3.8.5 \(\int \sqrt {3 x+\sqrt {-7+8 x}} \, dx\) [705]

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

[Out]

-47/216*arcsinh(1/47*(4+3*(-7+8*x)^(1/2))*47^(1/2))*6^(1/2)+1/144*(24*x+8*(-7+8*x)^(1/2))^(3/2)*2^(1/2)-1/36*(
4+3*(-7+8*x)^(1/2))*(6*x+2*(-7+8*x)^(1/2))^(1/2)*2^(1/2)

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Rubi [A]
time = 0.05, 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 = {654, 626, 633, 221} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3*x + Sqrt[-7 + 8*x]],x]

[Out]

-1/36*((4 + 3*Sqrt[-7 + 8*x])*Sqrt[21 - 3*(7 - 8*x) + 8*Sqrt[-7 + 8*x]])/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 221

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]

Rule 626

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 633

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 654

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]

Rubi steps

\begin {align*} \int \sqrt {3 x+\sqrt {-7+8 x}} \, dx &=\frac {1}{4} \text {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} \text {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} \text {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}} \text {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 = 0.20, size = 83, normalized size = 0.76 \begin {gather*} \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}} \end {gather*}

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*Log[-4 - 3*Sqrt[-7 + 8*x] + 2*Sqrt[6]*Sqrt[
3*x + Sqrt[-7 + 8*x]]])/(36*Sqrt[6])

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Maple [A]
time = 0.02, size = 67, normalized size = 0.61

method	result	size
derivativedivides	\(\frac {\left (48 x +16 \sqrt {-7+8 x}\right )^{\frac {3}{2}}}{288}-\frac {\left (12 \sqrt {-7+8 x}+16\right ) \sqrt {48 x +16 \sqrt {-7+8 x}}}{288}-\frac {47 \sqrt {6}\, \arcsinh \left (\frac {3 \sqrt {47}\, \left (\sqrt {-7+8 x}+\frac {4}{3}\right )}{47}\right )}{216}\)	\(67\)
default	\(\frac {\left (48 x +16 \sqrt {-7+8 x}\right )^{\frac {3}{2}}}{288}-\frac {\left (12 \sqrt {-7+8 x}+16\right ) \sqrt {48 x +16 \sqrt {-7+8 x}}}{288}-\frac {47 \sqrt {6}\, \arcsinh \left (\frac {3 \sqrt {47}\, \left (\sqrt {-7+8 x}+\frac {4}{3}\right )}{47}\right )}{216}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+(-7+8*x)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.93, size = 101, normalized size = 0.93 \begin {gather*} \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 {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {3 x + \sqrt {8 x - 7}}\, dx \end {gather*}

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 [A]
time = 3.31, size = 88, normalized size = 0.81 \begin {gather*} \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 )} \end {gather*}

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]

1/216*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*sqrt(2)*sqrt(3*x + sqrt(8*x - 7))) - 4))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {3\,x+\sqrt {8\,x-7}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + (8*x - 7)^(1/2))^(1/2),x)

[Out]

int((3*x + (8*x - 7)^(1/2))^(1/2), x)

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