3.1687 \(\int \frac{1}{(a+\frac{b}{x})^3 x^{7/2}} \, dx\)

Optimal. Leaf size=70 \[ \frac{3 \sqrt{x}}{4 b^2 (a x+b)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{\sqrt{x}}{2 b (a x+b)^2} \]

[Out]

Sqrt[x]/(2*b*(b + a*x)^2) + (3*Sqrt[x])/(4*b^2*(b + a*x)) + (3*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*Sqrt[a]*b
^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0233464, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac{3 \sqrt{x}}{4 b^2 (a x+b)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}+\frac{\sqrt{x}}{2 b (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^(7/2)),x]

[Out]

Sqrt[x]/(2*b*(b + a*x)^2) + (3*Sqrt[x])/(4*b^2*(b + a*x)) + (3*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(4*Sqrt[a]*b
^(5/2))

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{7/2}} \, dx &=\int \frac{1}{\sqrt{x} (b+a x)^3} \, dx\\ &=\frac{\sqrt{x}}{2 b (b+a x)^2}+\frac{3 \int \frac{1}{\sqrt{x} (b+a x)^2} \, dx}{4 b}\\ &=\frac{\sqrt{x}}{2 b (b+a x)^2}+\frac{3 \sqrt{x}}{4 b^2 (b+a x)}+\frac{3 \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{8 b^2}\\ &=\frac{\sqrt{x}}{2 b (b+a x)^2}+\frac{3 \sqrt{x}}{4 b^2 (b+a x)}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{4 b^2}\\ &=\frac{\sqrt{x}}{2 b (b+a x)^2}+\frac{3 \sqrt{x}}{4 b^2 (b+a x)}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{a} b^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0051713, size = 25, normalized size = 0.36 \[ \frac{2 \sqrt{x} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};-\frac{a x}{b}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^(7/2)),x]

[Out]

(2*Sqrt[x]*Hypergeometric2F1[1/2, 3, 3/2, -((a*x)/b)])/b^3

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 53, normalized size = 0.8 \begin{align*}{\frac{1}{2\,b \left ( ax+b \right ) ^{2}}\sqrt{x}}+{\frac{3}{4\,{b}^{2} \left ( ax+b \right ) }\sqrt{x}}+{\frac{3}{4\,{b}^{2}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^3/x^(7/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.75059, size = 423, normalized size = 6.04 \begin{align*} \left [-\frac{3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{-a b} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right ) - 2 \,{\left (3 \, a^{2} b x + 5 \, a b^{2}\right )} \sqrt{x}}{8 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{2} b^{4} x + a b^{5}\right )}}, -\frac{3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right ) -{\left (3 \, a^{2} b x + 5 \, a b^{2}\right )} \sqrt{x}}{4 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{2} b^{4} x + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^(7/2),x, algorithm="fricas")

[Out]

[-1/8*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-a*b)*log((a*x - b - 2*sqrt(-a*b)*sqrt(x))/(a*x + b)) - 2*(3*a^2*b*x +
5*a*b^2)*sqrt(x))/(a^3*b^3*x^2 + 2*a^2*b^4*x + a*b^5), -1/4*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(a*b)*arctan(sqr
t(a*b)/(a*sqrt(x))) - (3*a^2*b*x + 5*a*b^2)*sqrt(x))/(a^3*b^3*x^2 + 2*a^2*b^4*x + a*b^5)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**3/x**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.09362, size = 63, normalized size = 0.9 \begin{align*} \frac{3 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{2}} + \frac{3 \, a x^{\frac{3}{2}} + 5 \, b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^(7/2),x, algorithm="giac")

[Out]

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