3.2.13 \(\int \frac {1}{(b \sqrt {x}+a x)^{3/2}} \, dx\) [113]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {2025} \begin {gather*} \frac {4 \sqrt {x}}{b \sqrt {a x+b \sqrt {x}}} \end {gather*}

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 2025

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 = 0.12, size = 31, normalized size = 1.24 \begin {gather*} \frac {4 \sqrt {b \sqrt {x}+a x}}{b \left (b+a \sqrt {x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Sqrt[x] + a*x)^(-3/2),x]

[Out]

(4*Sqrt[b*Sqrt[x] + a*x])/(b*(b + a*Sqrt[x]))

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.38, size = 404, normalized size = 16.16

method	result	size
derivativedivides	\(-\frac {2}{a \sqrt {b \sqrt {x}+a x}}+\frac {2 b +4 a \sqrt {x}}{b a \sqrt {b \sqrt {x}+a x}}\)	\(45\)
default	\(\frac {\sqrt {b \sqrt {x}+a x}\, \left (2 \sqrt {b \sqrt {x}+a x}\, x \,a^{\frac {5}{2}}+2 x \,a^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}+x \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b -x \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{2} b +4 \sqrt {b \sqrt {x}+a x}\, \sqrt {x}\, a^{\frac {3}{2}} b +4 \sqrt {x}\, a^{\frac {3}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b +2 \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{2}-2 \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{2}-4 a^{\frac {3}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}\, b^{2}+2 \sqrt {a}\, \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{2}+\ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{3}-\ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{3}\right )}{\sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{2} \left (a \sqrt {x}+b \right )^{2} \sqrt {a}}\)	\(404\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^(1/2)+a*x)^(1/2)*(2*(b*x^(1/2)+a*x)^(1/2)*x*a^(5/2)+2*x*a^(5/2)*(x^(1/2)*(a*x^(1/2)+b))^(1/2)+x*ln(1/2*(2
*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*a^2*b-x*ln(1/2*(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(
1/2)*a^(1/2)+b)/a^(1/2))*a^2*b+4*(b*x^(1/2)+a*x)^(1/2)*x^(1/2)*a^(3/2)*b+4*x^(1/2)*a^(3/2)*(x^(1/2)*(a*x^(1/2)
+b))^(1/2)*b+2*x^(1/2)*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*a*b^2-2*x^(1/2)*ln(1/2*
(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))*a*b^2-4*a^(3/2)*(x^(1/2)*(a*x^(1/2)+b))^(3/2)
+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)*b^2+2*a^(1/2)*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*b^2+ln(1/2*(2*a*x^(1/2)+2*(b*x^(1
/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*b^3-ln(1/2*(2*a*x^(1/2)+2*(x^(1/2)*(a*x^(1/2)+b))^(1/2)*a^(1/2)+b)/a^(1/2))
*b^3)/(x^(1/2)*(a*x^(1/2)+b))^(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 6.19, size = 36, normalized size = 1.44 \begin {gather*} \frac {4 \, \sqrt {a x + b \sqrt {x}} {\left (a \sqrt {x} - b\right )}}{a^{2} b x - b^{3}} \end {gather*}

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

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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Giac [A]
time = 0.59, size = 34, normalized size = 1.36 \begin {gather*} \frac {4}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} \sqrt {a}} \end {gather*}

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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Mupad [B]
time = 5.43, size = 40, normalized size = 1.60 \begin {gather*} -\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 )} \end {gather*}

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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