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

Optimal. Leaf size=81 \[ \frac {5 b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 b}{a^3 \sqrt {b x-a}}-\frac {5 b}{3 a^2 (b x-a)^{3/2}}+\frac {1}{a x (b x-a)^{3/2}} \]

[Out]

-5/3*b/a^2/(b*x-a)^(3/2)+1/a/x/(b*x-a)^(3/2)+5*b*arctan((b*x-a)^(1/2)/a^(1/2))/a^(7/2)+5*b/a^3/(b*x-a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {51, 63, 205} \[ \frac {5 \sqrt {b x-a}}{a^3 x}+\frac {10}{3 a^2 x \sqrt {b x-a}}+\frac {5 b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {2}{3 a x (b x-a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(-a + b*x)^(5/2)),x]

[Out]

-2/(3*a*x*(-a + b*x)^(3/2)) + 10/(3*a^2*x*Sqrt[-a + b*x]) + (5*Sqrt[-a + b*x])/(a^3*x) + (5*b*ArcTan[Sqrt[-a +
b*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 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}{x^2 (-a+b x)^{5/2}} \, dx &=-\frac {2}{3 a x (-a+b x)^{3/2}}-\frac {5 \int \frac {1}{x^2 (-a+b x)^{3/2}} \, dx}{3 a}\\ &=-\frac {2}{3 a x (-a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {-a+b x}}+\frac {5 \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{a^2}\\ &=-\frac {2}{3 a x (-a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {-a+b x}}+\frac {5 \sqrt {-a+b x}}{a^3 x}+\frac {(5 b) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{2 a^3}\\ &=-\frac {2}{3 a x (-a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {-a+b x}}+\frac {5 \sqrt {-a+b x}}{a^3 x}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{a^3}\\ &=-\frac {2}{3 a x (-a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {-a+b x}}+\frac {5 \sqrt {-a+b x}}{a^3 x}+\frac {5 b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 36, normalized size = 0.44 \[ -\frac {2 b \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};1-\frac {b x}{a}\right )}{3 a^2 (b x-a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(-a + b*x)^(5/2)),x]

[Out]

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

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fricas [A]  time = 0.48, size = 226, normalized size = 2.79 \[ \left [-\frac {15 \, {\left (b^{3} x^{3} - 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (15 \, a b^{2} x^{2} - 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt {b x - a}}{6 \, {\left (a^{4} b^{2} x^{3} - 2 \, a^{5} b x^{2} + a^{6} x\right )}}, \frac {15 \, {\left (b^{3} x^{3} - 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (15 \, a b^{2} x^{2} - 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt {b x - a}}{3 \, {\left (a^{4} b^{2} x^{3} - 2 \, a^{5} b x^{2} + a^{6} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(15*(b^3*x^3 - 2*a*b^2*x^2 + a^2*b*x)*sqrt(-a)*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) - 2*(15*a*b
^2*x^2 - 20*a^2*b*x + 3*a^3)*sqrt(b*x - a))/(a^4*b^2*x^3 - 2*a^5*b*x^2 + a^6*x), 1/3*(15*(b^3*x^3 - 2*a*b^2*x^
2 + a^2*b*x)*sqrt(a)*arctan(sqrt(b*x - a)/sqrt(a)) + (15*a*b^2*x^2 - 20*a^2*b*x + 3*a^3)*sqrt(b*x - a))/(a^4*b
^2*x^3 - 2*a^5*b*x^2 + a^6*x)]

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giac [A]  time = 1.00, size = 66, normalized size = 0.81 \[ \frac {5 \, b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}} + \frac {2 \, {\left (6 \, {\left (b x - a\right )} b - a b\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{3}} + \frac {\sqrt {b x - a}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*b*arctan(sqrt(b*x - a)/sqrt(a))/a^(7/2) + 2/3*(6*(b*x - a)*b - a*b)/((b*x - a)^(3/2)*a^3) + sqrt(b*x - a)/(a
^3*x)

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maple [A]  time = 0.01, size = 68, normalized size = 0.84 \[ -\frac {2 b}{3 \left (b x -a \right )^{\frac {3}{2}} a^{2}}+\frac {5 b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}}+\frac {4 b}{\sqrt {b x -a}\, a^{3}}+\frac {\sqrt {b x -a}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*b/a^2/(b*x-a)^(3/2)+4*b/a^3/(b*x-a)^(1/2)+1/a^3*(b*x-a)^(1/2)/x+5*b*arctan((b*x-a)^(1/2)/a^(1/2))/a^(7/2)

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maxima [A]  time = 2.93, size = 82, normalized size = 1.01 \[ \frac {15 \, {\left (b x - a\right )}^{2} b + 10 \, {\left (b x - a\right )} a b - 2 \, a^{2} b}{3 \, {\left ({\left (b x - a\right )}^{\frac {5}{2}} a^{3} + {\left (b x - a\right )}^{\frac {3}{2}} a^{4}\right )}} + \frac {5 \, b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(15*(b*x - a)^2*b + 10*(b*x - a)*a*b - 2*a^2*b)/((b*x - a)^(5/2)*a^3 + (b*x - a)^(3/2)*a^4) + 5*b*arctan(s
qrt(b*x - a)/sqrt(a))/a^(7/2)

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mupad [B]  time = 0.12, size = 70, normalized size = 0.86 \[ \frac {1}{a\,x\,{\left (b\,x-a\right )}^{3/2}}-\frac {20\,b}{3\,a^2\,{\left (b\,x-a\right )}^{3/2}}+\frac {5\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5\,b^2\,x}{a^3\,{\left (b\,x-a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(b*x - a)^(5/2)),x)

[Out]

1/(a*x*(b*x - a)^(3/2)) - (20*b)/(3*a^2*(b*x - a)^(3/2)) + (5*b*atan((b*x - a)^(1/2)/a^(1/2)))/a^(7/2) + (5*b^
2*x)/(a^3*(b*x - a)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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