3.117 \(\int \frac{x^{5/2}}{\sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=204 \[ \frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{256 a^{13/2}}-\frac{231 b^5 \sqrt{a x+b \sqrt{x}}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{128 a^5}-\frac{77 b^3 x \sqrt{a x+b \sqrt{x}}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{a x+b \sqrt{x}}}{80 a^3}-\frac{11 b x^2 \sqrt{a x+b \sqrt{x}}}{30 a^2}+\frac{x^{5/2} \sqrt{a x+b \sqrt{x}}}{3 a} \]

[Out]

(-231*b^5*Sqrt[b*Sqrt[x] + a*x])/(256*a^6) + (77*b^4*Sqrt[x]*Sqrt[b*Sqrt[x] + a*
x])/(128*a^5) - (77*b^3*x*Sqrt[b*Sqrt[x] + a*x])/(160*a^4) + (33*b^2*x^(3/2)*Sqr
t[b*Sqrt[x] + a*x])/(80*a^3) - (11*b*x^2*Sqrt[b*Sqrt[x] + a*x])/(30*a^2) + (x^(5
/2)*Sqrt[b*Sqrt[x] + a*x])/(3*a) + (231*b^6*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqr
t[x] + a*x]])/(256*a^(13/2))

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Rubi [A]  time = 0.388035, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{231 b^6 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{256 a^{13/2}}-\frac{231 b^5 \sqrt{a x+b \sqrt{x}}}{256 a^6}+\frac{77 b^4 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{128 a^5}-\frac{77 b^3 x \sqrt{a x+b \sqrt{x}}}{160 a^4}+\frac{33 b^2 x^{3/2} \sqrt{a x+b \sqrt{x}}}{80 a^3}-\frac{11 b x^2 \sqrt{a x+b \sqrt{x}}}{30 a^2}+\frac{x^{5/2} \sqrt{a x+b \sqrt{x}}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

(-231*b^5*Sqrt[b*Sqrt[x] + a*x])/(256*a^6) + (77*b^4*Sqrt[x]*Sqrt[b*Sqrt[x] + a*
x])/(128*a^5) - (77*b^3*x*Sqrt[b*Sqrt[x] + a*x])/(160*a^4) + (33*b^2*x^(3/2)*Sqr
t[b*Sqrt[x] + a*x])/(80*a^3) - (11*b*x^2*Sqrt[b*Sqrt[x] + a*x])/(30*a^2) + (x^(5
/2)*Sqrt[b*Sqrt[x] + a*x])/(3*a) + (231*b^6*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqr
t[x] + a*x]])/(256*a^(13/2))

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Rubi in Sympy [A]  time = 40.8057, size = 190, normalized size = 0.93 \[ \frac{x^{\frac{5}{2}} \sqrt{a x + b \sqrt{x}}}{3 a} - \frac{11 b x^{2} \sqrt{a x + b \sqrt{x}}}{30 a^{2}} + \frac{33 b^{2} x^{\frac{3}{2}} \sqrt{a x + b \sqrt{x}}}{80 a^{3}} - \frac{77 b^{3} x \sqrt{a x + b \sqrt{x}}}{160 a^{4}} + \frac{77 b^{4} \sqrt{x} \sqrt{a x + b \sqrt{x}}}{128 a^{5}} - \frac{231 b^{5} \sqrt{a x + b \sqrt{x}}}{256 a^{6}} + \frac{231 b^{6} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{256 a^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(5/2)*sqrt(a*x + b*sqrt(x))/(3*a) - 11*b*x**2*sqrt(a*x + b*sqrt(x))/(30*a**2)
+ 33*b**2*x**(3/2)*sqrt(a*x + b*sqrt(x))/(80*a**3) - 77*b**3*x*sqrt(a*x + b*sqr
t(x))/(160*a**4) + 77*b**4*sqrt(x)*sqrt(a*x + b*sqrt(x))/(128*a**5) - 231*b**5*s
qrt(a*x + b*sqrt(x))/(256*a**6) + 231*b**6*atanh(sqrt(a)*sqrt(x)/sqrt(a*x + b*sq
rt(x)))/(256*a**(13/2))

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Mathematica [A]  time = 0.173418, size = 128, normalized size = 0.63 \[ \frac{2 \sqrt{a} \sqrt{a x+b \sqrt{x}} \left (1280 a^5 x^{5/2}-1408 a^4 b x^2+1584 a^3 b^2 x^{3/2}-1848 a^2 b^3 x+2310 a b^4 \sqrt{x}-3465 b^5\right )+3465 b^6 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{7680 a^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

(2*Sqrt[a]*Sqrt[b*Sqrt[x] + a*x]*(-3465*b^5 + 2310*a*b^4*Sqrt[x] - 1848*a^2*b^3*
x + 1584*a^3*b^2*x^(3/2) - 1408*a^4*b*x^2 + 1280*a^5*x^(5/2)) + 3465*b^6*Log[b +
2*a*Sqrt[x] + 2*Sqrt[a]*Sqrt[b*Sqrt[x] + a*x]])/(7680*a^(13/2))

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Maple [A]  time = 0.013, size = 249, normalized size = 1.2 \[ -{\frac{1}{7680}\sqrt{b\sqrt{x}+ax} \left ( -2560\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{21/2}-8544\,{b}^{2}\sqrt{x} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{17/2}+5376\,bx \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{19/2}+12240\,{b}^{3} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{15/2}-16860\,{b}^{4}\sqrt{b\sqrt{x}+ax}\sqrt{x}{a}^{15/2}+15360\,{b}^{5}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{a}^{13/2}-8430\,{b}^{5}\sqrt{b\sqrt{x}+ax}{a}^{13/2}+4215\,{b}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{6}-7680\,{b}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{6} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{a}^{-{\frac{25}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x)

[Out]

-1/7680*(b*x^(1/2)+a*x)^(1/2)*(-2560*x^(3/2)*(b*x^(1/2)+a*x)^(3/2)*a^(21/2)-8544
*b^2*x^(1/2)*(b*x^(1/2)+a*x)^(3/2)*a^(17/2)+5376*b*x*(b*x^(1/2)+a*x)^(3/2)*a^(19
/2)+12240*b^3*(b*x^(1/2)+a*x)^(3/2)*a^(15/2)-16860*b^4*(b*x^(1/2)+a*x)^(1/2)*x^(
1/2)*a^(15/2)+15360*b^5*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(13/2)-8430*b^5*(b*x^(1/
2)+a*x)^(1/2)*a^(13/2)+4215*b^6*ln(1/2*(2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+2*x^(1/2
)*a+b)/a^(1/2))*a^6-7680*b^6*ln(1/2*(2*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(1/2)+2*x
^(1/2)*a+b)/a^(1/2))*a^6)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/a^(25/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(a*x + b*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}}}{\sqrt{a x + b \sqrt{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x**(1/2)+a*x)**(1/2),x)

[Out]

Integral(x**(5/2)/sqrt(a*x + b*sqrt(x)), x)

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GIAC/XCAS [A]  time = 0.279129, size = 169, normalized size = 0.83 \[ \frac{1}{3840} \, \sqrt{a x + b \sqrt{x}}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, \sqrt{x}{\left (\frac{10 \, \sqrt{x}}{a} - \frac{11 \, b}{a^{2}}\right )} + \frac{99 \, b^{2}}{a^{3}}\right )} \sqrt{x} - \frac{231 \, b^{3}}{a^{4}}\right )} \sqrt{x} + \frac{1155 \, b^{4}}{a^{5}}\right )} \sqrt{x} - \frac{3465 \, b^{5}}{a^{6}}\right )} - \frac{231 \, b^{6}{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{512 \, a^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(a*x + b*sqrt(x)),x, algorithm="giac")

[Out]

1/3840*sqrt(a*x + b*sqrt(x))*(2*(4*(2*(8*sqrt(x)*(10*sqrt(x)/a - 11*b/a^2) + 99*
b^2/a^3)*sqrt(x) - 231*b^3/a^4)*sqrt(x) + 1155*b^4/a^5)*sqrt(x) - 3465*b^5/a^6)
- 231/512*b^6*ln(abs(-2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/
a^(13/2)