3.7.40 \(\int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx\)

Optimal. Leaf size=135 \[ \frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}-\frac {1}{24} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{96} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {5}{64} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {280, 323, 330, 52} \begin {gather*} \frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{7/2}-\frac {1}{24} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{96} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {5}{64} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2),x]

[Out]

(-5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/64 - (5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/96 - (
Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/24 + (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2))/4 - (5*ArcCo
sh[Sqrt[x]])/64

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 280

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*
x)^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(c*(m + 2*n*p + 1)), x] + Dist[(2*a1*a2*n*p)/(m + 2*n*p + 1), Int[
(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 +
a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && NeQ[m + 2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x
]

Rule 323

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(2
*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a
1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x],
x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m +
2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 330

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x]
, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rubi steps

\begin {align*} \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2} \, dx &=\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {1}{8} \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{48} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{128} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{64} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{96} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{24} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{7/2}-\frac {5}{64} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 111, normalized size = 0.82 \begin {gather*} \frac {\sqrt {\sqrt {x}+1} \sqrt {x} \left (48 x^{7/2}-8 x^{5/2}-10 x^{3/2}-48 x^3+8 x^2+10 x-15 \sqrt {x}+15\right )+30 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )}{192 \sqrt {\sqrt {x}-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2),x]

[Out]

(Sqrt[1 + Sqrt[x]]*Sqrt[x]*(15 - 15*Sqrt[x] + 10*x - 10*x^(3/2) + 8*x^2 - 8*x^(5/2) - 48*x^3 + 48*x^(7/2)) + 3
0*Sqrt[1 - Sqrt[x]]*ArcSin[Sqrt[1 - Sqrt[x]]/Sqrt[2]])/(192*Sqrt[-1 + Sqrt[x]])

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IntegrateAlgebraic [A]  time = 2.44, size = 216, normalized size = 1.60 \begin {gather*} \frac {\left (\frac {15 \left (\sqrt {x}-1\right )^7}{\left (\sqrt {x}+1\right )^7}+\frac {397 \left (\sqrt {x}-1\right )^6}{\left (\sqrt {x}+1\right )^6}+\frac {895 \left (\sqrt {x}-1\right )^5}{\left (\sqrt {x}+1\right )^5}+\frac {1765 \left (\sqrt {x}-1\right )^4}{\left (\sqrt {x}+1\right )^4}+\frac {1765 \left (\sqrt {x}-1\right )^3}{\left (\sqrt {x}+1\right )^3}+\frac {895 \left (\sqrt {x}-1\right )^2}{\left (\sqrt {x}+1\right )^2}+\frac {397 \left (\sqrt {x}-1\right )}{\sqrt {x}+1}+15\right ) \sqrt {\sqrt {x}-1}}{96 \left (\frac {\sqrt {x}-1}{\sqrt {x}+1}-1\right )^8 \sqrt {\sqrt {x}+1}}-\frac {5}{32} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}}{\sqrt {\sqrt {x}+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2),x]

[Out]

((15 + (15*(-1 + Sqrt[x])^7)/(1 + Sqrt[x])^7 + (397*(-1 + Sqrt[x])^6)/(1 + Sqrt[x])^6 + (895*(-1 + Sqrt[x])^5)
/(1 + Sqrt[x])^5 + (1765*(-1 + Sqrt[x])^4)/(1 + Sqrt[x])^4 + (1765*(-1 + Sqrt[x])^3)/(1 + Sqrt[x])^3 + (895*(-
1 + Sqrt[x])^2)/(1 + Sqrt[x])^2 + (397*(-1 + Sqrt[x]))/(1 + Sqrt[x]))*Sqrt[-1 + Sqrt[x]])/(96*(-1 + (-1 + Sqrt
[x])/(1 + Sqrt[x]))^8*Sqrt[1 + Sqrt[x]]) - (5*ArcTanh[Sqrt[-1 + Sqrt[x]]/Sqrt[1 + Sqrt[x]]])/32

