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

Optimal. Leaf size=54 \[ \frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x} \]

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Rubi [A]  time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \begin {gather*} \frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2014

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

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}-\frac {(2 a) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{3 b}\\ &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {8 a \sqrt {b \sqrt {x}+a x}}{3 b^2 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 37, normalized size = 0.69 \begin {gather*} \frac {4 \left (2 a \sqrt {x}-b\right ) \sqrt {a x+b \sqrt {x}}}{3 b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.16, size = 37, normalized size = 0.69 \begin {gather*} \frac {4 \left (2 a \sqrt {x}-b\right ) \sqrt {a x+b \sqrt {x}}}{3 b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 29, normalized size = 0.54 \begin {gather*} \frac {4 \, \sqrt {a x + b \sqrt {x}} {\left (2 \, a \sqrt {x} - b\right )}}{3 \, b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 53, normalized size = 0.98 \begin {gather*} \frac {4 \, {\left (3 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )}}{3 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.06, size = 194, normalized size = 3.59 \begin {gather*} -\frac {\sqrt {a x +b \sqrt {x}}\, \left (-3 a^{2} b \,x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+3 a^{2} b \,x^{\frac {5}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} x^{\frac {5}{2}}+6 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} x^{\frac {5}{2}}-12 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} x^{\frac {3}{2}}+4 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b x \right )}{3 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{3} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b \sqrt {x}} x^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^{3/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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