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

Optimal. Leaf size=75 \[ \frac{1}{4} x^{3/2} (x+1)^{5/2}+\frac{5}{24} x^{3/2} (x+1)^{3/2}+\frac{5}{32} x^{3/2} \sqrt{x+1}+\frac{5}{64} \sqrt{x} \sqrt{x+1}-\frac{5}{64} \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

(5*Sqrt[x]*Sqrt[1 + x])/64 + (5*x^(3/2)*Sqrt[1 + x])/32 + (5*x^(3/2)*(1 + x)^(3/2))/24 + (x^(3/2)*(1 + x)^(5/2
))/4 - (5*ArcSinh[Sqrt[x]])/64

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Rubi [A]  time = 0.0130271, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 54, 215} \[ \frac{1}{4} x^{3/2} (x+1)^{5/2}+\frac{5}{24} x^{3/2} (x+1)^{3/2}+\frac{5}{32} x^{3/2} \sqrt{x+1}+\frac{5}{64} \sqrt{x} \sqrt{x+1}-\frac{5}{64} \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(1 + x)^(5/2),x]

[Out]

(5*Sqrt[x]*Sqrt[1 + x])/64 + (5*x^(3/2)*Sqrt[1 + x])/32 + (5*x^(3/2)*(1 + x)^(3/2))/24 + (x^(3/2)*(1 + x)^(5/2
))/4 - (5*ArcSinh[Sqrt[x]])/64

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 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{x} (1+x)^{5/2} \, dx &=\frac{1}{4} x^{3/2} (1+x)^{5/2}+\frac{5}{8} \int \sqrt{x} (1+x)^{3/2} \, dx\\ &=\frac{5}{24} x^{3/2} (1+x)^{3/2}+\frac{1}{4} x^{3/2} (1+x)^{5/2}+\frac{5}{16} \int \sqrt{x} \sqrt{1+x} \, dx\\ &=\frac{5}{32} x^{3/2} \sqrt{1+x}+\frac{5}{24} x^{3/2} (1+x)^{3/2}+\frac{1}{4} x^{3/2} (1+x)^{5/2}+\frac{5}{64} \int \frac{\sqrt{x}}{\sqrt{1+x}} \, dx\\ &=\frac{5}{64} \sqrt{x} \sqrt{1+x}+\frac{5}{32} x^{3/2} \sqrt{1+x}+\frac{5}{24} x^{3/2} (1+x)^{3/2}+\frac{1}{4} x^{3/2} (1+x)^{5/2}-\frac{5}{128} \int \frac{1}{\sqrt{x} \sqrt{1+x}} \, dx\\ &=\frac{5}{64} \sqrt{x} \sqrt{1+x}+\frac{5}{32} x^{3/2} \sqrt{1+x}+\frac{5}{24} x^{3/2} (1+x)^{3/2}+\frac{1}{4} x^{3/2} (1+x)^{5/2}-\frac{5}{64} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{64} \sqrt{x} \sqrt{1+x}+\frac{5}{32} x^{3/2} \sqrt{1+x}+\frac{5}{24} x^{3/2} (1+x)^{3/2}+\frac{1}{4} x^{3/2} (1+x)^{5/2}-\frac{5}{64} \sinh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0194251, size = 41, normalized size = 0.55 \[ \frac{1}{192} \left (\sqrt{x} \sqrt{x+1} \left (48 x^3+136 x^2+118 x+15\right )-15 \sinh ^{-1}\left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(1 + x)^(5/2),x]

[Out]

(Sqrt[x]*Sqrt[1 + x]*(15 + 118*x + 136*x^2 + 48*x^3) - 15*ArcSinh[Sqrt[x]])/192

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Maple [A]  time = 0.003, size = 70, normalized size = 0.9 \begin{align*}{\frac{1}{4}\sqrt{x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{96}\sqrt{x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{64}\sqrt{x}\sqrt{1+x}}-{\frac{5}{128}\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^(1/2)*(1+x)^(7/2)-1/24*x^(1/2)*(1+x)^(5/2)-5/96*x^(1/2)*(1+x)^(3/2)-5/64*x^(1/2)*(1+x)^(1/2)-5/128*(x*(1
+x))^(1/2)/(1+x)^(1/2)/x^(1/2)*ln(1/2+x+(x^2+x)^(1/2))