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fricas [A]  time = 0.39, size = 62, normalized size = 0.46 \begin {gather*} \frac {1}{192} \, {\left (48 \, x^{3} - 8 \, x^{2} - 10 \, x - 15\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{128} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/192*(48*x^3 - 8*x^2 - 10*x - 15)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 5/128*log(2*sqrt(x)*sqrt(sqrt
(x) + 1)*sqrt(sqrt(x) - 1) - 2*x + 1)

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giac [A]  time = 0.29, size = 162, normalized size = 1.20 \begin {gather*} \frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, \sqrt {x} - 50\right )} {\left (\sqrt {x} + 1\right )} + 1219\right )} {\left (\sqrt {x} + 1\right )} - 12463\right )} {\left (\sqrt {x} + 1\right )} + 64233\right )} {\left (\sqrt {x} + 1\right )} - 53963\right )} {\left (\sqrt {x} + 1\right )} + 59465\right )} {\left (\sqrt {x} + 1\right )} - 23205\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{840} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, \sqrt {x} - 37\right )} {\left (\sqrt {x} + 1\right )} + 661\right )} {\left (\sqrt {x} + 1\right )} - 4551\right )} {\left (\sqrt {x} + 1\right )} + 4781\right )} {\left (\sqrt {x} + 1\right )} - 6335\right )} {\left (\sqrt {x} + 1\right )} + 2835\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {5}{32} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x, algorithm="giac")

[Out]

1/6720*((2*((4*(5*(6*(7*sqrt(x) - 50)*(sqrt(x) + 1) + 1219)*(sqrt(x) + 1) - 12463)*(sqrt(x) + 1) + 64233)*(sqr
t(x) + 1) - 53963)*(sqrt(x) + 1) + 59465)*(sqrt(x) + 1) - 23205)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 1/840*(
(2*((4*(5*(6*sqrt(x) - 37)*(sqrt(x) + 1) + 661)*(sqrt(x) + 1) - 4551)*(sqrt(x) + 1) + 4781)*(sqrt(x) + 1) - 63
35)*(sqrt(x) + 1) + 2835)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 5/32*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1)
)

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maple [A]  time = 0.05, size = 75, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-48 \sqrt {x -1}\, x^{\frac {7}{2}}+8 \sqrt {x -1}\, x^{\frac {5}{2}}+10 \sqrt {x -1}\, x^{\frac {3}{2}}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )+15 \sqrt {x -1}\, \sqrt {x}\right )}{192 \sqrt {x -1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2),x)

[Out]

-1/192*(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)*(-48*x^(7/2)*(x-1)^(1/2)+8*x^(5/2)*(x-1)^(1/2)+10*x^(3/2)*(x-1)^(1/
2)+15*(x-1)^(1/2)*x^(1/2)+15*ln(x^(1/2)+(x-1)^(1/2)))/(x-1)^(1/2)

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maxima [A]  time = 0.55, size = 57, normalized size = 0.42 \begin {gather*} \frac {1}{4} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {5}{2}} + \frac {5}{24} \, {\left (x - 1\right )}^{\frac {3}{2}} x^{\frac {3}{2}} + \frac {5}{32} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {5}{64} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{64} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

1/4*(x - 1)^(3/2)*x^(5/2) + 5/24*(x - 1)^(3/2)*x^(3/2) + 5/32*(x - 1)^(3/2)*sqrt(x) + 5/64*sqrt(x - 1)*sqrt(x)
- 5/64*log(2*sqrt(x - 1) + 2*sqrt(x))

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mupad [B]  time = 52.03, size = 831, normalized size = 6.16

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(x^(1/2) - 1)^(1/2)*(x^(1/2) + 1)^(1/2),x)

[Out]

