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 = 0.0365901, 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 \[ \frac{1}{4} x^{3/2} (x+1)^{5/2}+\frac{5}{24} x^{3/2} (x+1)^{3/2}+\frac{5}{32} x^{3/2} \sqrt{x+1}+\frac{5}{64} \sqrt{x} \sqrt{x+1}-\frac{5}{64} \sinh ^{-1}\left (\sqrt{x}\right ) \]

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

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Rubi in Sympy [A]  time = 2.66932, size = 66, normalized size = 0.88 \[ \frac{\sqrt{x} \left (x + 1\right )^{\frac{7}{2}}}{4} - \frac{\sqrt{x} \left (x + 1\right )^{\frac{5}{2}}}{24} - \frac{5 \sqrt{x} \left (x + 1\right )^{\frac{3}{2}}}{96} - \frac{5 \sqrt{x} \sqrt{x + 1}}{64} - \frac{5 \operatorname{asinh}{\left (\sqrt{x} \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(1+x)**(5/2),x)

[Out]

sqrt(x)*(x + 1)**(7/2)/4 - sqrt(x)*(x + 1)**(5/2)/24 - 5*sqrt(x)*(x + 1)**(3/2)/
96 - 5*sqrt(x)*sqrt(x + 1)/64 - 5*asinh(sqrt(x))/64

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Mathematica [A]  time = 0.0395441, 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 = 0.006, size = 70, normalized size = 0.9 \[{\frac{1}{4}\sqrt{x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{96}\sqrt{x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{64}\sqrt{x}\sqrt{1+x}}-{\frac{5}{128}\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \]

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 [A]  time = 1.34759, size = 153, normalized size = 2.04 \[ \frac{\frac{15 \,{\left (x + 1\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}} + \frac{73 \,{\left (x + 1\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}} - \frac{55 \,{\left (x + 1\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}} + \frac{15 \, \sqrt{x + 1}}{\sqrt{x}}}{192 \,{\left (\frac{{\left (x + 1\right )}^{4}}{x^{4}} - \frac{4 \,{\left (x + 1\right )}^{3}}{x^{3}} + \frac{6 \,{\left (x + 1\right )}^{2}}{x^{2}} - \frac{4 \,{\left (x + 1\right )}}{x} + 1\right )}} - \frac{5}{128} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) + \frac{5}{128} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*sqrt(x),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 = 0.214251, size = 266, normalized size = 3.55 \[ -\frac{98304 \, x^{8} + 524288 \, x^{7} + 1146880 \, x^{6} + 1294336 \, x^{5} + 788608 \, x^{4} + 250112 \, x^{3} - 8 \,{\left (12288 \, x^{7} + 59392 \, x^{6} + 115200 \, x^{5} + 110848 \, x^{4} + 54320 \, x^{3} + 12744 \, x^{2} + 1246 \, x + 35\right )} \sqrt{x + 1} \sqrt{x} + 37024 \, x^{2} - 120 \,{\left (128 \, x^{4} + 256 \, x^{3} - 8 \,{\left (16 \, x^{3} + 24 \, x^{2} + 10 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + 160 \, x^{2} + 32 \, x + 1\right )} \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) + 2080 \, x + 5}{3072 \,{\left (128 \, x^{4} + 256 \, x^{3} - 8 \,{\left (16 \, x^{3} + 24 \, x^{2} + 10 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + 160 \, x^{2} + 32 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3072*(98304*x^8 + 524288*x^7 + 1146880*x^6 + 1294336*x^5 + 788608*x^4 + 25011
2*x^3 - 8*(12288*x^7 + 59392*x^6 + 115200*x^5 + 110848*x^4 + 54320*x^3 + 12744*x
^2 + 1246*x + 35)*sqrt(x + 1)*sqrt(x) + 37024*x^2 - 120*(128*x^4 + 256*x^3 - 8*(
16*x^3 + 24*x^2 + 10*x + 1)*sqrt(x + 1)*sqrt(x) + 160*x^2 + 32*x + 1)*log(2*sqrt
(x + 1)*sqrt(x) - 2*x - 1) + 2080*x + 5)/(128*x^4 + 256*x^3 - 8*(16*x^3 + 24*x^2
+ 10*x + 1)*sqrt(x + 1)*sqrt(x) + 160*x^2 + 32*x + 1)

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Sympy [A]  time = 39.1646, size = 190, normalized size = 2.53 \[ \begin{cases} - \frac{5 \operatorname{acosh}{\left (\sqrt{x + 1} \right )}}{64} + \frac{\left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{x}} - \frac{7 \left (x + 1\right )^{\frac{7}{2}}}{24 \sqrt{x}} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{96 \sqrt{x}} - \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{192 \sqrt{x}} + \frac{5 \sqrt{x + 1}}{64 \sqrt{x}} & \text{for}\: \left |{x + 1}\right | > 1 \\\frac{5 i \operatorname{asin}{\left (\sqrt{x + 1} \right )}}{64} - \frac{i \left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{- x}} + \frac{7 i \left (x + 1\right )^{\frac{7}{2}}}{24 \sqrt{- x}} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{96 \sqrt{- x}} + \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{192 \sqrt{- x}} - \frac{5 i \sqrt{x + 1}}{64 \sqrt{- x}} & \text{otherwise} \end{cases} \]

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(sqrt(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/XCAS [A]  time = 0.28991, size = 111, normalized size = 1.48 \[ \frac{1}{192} \,{\left (2 \,{\left (4 \,{\left (6 \, x - 11\right )}{\left (x + 1\right )} + 59\right )}{\left (x + 1\right )} - 15\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{12} \,{\left (2 \,{\left (4 \, x - 3\right )}{\left (x + 1\right )} + 3\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{4} \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + \frac{5}{64} \,{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*sqrt(x),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*ln(abs(-sqrt(x + 1) + sqrt(x)))