85 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\frac {1}{18} \, {\left (12 \, x + \sqrt {8 \, x - 7} - 4\right )} \sqrt {3 \, x + \sqrt {8 \, x - 7}} + \frac {47}{864} \, \sqrt {6} \log \left (-41472 \, x^{2} - 192 \, {\left (144 \, x - 47\right )} \sqrt {8 \, x - 7} + 8 \, {\left (3 \, \sqrt {6} {\left (144 \, x + 17\right )} \sqrt {8 \, x - 7} + 4 \, \sqrt {6} {\left (432 \, x - 299\right )}\right )} \sqrt {3 \, x + \sqrt {8 \, x - 7}} - 9792 \, x + 30047\right ) \] Input:
integrate((3*x+(-7+8*x)^(1/2))^(1/2),x, algorithm="fricas")
Output:
1/18*(12*x + sqrt(8*x - 7) - 4)*sqrt(3*x + sqrt(8*x - 7)) + 47/864*sqrt(6) *log(-41472*x^2 - 192*(144*x - 47)*sqrt(8*x - 7) + 8*(3*sqrt(6)*(144*x + 1 7)*sqrt(8*x - 7) + 4*sqrt(6)*(432*x - 299))*sqrt(3*x + sqrt(8*x - 7)) - 97 92*x + 30047)
Time = 0.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\frac {\sqrt {3 x + \sqrt {8 x - 7}} \cdot \left (\frac {8 x}{3} + \frac {2 \sqrt {8 x - 7}}{9} - \frac {8}{9}\right )}{4} - \frac {47 \sqrt {6} \operatorname {asinh}{\left (\frac {3 \sqrt {47} \left (\sqrt {8 x - 7} + \frac {4}{3}\right )}{47} \right )}}{216} \] Input:
integrate((3*x+(-7+8*x)**(1/2))**(1/2),x)
Output:
sqrt(3*x + sqrt(8*x - 7))*(8*x/3 + 2*sqrt(8*x - 7)/9 - 8/9)/4 - 47*sqrt(6) *asinh(3*sqrt(47)*(sqrt(8*x - 7) + 4/3)/47)/216
\[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\int { \sqrt {3 \, x + \sqrt {8 \, x - 7}} \,d x } \] Input:
integrate((3*x+(-7+8*x)^(1/2))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(3*x + sqrt(8*x - 7)), x)
Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\frac {1}{216} \, \sqrt {2} {\left (3 \, \sqrt {2} {\left (\sqrt {8 \, x - 7} {\left (3 \, \sqrt {8 \, x - 7} + 2\right )} + 13\right )} \sqrt {3 \, x + \sqrt {8 \, x - 7}} + 47 \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} \sqrt {8 \, x - 7} - 2 \, \sqrt {2} \sqrt {3 \, x + \sqrt {8 \, x - 7}}\right )} - 4\right )\right )} \] Input:
integrate((3*x+(-7+8*x)^(1/2))^(1/2),x, algorithm="giac")
Output:
1/216*sqrt(2)*(3*sqrt(2)*(sqrt(8*x - 7)*(3*sqrt(8*x - 7) + 2) + 13)*sqrt(3 *x + sqrt(8*x - 7)) + 47*sqrt(3)*log(-sqrt(3)*(sqrt(3)*sqrt(8*x - 7) - 2*s qrt(2)*sqrt(3*x + sqrt(8*x - 7))) - 4))
Timed out. \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\int \sqrt {3\,x+\sqrt {8\,x-7}} \,d x \] Input:
int((3*x + (8*x - 7)^(1/2))^(1/2),x)
Output:
int((3*x + (8*x - 7)^(1/2))^(1/2), x)
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75 \[ \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx=\frac {\sqrt {8 x -7}\, \sqrt {\sqrt {8 x -7}+3 x}}{18}+\frac {2 \sqrt {\sqrt {8 x -7}+3 x}\, x}{3}-\frac {2 \sqrt {\sqrt {8 x -7}+3 x}}{9}-\frac {47 \sqrt {6}\, \mathrm {log}\left (\frac {2 \sqrt {\sqrt {8 x -7}+3 x}\, \sqrt {6}+3 \sqrt {8 x -7}+4}{\sqrt {47}}\right )}{216} \] Input:
int((3*x+(-7+8*x)^(1/2))^(1/2),x)
Output:
(12*sqrt(8*x - 7)*sqrt(sqrt(8*x - 7) + 3*x) + 144*sqrt(sqrt(8*x - 7) + 3*x )*x - 48*sqrt(sqrt(8*x - 7) + 3*x) - 47*sqrt(6)*log((2*sqrt(sqrt(8*x - 7) + 3*x)*sqrt(6) + 3*sqrt(8*x - 7) + 4)/sqrt(47)))/216