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

Optimal. Leaf size=60 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}}-\frac{4 \sqrt{x}}{a \sqrt{a x+b \sqrt{x}}} \]

[Out]

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

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Rubi [A]  time = 0.140339, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}}-\frac{4 \sqrt{x}}{a \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.2549, size = 53, normalized size = 0.88 \[ - \frac{4 \sqrt{x}}{a \sqrt{a x + b \sqrt{x}}} + \frac{4 \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0732259, size = 72, normalized size = 1.2 \[ \frac{2 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{a^{3/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{a \left (a \sqrt{x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 238, normalized size = 4. \[ 2\,{\frac{\sqrt{b\sqrt{x}+ax}}{{a}^{3/2}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }b \left ( b+\sqrt{x}a \right ) ^{2}} \left ( x\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{2}b-2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }x{a}^{5/2}+2\,\sqrt{x}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) a{b}^{2}+2\,{a}^{3/2} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}-4\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{x}{a}^{3/2}b+\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){b}^{3}-2\,\sqrt{a}{b}^{2}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*x^(1/2)+a*x)^(1/2)/a^(3/2)*(x*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^2*b-2*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x*a^(5/2)+2*x^(
1/2)*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*b
^2+2*a^(3/2)*(x^(1/2)*(b+x^(1/2)*a))^(3/2)-4*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(1/
2)*a^(3/2)*b+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^3-2*a^(1/2)*b^2*(x^(1/2)*(b+x^(1/2)*a))^(1/2))/(x^(1/2)*(b+x^(1/2)*a))^(1
/2)/b/(b+x^(1/2)*a)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(sqrt(x)/(a*x + b*sqrt(x))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError