3.122 \(\int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=112 \[ \frac{64 a^3 \sqrt{a x+b \sqrt{x}}}{35 b^4 \sqrt{x}}-\frac{32 a^2 \sqrt{a x+b \sqrt{x}}}{35 b^3 x}+\frac{24 a \sqrt{a x+b \sqrt{x}}}{35 b^2 x^{3/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{7 b x^2} \]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) + (24*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^2*x^(3
/2)) - (32*a^2*Sqrt[b*Sqrt[x] + a*x])/(35*b^3*x) + (64*a^3*Sqrt[b*Sqrt[x] + a*x]
)/(35*b^4*Sqrt[x])

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Rubi [A]  time = 0.26779, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{64 a^3 \sqrt{a x+b \sqrt{x}}}{35 b^4 \sqrt{x}}-\frac{32 a^2 \sqrt{a x+b \sqrt{x}}}{35 b^3 x}+\frac{24 a \sqrt{a x+b \sqrt{x}}}{35 b^2 x^{3/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{7 b x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) + (24*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^2*x^(3
/2)) - (32*a^2*Sqrt[b*Sqrt[x] + a*x])/(35*b^3*x) + (64*a^3*Sqrt[b*Sqrt[x] + a*x]
)/(35*b^4*Sqrt[x])

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Rubi in Sympy [A]  time = 24.654, size = 102, normalized size = 0.91 \[ \frac{64 a^{3} \sqrt{a x + b \sqrt{x}}}{35 b^{4} \sqrt{x}} - \frac{32 a^{2} \sqrt{a x + b \sqrt{x}}}{35 b^{3} x} + \frac{24 a \sqrt{a x + b \sqrt{x}}}{35 b^{2} x^{\frac{3}{2}}} - \frac{4 \sqrt{a x + b \sqrt{x}}}{7 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

64*a**3*sqrt(a*x + b*sqrt(x))/(35*b**4*sqrt(x)) - 32*a**2*sqrt(a*x + b*sqrt(x))/
(35*b**3*x) + 24*a*sqrt(a*x + b*sqrt(x))/(35*b**2*x**(3/2)) - 4*sqrt(a*x + b*sqr
t(x))/(7*b*x**2)

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Mathematica [A]  time = 0.0379007, size = 59, normalized size = 0.53 \[ \frac{4 \sqrt{a x+b \sqrt{x}} \left (16 a^3 x^{3/2}-8 a^2 b x+6 a b^2 \sqrt{x}-5 b^3\right )}{35 b^4 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[b*Sqrt[x] + a*x]*(-5*b^3 + 6*a*b^2*Sqrt[x] - 8*a^2*b*x + 16*a^3*x^(3/2))
)/(35*b^4*x^2)

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Maple [C]  time = 0.015, size = 232, normalized size = 2.1 \[ -{\frac{1}{35\,{b}^{5}}\sqrt{b\sqrt{x}+ax} \left ( 35\,{a}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{9/2}-35\,{a}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{9/2}+70\,{a}^{4}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{9/2}-140\,{a}^{3} \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{7/2}+70\,{a}^{4}\sqrt{b\sqrt{x}+ax}{x}^{9/2}-44\,{x}^{5/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}a{b}^{2}+76\,{a}^{2} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{3}+20\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{2}{b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(b*x^(1/2)+a*x)^(1/2)*(35*a^(7/2)*ln(1/2*(2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+
2*x^(1/2)*a+b)/a^(1/2))*b*x^(9/2)-35*a^(7/2)*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))*b*x^(9/2)+70*a^4*(x^(1/2)*(b+x^(1/2)*a))^(1
/2)*x^(9/2)-140*a^3*(b*x^(1/2)+a*x)^(3/2)*x^(7/2)+70*a^4*(b*x^(1/2)+a*x)^(1/2)*x
^(9/2)-44*x^(5/2)*(b*x^(1/2)+a*x)^(3/2)*a*b^2+76*a^2*(b*x^(1/2)+a*x)^(3/2)*b*x^3
+20*(b*x^(1/2)+a*x)^(3/2)*x^2*b^3)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/b^5/x^(9/2)

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Maxima [A]  time = 1.46194, size = 93, normalized size = 0.83 \[ \frac{4 \,{\left (\frac{35 \, \sqrt{a \sqrt{x} + b} a^{3}}{x^{\frac{1}{4}}} - \frac{35 \,{\left (a \sqrt{x} + b\right )}^{\frac{3}{2}} a^{2}}{x^{\frac{3}{4}}} + \frac{21 \,{\left (a \sqrt{x} + b\right )}^{\frac{5}{2}} a}{x^{\frac{5}{4}}} - \frac{5 \,{\left (a \sqrt{x} + b\right )}^{\frac{7}{2}}}{x^{\frac{7}{4}}}\right )}}{35 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/35*(35*sqrt(a*sqrt(x) + b)*a^3/x^(1/4) - 35*(a*sqrt(x) + b)^(3/2)*a^2/x^(3/4)
+ 21*(a*sqrt(x) + b)^(5/2)*a/x^(5/4) - 5*(a*sqrt(x) + b)^(7/2)/x^(7/4))/b^4

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Fricas [A]  time = 0.262855, size = 68, normalized size = 0.61 \[ -\frac{4 \,{\left (8 \, a^{2} b x + 5 \, b^{3} - 2 \,{\left (8 \, a^{3} x + 3 \, a b^{2}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{35 \, b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/35*(8*a^2*b*x + 5*b^3 - 2*(8*a^3*x + 3*a*b^2)*sqrt(x))*sqrt(a*x + b*sqrt(x))/
(b^4*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{5}{2}} \sqrt{a x + b \sqrt{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225601, size = 155, normalized size = 1.38 \[ \frac{4 \,{\left (70 \, a^{\frac{3}{2}}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3} + 84 \, a b{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{2} + 35 \, \sqrt{a} b^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + 5 \, b^{3}\right )}}{35 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/35*(70*a^(3/2)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 84*a*b*(sqrt(a)*s
qrt(x) - sqrt(a*x + b*sqrt(x)))^2 + 35*sqrt(a)*b^2*(sqrt(a)*sqrt(x) - sqrt(a*x +
b*sqrt(x))) + 5*b^3)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^7