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

Optimal. Leaf size=25 \[ \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2000} \[ \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sqrt[x] + a*x)^(-3/2),x]

[Out]

(4*Sqrt[x])/(b*Sqrt[b*Sqrt[x] + a*x])

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4 \sqrt {x}}{b \sqrt {b \sqrt {x}+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \[ \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sqrt[x] + a*x)^(-3/2),x]

[Out]

(4*Sqrt[x])/(b*Sqrt[b*Sqrt[x] + a*x])

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fricas [A]  time = 0.88, size = 36, normalized size = 1.44 \[ \frac {4 \, \sqrt {a x + b \sqrt {x}} {\left (a \sqrt {x} - b\right )}}{a^{2} b x - b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 34, normalized size = 1.36 \[ \frac {4}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/((sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + b)*sqrt(a))

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maple [C]  time = 0.06, size = 404, normalized size = 16.16 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (-a^{2} b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+a^{2} b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2 a \,b^{2} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 a \,b^{2} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} x +2 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} x +4 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b \sqrt {x}+4 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b \sqrt {x}+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{2}+2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}\, b^{2}-4 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.43, size = 40, normalized size = 1.60 \[ -\frac {4\,x\,\left (\frac {b}{a\,\sqrt {x}}+1\right )}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}\,\left (\sqrt {\frac {b}{a\,\sqrt {x}}+1}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b*x^(1/2))^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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