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Maxima [B]  time = 0.943377, size = 153, normalized size = 2.04 \begin{align*} \frac{\frac{15 \,{\left (x + 1\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}} + \frac{73 \,{\left (x + 1\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}} - \frac{55 \,{\left (x + 1\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}} + \frac{15 \, \sqrt{x + 1}}{\sqrt{x}}}{192 \,{\left (\frac{{\left (x + 1\right )}^{4}}{x^{4}} - \frac{4 \,{\left (x + 1\right )}^{3}}{x^{3}} + \frac{6 \,{\left (x + 1\right )}^{2}}{x^{2}} - \frac{4 \,{\left (x + 1\right )}}{x} + 1\right )}} - \frac{5}{128} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) + \frac{5}{128} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(15*(x + 1)^(7/2)/x^(7/2) + 73*(x + 1)^(5/2)/x^(5/2) - 55*(x + 1)^(3/2)/x^(3/2) + 15*sqrt(x + 1)/sqrt(x)
)/((x + 1)^4/x^4 - 4*(x + 1)^3/x^3 + 6*(x + 1)^2/x^2 - 4*(x + 1)/x + 1) - 5/128*log(sqrt(x + 1)/sqrt(x) + 1) +
5/128*log(sqrt(x + 1)/sqrt(x) - 1)

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Fricas [A]  time = 1.79587, size = 140, normalized size = 1.87 \begin{align*} \frac{1}{192} \,{\left (48 \, x^{3} + 136 \, x^{2} + 118 \, x + 15\right )} \sqrt{x + 1} \sqrt{x} + \frac{5}{128} \, \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(48*x^3 + 136*x^2 + 118*x + 15)*sqrt(x + 1)*sqrt(x) + 5/128*log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1)

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Sympy [A]  time = 11.9038, size = 190, normalized size = 2.53 \begin{align*} \begin{cases} - \frac{5 \operatorname{acosh}{\left (\sqrt{x + 1} \right )}}{64} + \frac{\left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{x}} - \frac{7 \left (x + 1\right )^{\frac{7}{2}}}{24 \sqrt{x}} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{96 \sqrt{x}} - \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{192 \sqrt{x}} + \frac{5 \sqrt{x + 1}}{64 \sqrt{x}} & \text{for}\: \left |{x + 1}\right | > 1 \\\frac{5 i \operatorname{asin}{\left (\sqrt{x + 1} \right )}}{64} - \frac{i \left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{- x}} + \frac{7 i \left (x + 1\right )^{\frac{7}{2}}}{24 \sqrt{- x}} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{96 \sqrt{- x}} + \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{192 \sqrt{- x}} - \frac{5 i \sqrt{x + 1}}{64 \sqrt{- x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-5*acosh(sqrt(x + 1))/64 + (x + 1)**(9/2)/(4*sqrt(x)) - 7*(x + 1)**(7/2)/(24*sqrt(x)) - (x + 1)**(5
/2)/(96*sqrt(x)) - 5*(x + 1)**(3/2)/(192*sqrt(x)) + 5*sqrt(x + 1)/(64*sqrt(x)), Abs(x + 1) > 1), (5*I*asin(sqr
t(x + 1))/64 - I*(x + 1)**(9/2)/(4*sqrt(-x)) + 7*I*(x + 1)**(7/2)/(24*sqrt(-x)) + I*(x + 1)**(5/2)/(96*sqrt(-x
)) + 5*I*(x + 1)**(3/2)/(192*sqrt(-x)) - 5*I*sqrt(x + 1)/(64*sqrt(-x)), True))

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Giac [A]  time = 1.27791, size = 111, normalized size = 1.48 \begin{align*} \frac{1}{192} \,{\left (2 \,{\left (4 \,{\left (6 \, x - 11\right )}{\left (x + 1\right )} + 59\right )}{\left (x + 1\right )} - 15\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{12} \,{\left (2 \,{\left (4 \, x - 3\right )}{\left (x + 1\right )} + 3\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{4} \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + \frac{5}{64} \, \log \left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(2*(4*(6*x - 11)*(x + 1) + 59)*(x + 1) - 15)*sqrt(x + 1)*sqrt(x) + 1/12*(2*(4*x - 3)*(x + 1) + 3)*sqrt(x
+ 1)*sqrt(x) + 1/4*(2*x + 1)*sqrt(x + 1)*sqrt(x) + 5/64*log(abs(-sqrt(x + 1) + sqrt(x)))