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

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])

________________________________________________________________________________________

Rubi [A]  time = 0.06955, 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, 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.

[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])

Rule 640

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

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] - Dist[(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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[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]

Rule 215

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]

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*}

Mathematica [A]  time = 0.0519043, size = 65, normalized size = 0.6 \[ \frac{1}{216} \left (12 \sqrt{3 x+\sqrt{8 x-7}} \left (12 x+\sqrt{8 x-7}-4\right )-47 \sqrt{6} \sinh ^{-1}\left (\frac{3 \sqrt{8 x-7}+4}{\sqrt{47}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*Sqrt[3*x + Sqrt[-7 + 8*x]]*(-4 + 12*x + Sqrt[-7 + 8*x]) - 47*Sqrt[6]*ArcSinh[(4 + 3*Sqrt[-7 + 8*x])/Sqrt[4
7]])/216

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 67, normalized size = 0.6 \begin{align*}{\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 ) } \end{align*}

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))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+(-7+8*x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 8.64316, size = 320, normalized size = 2.94 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+(-7+8*x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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)*s
qrt(8*x - 7) + 8*(3*sqrt(6)*(144*x + 17)*sqrt(8*x - 7) + 4*sqrt(6)*(432*x - 299))*sqrt(3*x + sqrt(8*x - 7)) -
9792*x + 30047)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]  time = 1.19073, size = 174, normalized size = 1.6 \begin{align*} \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} \log \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} \log \left (\sqrt{21} \sqrt{3} - 4\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+(-7+8*x)^(1/2))^(1/2),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)*log(-sqrt(3)*(sqrt(3)*sqrt(8*x - 7) - 2*sqrt(2)*sqrt(3*x + sqrt(8*x - 7))) - 4) - 1/432*s
qrt(3)*(13*sqrt(21)*sqrt(3)*sqrt(2) + 94*sqrt(2)*log(sqrt(21)*sqrt(3) - 4))