3.122 \(\int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=112 \[ \frac{64 a^3 \sqrt{a x+b \sqrt{x}}}{35 b^4 \sqrt{x}}-\frac{32 a^2 \sqrt{a x+b \sqrt{x}}}{35 b^3 x}+\frac{24 a \sqrt{a x+b \sqrt{x}}}{35 b^2 x^{3/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{7 b x^2} \]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) + (24*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^2*x^(3/2)) - (32*a^2*Sqrt[b*Sqrt[x]
+ a*x])/(35*b^3*x) + (64*a^3*Sqrt[b*Sqrt[x] + a*x])/(35*b^4*Sqrt[x])

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Rubi [A]  time = 0.153552, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac{64 a^3 \sqrt{a x+b \sqrt{x}}}{35 b^4 \sqrt{x}}-\frac{32 a^2 \sqrt{a x+b \sqrt{x}}}{35 b^3 x}+\frac{24 a \sqrt{a x+b \sqrt{x}}}{35 b^2 x^{3/2}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{7 b x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) + (24*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^2*x^(3/2)) - (32*a^2*Sqrt[b*Sqrt[x]
+ a*x])/(35*b^3*x) + (64*a^3*Sqrt[b*Sqrt[x] + a*x])/(35*b^4*Sqrt[x])

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])

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])

Rubi steps

\begin{align*} \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{7 b x^2}-\frac{(6 a) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{7 b}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{7 b x^2}+\frac{24 a \sqrt{b \sqrt{x}+a x}}{35 b^2 x^{3/2}}+\frac{\left (24 a^2\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{35 b^2}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{7 b x^2}+\frac{24 a \sqrt{b \sqrt{x}+a x}}{35 b^2 x^{3/2}}-\frac{32 a^2 \sqrt{b \sqrt{x}+a x}}{35 b^3 x}-\frac{\left (16 a^3\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{35 b^3}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{7 b x^2}+\frac{24 a \sqrt{b \sqrt{x}+a x}}{35 b^2 x^{3/2}}-\frac{32 a^2 \sqrt{b \sqrt{x}+a x}}{35 b^3 x}+\frac{64 a^3 \sqrt{b \sqrt{x}+a x}}{35 b^4 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0499462, size = 59, normalized size = 0.53 \[ \frac{4 \sqrt{a x+b \sqrt{x}} \left (-8 a^2 b x+16 a^3 x^{3/2}+6 a b^2 \sqrt{x}-5 b^3\right )}{35 b^4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(4*Sqrt[b*Sqrt[x] + a*x]*(-5*b^3 + 6*a*b^2*Sqrt[x] - 8*a^2*b*x + 16*a^3*x^(3/2)))/(35*b^4*x^2)

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Maple [C]  time = 0.008, size = 240, normalized size = 2.1 \begin{align*}{\frac{1}{35\,{b}^{5}}\sqrt{b\sqrt{x}+ax} \left ( 140\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}{x}^{7/2}-70\,\sqrt{b\sqrt{x}+ax}{a}^{9/2}{x}^{9/2}-35\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{9/2}{a}^{4}b-70\,{a}^{9/2}{x}^{9/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }+35\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{9/2}{a}^{4}b+44\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{5/2}{b}^{2}-76\,{a}^{5/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{3}-20\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}{x}^{2}{b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}{x}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x)

[Out]

1/35*(b*x^(1/2)+a*x)^(1/2)*(140*(b*x^(1/2)+a*x)^(3/2)*a^(7/2)*x^(7/2)-70*(b*x^(1/2)+a*x)^(1/2)*a^(9/2)*x^(9/2)
-35*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*x^(9/2)*a^4*b-70*a^(9/2)*x^(9/2)*(x^(1/2)*
(b+a*x^(1/2)))^(1/2)+35*ln(1/2*(2*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2)+b)/a^(1/2))*x^(9/2)*a^4*b+
44*(b*x^(1/2)+a*x)^(3/2)*a^(3/2)*x^(5/2)*b^2-76*a^(5/2)*(b*x^(1/2)+a*x)^(3/2)*b*x^3-20*(b*x^(1/2)+a*x)^(3/2)*a
^(1/2)*x^2*b^3)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/b^5/a^(1/2)/x^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b \sqrt{x}} x^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a*x + b*sqrt(x))*x^(5/2)), x)

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Fricas [A]  time = 2.36806, size = 123, normalized size = 1.1 \begin{align*} -\frac{4 \,{\left (8 \, a^{2} b x + 5 \, b^{3} - 2 \,{\left (8 \, a^{3} x + 3 \, a b^{2}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{35 \, b^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="fricas")

[Out]

-4/35*(8*a^2*b*x + 5*b^3 - 2*(8*a^3*x + 3*a*b^2)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(b^4*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{5}{2}} \sqrt{a x + b \sqrt{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(b*x**(1/2)+a*x)**(1/2),x)

[Out]

Integral(1/(x**(5/2)*sqrt(a*x + b*sqrt(x))), x)

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Giac [A]  time = 1.22808, size = 155, normalized size = 1.38 \begin{align*} \frac{4 \,{\left (70 \, a^{\frac{3}{2}}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3} + 84 \, a b{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{2} + 35 \, \sqrt{a} b^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + 5 \, b^{3}\right )}}{35 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="giac")

[Out]

4/35*(70*a^(3/2)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 84*a*b*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))
)^2 + 35*sqrt(a)*b^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + 5*b^3)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x
)))^7