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

Optimal. Leaf size=70 \[ \frac{5 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a-b x)} \]

[Out]

-5/(3*a^2*x^(3/2)) - (5*b)/(a^3*Sqrt[x]) + 1/(a*x^(3/2)*(a - b*x)) + (5*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt
[a]])/a^(7/2)

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Rubi [A]  time = 0.0224064, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 208} \[ \frac{5 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a-b x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(-a + b*x)^2),x]

[Out]

-5/(3*a^2*x^(3/2)) - (5*b)/(a^3*Sqrt[x]) + 1/(a*x^(3/2)*(a - b*x)) + (5*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt
[a]])/a^(7/2)

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 208

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

Rubi steps

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

Mathematica [C]  time = 0.0058593, size = 26, normalized size = 0.37 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{b x}{a}\right )}{3 a^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(-a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 60, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-4\,{\frac{b}{{a}^{3}\sqrt{x}}}-2\,{\frac{{b}^{2}}{{a}^{3}} \left ( 1/2\,{\frac{\sqrt{x}}{bx-a}}-5/2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/a^2/x^(3/2)-4*b/a^3/x^(1/2)-2*b^2/a^3*(1/2*x^(1/2)/(b*x-a)-5/2/(a*b)^(1/2)*arctanh(b*x^(1/2)/(a*b)^(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/x^(5/2)/(b*x-a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55767, size = 402, normalized size = 5.74 \begin{align*} \left [\frac{15 \,{\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt{\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) - 2 \,{\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt{x}}{6 \,{\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}, -\frac{15 \,{\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*(b^2*x^3 - a*b*x^2)*sqrt(b/a)*log((b*x + 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) - 2*(15*b^2*x^2 - 10*a
*b*x - 2*a^2)*sqrt(x))/(a^3*b*x^3 - a^4*x^2), -1/3*(15*(b^2*x^3 - a*b*x^2)*sqrt(-b/a)*arctan(a*sqrt(-b/a)/(b*s
qrt(x))) + (15*b^2*x^2 - 10*a*b*x - 2*a^2)*sqrt(x))/(a^3*b*x^3 - a^4*x^2)]

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Sympy [A]  time = 139.005, size = 471, normalized size = 6.73 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b^{2} x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{2}{3 a^{2} x^{\frac{3}{2}}} & \text{for}\: b = 0 \\- \frac{4 a^{\frac{5}{2}} \sqrt{\frac{1}{b}}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{20 a^{\frac{3}{2}} b x \sqrt{\frac{1}{b}}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{30 \sqrt{a} b^{2} x^{2} \sqrt{\frac{1}{b}}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{15 a b x^{\frac{3}{2}} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{15 a b x^{\frac{3}{2}} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{15 b^{2} x^{\frac{5}{2}} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{15 b^{2} x^{\frac{5}{2}} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b**2*x**(7/2)), Eq(a, 0)), (-2/(3*a**2*x**(3/2)), Eq(b,
0)), (-4*a**(5/2)*sqrt(1/b)/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)) - 20*a**(3/2)*b*
x*sqrt(1/b)/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)) + 30*sqrt(a)*b**2*x**2*sqrt(1/b)
/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)) - 15*a*b*x**(3/2)*log(-sqrt(a)*sqrt(1/b) +
sqrt(x))/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)) + 15*a*b*x**(3/2)*log(sqrt(a)*sqrt(
1/b) + sqrt(x))/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)) + 15*b**2*x**(5/2)*log(-sqrt
(a)*sqrt(1/b) + sqrt(x))/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)) - 15*b**2*x**(5/2)*
log(sqrt(a)*sqrt(1/b) + sqrt(x))/(6*a**(9/2)*x**(3/2)*sqrt(1/b) - 6*a**(7/2)*b*x**(5/2)*sqrt(1/b)), True))

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Giac [A]  time = 1.19526, size = 82, normalized size = 1.17 \begin{align*} -\frac{5 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a^{3}} - \frac{b^{2} \sqrt{x}}{{\left (b x - a\right )} a^{3}} - \frac{2 \,{\left (6 \, b x + a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*b^2*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^3) - b^2*sqrt(x)/((b*x - a)*a^3) - 2/3*(6*b*x + a)/(a^3*x^(3
/2))