3.1679 \(\int \frac{1}{(a+\frac{b}{x})^2 x^{7/2}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{1}{b \sqrt{x} (a x+b)}-\frac{3}{b^2 \sqrt{x}} \]

[Out]

-3/(b^2*Sqrt[x]) + 1/(b*Sqrt[x]*(b + a*x)) - (3*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(5/2)

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Rubi [A]  time = 0.0191806, antiderivative size = 56, 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} \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}+\frac{1}{b \sqrt{x} (a x+b)}-\frac{3}{b^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^2*x^(7/2)),x]

[Out]

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

Mathematica [C]  time = 0.0044137, size = 25, normalized size = 0.45 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{a x}{b}\right )}{b^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^2*x^(7/2)),x]

[Out]

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

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Maple [A]  time = 0.011, size = 48, normalized size = 0.9 \begin{align*} -{\frac{a}{{b}^{2} \left ( ax+b \right ) }\sqrt{x}}-3\,{\frac{a}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{1}{{b}^{2}\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2/x^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74837, size = 323, normalized size = 5.77 \begin{align*} \left [\frac{3 \,{\left (a x^{2} + b x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 2 \,{\left (3 \, a x + 2 \, b\right )} \sqrt{x}}{2 \,{\left (a b^{2} x^{2} + b^{3} x\right )}}, \frac{3 \,{\left (a x^{2} + b x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) -{\left (3 \, a x + 2 \, b\right )} \sqrt{x}}{a b^{2} x^{2} + b^{3} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*(a*x^2 + b*x)*sqrt(-a/b)*log((a*x - 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)) - 2*(3*a*x + 2*b)*sqrt(x))/
(a*b^2*x^2 + b^3*x), (3*(a*x^2 + b*x)*sqrt(a/b)*arctan(b*sqrt(a/b)/(a*sqrt(x))) - (3*a*x + 2*b)*sqrt(x))/(a*b^
2*x^2 + b^3*x)]

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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)**2/x**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12187, size = 66, normalized size = 1.18 \begin{align*} -\frac{3 \, a \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} - \frac{3 \, a x + 2 \, b}{{\left (a x^{\frac{3}{2}} + b \sqrt{x}\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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