((1723*((x^(1/2) - 1)^(1/2) - 1i)^5)/(48*((x^(1/2) + 1)^(1/2) - 1)^5) - (235*((x^(1/2) - 1)^(1/2) - 1i)^3)/(48
*((x^(1/2) + 1)^(1/2) - 1)^3) + (72283*((x^(1/2) - 1)^(1/2) - 1i)^7)/(16*((x^(1/2) + 1)^(1/2) - 1)^7) + (84880
1*((x^(1/2) - 1)^(1/2) - 1i)^9)/(16*((x^(1/2) + 1)^(1/2) - 1)^9) + (4181067*((x^(1/2) - 1)^(1/2) - 1i)^11)/(16
*((x^(1/2) + 1)^(1/2) - 1)^11) + (10994181*((x^(1/2) - 1)^(1/2) - 1i)^13)/(16*((x^(1/2) + 1)^(1/2) - 1)^13) +
(17457599*((x^(1/2) - 1)^(1/2) - 1i)^15)/(16*((x^(1/2) + 1)^(1/2) - 1)^15) + (17457599*((x^(1/2) - 1)^(1/2) -
1i)^17)/(16*((x^(1/2) + 1)^(1/2) - 1)^17) + (10994181*((x^(1/2) - 1)^(1/2) - 1i)^19)/(16*((x^(1/2) + 1)^(1/2)
- 1)^19) + (4181067*((x^(1/2) - 1)^(1/2) - 1i)^21)/(16*((x^(1/2) + 1)^(1/2) - 1)^21) + (848801*((x^(1/2) - 1)^
(1/2) - 1i)^23)/(16*((x^(1/2) + 1)^(1/2) - 1)^23) + (72283*((x^(1/2) - 1)^(1/2) - 1i)^25)/(16*((x^(1/2) + 1)^(
1/2) - 1)^25) + (1723*((x^(1/2) - 1)^(1/2) - 1i)^27)/(48*((x^(1/2) + 1)^(1/2) - 1)^27) - (235*((x^(1/2) - 1)^(
1/2) - 1i)^29)/(48*((x^(1/2) + 1)^(1/2) - 1)^29) + (5*((x^(1/2) - 1)^(1/2) - 1i)^31)/(16*((x^(1/2) + 1)^(1/2)
- 1)^31) + (5*((x^(1/2) - 1)^(1/2) - 1i))/(16*((x^(1/2) + 1)^(1/2) - 1)))/((120*((x^(1/2) - 1)^(1/2) - 1i)^4)/
((x^(1/2) + 1)^(1/2) - 1)^4 - (16*((x^(1/2) - 1)^(1/2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (560*((x^(1/2) -
1)^(1/2) - 1i)^6)/((x^(1/2) + 1)^(1/2) - 1)^6 + (1820*((x^(1/2) - 1)^(1/2) - 1i)^8)/((x^(1/2) + 1)^(1/2) - 1)
^8 - (4368*((x^(1/2) - 1)^(1/2) - 1i)^10)/((x^(1/2) + 1)^(1/2) - 1)^10 + (8008*((x^(1/2) - 1)^(1/2) - 1i)^12)/
((x^(1/2) + 1)^(1/2) - 1)^12 - (11440*((x^(1/2) - 1)^(1/2) - 1i)^14)/((x^(1/2) + 1)^(1/2) - 1)^14 + (12870*((x
^(1/2) - 1)^(1/2) - 1i)^16)/((x^(1/2) + 1)^(1/2) - 1)^16 - (11440*((x^(1/2) - 1)^(1/2) - 1i)^18)/((x^(1/2) + 1
)^(1/2) - 1)^18 + (8008*((x^(1/2) - 1)^(1/2) - 1i)^20)/((x^(1/2) + 1)^(1/2) - 1)^20 - (4368*((x^(1/2) - 1)^(1/
2) - 1i)^22)/((x^(1/2) + 1)^(1/2) - 1)^22 + (1820*((x^(1/2) - 1)^(1/2) - 1i)^24)/((x^(1/2) + 1)^(1/2) - 1)^24
- (560*((x^(1/2) - 1)^(1/2) - 1i)^26)/((x^(1/2) + 1)^(1/2) - 1)^26 + (120*((x^(1/2) - 1)^(1/2) - 1i)^28)/((x^(
1/2) + 1)^(1/2) - 1)^28 - (16*((x^(1/2) - 1)^(1/2) - 1i)^30)/((x^(1/2) + 1)^(1/2) - 1)^30 + ((x^(1/2) - 1)^(1/
2) - 1i)^32/((x^(1/2) + 1)^(1/2) - 1)^32 + 1) - (5*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1))
)/16

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{\frac {5}{2}} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2),x)

[Out]

Integral(x**(5/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1), x)

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