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

Optimal. Leaf size=79 \[ \frac{32 a \sqrt{a x+b \sqrt{x}}}{3 b^3 \sqrt{x}}-\frac{16 \sqrt{a x+b \sqrt{x}}}{3 b^2 x}+\frac{4}{b \sqrt{x} \sqrt{a x+b \sqrt{x}}} \]

[Out]

4/(b*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x]) - (16*Sqrt[b*Sqrt[x] + a*x])/(3*b^2*x) + (32
*a*Sqrt[b*Sqrt[x] + a*x])/(3*b^3*Sqrt[x])

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Rubi [A]  time = 0.199243, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{32 a \sqrt{a x+b \sqrt{x}}}{3 b^3 \sqrt{x}}-\frac{16 \sqrt{a x+b \sqrt{x}}}{3 b^2 x}+\frac{4}{b \sqrt{x} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

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

[Out]

4/(b*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x]) - (16*Sqrt[b*Sqrt[x] + a*x])/(3*b^2*x) + (32
*a*Sqrt[b*Sqrt[x] + a*x])/(3*b^3*Sqrt[x])

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Rubi in Sympy [A]  time = 18.3145, size = 70, normalized size = 0.89 \[ \frac{32 a \sqrt{a x + b \sqrt{x}}}{3 b^{3} \sqrt{x}} + \frac{4}{b \sqrt{x} \sqrt{a x + b \sqrt{x}}} - \frac{16 \sqrt{a x + b \sqrt{x}}}{3 b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32*a*sqrt(a*x + b*sqrt(x))/(3*b**3*sqrt(x)) + 4/(b*sqrt(x)*sqrt(a*x + b*sqrt(x))
) - 16*sqrt(a*x + b*sqrt(x))/(3*b**2*x)

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Mathematica [A]  time = 0.0431302, size = 57, normalized size = 0.72 \[ \frac{4 \sqrt{a x+b \sqrt{x}} \left (8 a^2 x+4 a b \sqrt{x}-b^2\right )}{3 b^3 x \left (a \sqrt{x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.018, size = 516, normalized size = 6.5 \[ -{\frac{1}{3\,{b}^{4}}\sqrt{b\sqrt{x}+ax} \left ( -3\,{x}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}b+3\,{x}^{7/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}b-6\,{x}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5/2}{b}^{2}+6\,{x}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5/2}{b}^{2}+6\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{7/2}{a}^{4}-24\,{x}^{5/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3}+6\,{x}^{7/2}\sqrt{b\sqrt{x}+ax}{a}^{4}-3\,{x}^{5/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{3/2}{b}^{3}+3\,{x}^{5/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{3/2}{b}^{3}+12\,{a}^{3} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}{x}^{5/2}+12\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{3}{a}^{3}b-44\,{x}^{2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{2}b+12\,{x}^{3}\sqrt{b\sqrt{x}+ax}{a}^{3}b+6\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{5/2}{a}^{2}{b}^{2}-16\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}a{b}^{2}+6\,{x}^{5/2}\sqrt{b\sqrt{x}+ax}{a}^{2}{b}^{2}+4\, \left ( b\sqrt{x}+ax \right ) ^{3/2}x{b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{5}{2}}} \left ( b+\sqrt{x}a \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b*x^(1/2)+a*x)^(1/2)*(-3*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^(7/2)*b+3*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^(7/2)*b-6*x^3*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^2+6*x^3*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^2+6*(x^(1/2)*(b+x^(1/2)*a
))^(1/2)*x^(7/2)*a^4-24*x^(5/2)*(b*x^(1/2)+a*x)^(3/2)*a^3+6*x^(7/2)*(b*x^(1/2)+a
*x)^(1/2)*a^4-3*x^(5/2)*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^(3/2)*b^3+3*x^(5/2)*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^(3/2)*b^3+12*a^3*(x^(1/2)*(b+x^(1/2)*a))^(3/2)*x^(5/2)+1
2*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^3*a^3*b-44*x^2*(b*x^(1/2)+a*x)^(3/2)*a^2*b+12*
x^3*(b*x^(1/2)+a*x)^(1/2)*a^3*b+6*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(5/2)*a^2*b^2-
16*x^(3/2)*(b*x^(1/2)+a*x)^(3/2)*a*b^2+6*x^(5/2)*(b*x^(1/2)+a*x)^(1/2)*a^2*b^2+4
*(b*x^(1/2)+a*x)^(3/2)*x*b^3)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/b^4/x^(5/2)/(b+x^(1/
2)*a)^2

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Maxima [A]  time = 1.49068, size = 49, normalized size = 0.62 \[ \frac{4 \,{\left (8 \, a^{2} x + 4 \, a b \sqrt{x} - b^{2}\right )}}{3 \, \sqrt{a \sqrt{x} + b} b^{3} x^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255438, size = 85, normalized size = 1.08 \[ -\frac{4 \,{\left (4 \, a^{2} b x - b^{3} -{\left (8 \, a^{3} x - 5 \, a b^{2}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{3 \,{\left (a^{2} b^{3} x^{2} - b^{5} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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