3.128 \(\int \frac{1}{x^{3/2} \left (b \sqrt{x}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{5 b^4 \sqrt{x}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{5 b^3 x}-\frac{24 \sqrt{a x+b \sqrt{x}}}{5 b^2 x^{3/2}}+\frac{4}{b x \sqrt{a x+b \sqrt{x}}} \]

[Out]

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

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Rubi [A]  time = 0.267868, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{5 b^4 \sqrt{x}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{5 b^3 x}-\frac{24 \sqrt{a x+b \sqrt{x}}}{5 b^2 x^{3/2}}+\frac{4}{b x \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.0075, size = 95, normalized size = 0.89 \[ - \frac{64 a^{2} \sqrt{a x + b \sqrt{x}}}{5 b^{4} \sqrt{x}} + \frac{32 a \sqrt{a x + b \sqrt{x}}}{5 b^{3} x} + \frac{4}{b x \sqrt{a x + b \sqrt{x}}} - \frac{24 \sqrt{a x + b \sqrt{x}}}{5 b^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(b*x**(1/2)+a*x)**(3/2),x)

[Out]

-64*a**2*sqrt(a*x + b*sqrt(x))/(5*b**4*sqrt(x)) + 32*a*sqrt(a*x + b*sqrt(x))/(5*
b**3*x) + 4/(b*x*sqrt(a*x + b*sqrt(x))) - 24*sqrt(a*x + b*sqrt(x))/(5*b**2*x**(3
/2))

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Mathematica [A]  time = 0.0505471, size = 70, normalized size = 0.65 \[ -\frac{4 \sqrt{a x+b \sqrt{x}} \left (16 a^3 x^{3/2}+8 a^2 b x-2 a b^2 \sqrt{x}+b^3\right )}{5 b^4 x^{3/2} \left (a \sqrt{x}+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

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

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Maple [C]  time = 0.017, size = 540, normalized size = 5.1 \[{\frac{2}{5\,{b}^{5}}\sqrt{b\sqrt{x}+ax} \left ( -5\,{x}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{9/2}b+5\,{x}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{9/2}b-10\,{x}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}{b}^{2}+10\,{x}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}{b}^{2}-5\,{x}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5/2}{b}^{3}+5\,{x}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5/2}{b}^{3}+10\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{9/2}{a}^{5}-30\,{x}^{7/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{4}+10\,{x}^{9/2}\sqrt{b\sqrt{x}+ax}{a}^{5}+10\,{a}^{4} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}{x}^{7/2}+20\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{4}{a}^{4}b-16\,{x}^{5/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{2}{b}^{2}-52\,{x}^{3} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3}b+20\,{x}^{4}\sqrt{b\sqrt{x}+ax}{a}^{4}b+10\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{7/2}{a}^{3}{b}^{2}+4\,{x}^{2} \left ( b\sqrt{x}+ax \right ) ^{3/2}a{b}^{3}+10\,{x}^{7/2}\sqrt{b\sqrt{x}+ax}{a}^{3}{b}^{2}-2\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{b}^{4} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{7}{2}}} \left ( b+\sqrt{x}a \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x^(1/2)+a*x)^(3/2),x)

[Out]

2/5*(b*x^(1/2)+a*x)^(1/2)*(-5*x^(9/2)*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))*a^(9/2)*b+5*x^(9/2)*ln(1/2*(2*(b*x^(1/2)+a*x)^(1/2
)*a^(1/2)+2*x^(1/2)*a+b)/a^(1/2))*a^(9/2)*b-10*x^4*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^(7/2)*b^2+10*x^4*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^(7/2)*b^2-5*x^(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))*a^(5/2)*b^3+5*x^(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))*a^(5/2)*b^3+10
*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(9/2)*a^5-30*x^(7/2)*(b*x^(1/2)+a*x)^(3/2)*a^4+
10*x^(9/2)*(b*x^(1/2)+a*x)^(1/2)*a^5+10*a^4*(x^(1/2)*(b+x^(1/2)*a))^(3/2)*x^(7/2
)+20*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^4*a^4*b-16*x^(5/2)*(b*x^(1/2)+a*x)^(3/2)*a^
2*b^2-52*x^3*(b*x^(1/2)+a*x)^(3/2)*a^3*b+20*x^4*(b*x^(1/2)+a*x)^(1/2)*a^4*b+10*(
x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(7/2)*a^3*b^2+4*x^2*(b*x^(1/2)+a*x)^(3/2)*a*b^3+1
0*x^(7/2)*(b*x^(1/2)+a*x)^(1/2)*a^3*b^2-2*x^(3/2)*(b*x^(1/2)+a*x)^(3/2)*b^4)/(x^
(1/2)*(b+x^(1/2)*a))^(1/2)/b^5/x^(7/2)/(b+x^(1/2)*a)^2

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Maxima [A]  time = 1.51746, size = 61, normalized size = 0.57 \[ -\frac{4 \,{\left (16 \, a^{3} x^{\frac{3}{2}} + 8 \, a^{2} b x - 2 \, a b^{2} \sqrt{x} + b^{3}\right )}}{5 \, \sqrt{a \sqrt{x} + b} b^{4} x^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)),x, algorithm="maxima")

[Out]

-4/5*(16*a^3*x^(3/2) + 8*a^2*b*x - 2*a*b^2*sqrt(x) + b^3)/(sqrt(a*sqrt(x) + b)*b
^4*x^(5/4))

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Fricas [A]  time = 0.25508, size = 107, normalized size = 1. \[ \frac{4 \,{\left (8 \, a^{3} b x^{2} - 3 \, a b^{3} x -{\left (16 \, a^{4} x^{2} - 10 \, a^{2} b^{2} x - b^{4}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{5 \,{\left (a^{2} b^{4} x^{3} - b^{6} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)),x, algorithm="fricas")

[Out]

4/5*(8*a^3*b*x^2 - 3*a*b^3*x - (16*a^4*x^2 - 10*a^2*b^2*x - b^4)*sqrt(x))*sqrt(a
*x + b*sqrt(x))/(a^2*b^4*x^3 - b^6*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{3}{2}} \left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(b*x**(1/2)+a*x)**(3/2),x)

[Out]

Integral(1/(x**(3/2)*(a*x + b*sqrt(x))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(3/2)), x)