3.365 \(\int \frac{1}{x (-a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=60 \[ \frac{2}{a^2 \sqrt{b x-a}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{3 a (b x-a)^{3/2}} \]
[Out]
-2/(3*a*(-a + b*x)^(3/2)) + 2/(a^2*Sqrt[-a + b*x]) + (2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(5/2)
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Rubi [A] time = 0.0151089, antiderivative size = 60, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{51, 63, 205} \[ \frac{2}{a^2 \sqrt{b x-a}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{3 a (b x-a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/(x*(-a + b*x)^(5/2)),x]
[Out]
-2/(3*a*(-a + b*x)^(3/2)) + 2/(a^2*Sqrt[-a + b*x]) + (2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(5/2)
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 (-a+b x)^{5/2}} \, dx &=-\frac{2}{3 a (-a+b x)^{3/2}}-\frac{\int \frac{1}{x (-a+b x)^{3/2}} \, dx}{a}\\ &=-\frac{2}{3 a (-a+b x)^{3/2}}+\frac{2}{a^2 \sqrt{-a+b x}}+\frac{\int \frac{1}{x \sqrt{-a+b x}} \, dx}{a^2}\\ &=-\frac{2}{3 a (-a+b x)^{3/2}}+\frac{2}{a^2 \sqrt{-a+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{a^2 b}\\ &=-\frac{2}{3 a (-a+b x)^{3/2}}+\frac{2}{a^2 \sqrt{-a+b x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0123223, size = 35, normalized size = 0.58 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};1-\frac{b x}{a}\right )}{3 a (b x-a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x*(-a + b*x)^(5/2)),x]
[Out]
(-2*Hypergeometric2F1[-3/2, 1, -1/2, 1 - (b*x)/a])/(3*a*(-a + b*x)^(3/2))
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Maple [A] time = 0.009, size = 49, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,a} \left ( bx-a \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{1}{{a}^{5/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{{a}^{2}\sqrt{bx-a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(b*x-a)^(5/2),x)
[Out]
-2/3/a/(b*x-a)^(3/2)+2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(5/2)+2/a^2/(b*x-a)^(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/(b*x-a)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.58615, size = 408, normalized size = 6.8 \begin{align*} \left [-\frac{3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{-a} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) - 2 \,{\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt{b x - a}}{3 \,{\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}, \frac{2 \,{\left (3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt{b x - a}\right )}}{3 \,{\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b*x-a)^(5/2),x, algorithm="fricas")
[Out]
[-1/3*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-a)*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) - 2*(3*a*b*x - 4*a^2
)*sqrt(b*x - a))/(a^3*b^2*x^2 - 2*a^4*b*x + a^5), 2/3*(3*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(a)*arctan(sqrt(b*x - a
)/sqrt(a)) + (3*a*b*x - 4*a^2)*sqrt(b*x - a))/(a^3*b^2*x^2 - 2*a^4*b*x + a^5)]
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Sympy [C] time = 4.29591, size = 1952, normalized size = 32.53 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b*x-a)**(5/2),x)
[Out]
Piecewise((8*a**7*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*
x**3) + 3*I*a**7*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) -
6*I*a**7*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b
**3*x**3) + 6*a**7*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3
*a**(13/2)*b**3*x**3) - 14*a**6*b*x*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 +
3*a**(13/2)*b**3*x**3) - 9*I*a**6*b*x*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*
a**(13/2)*b**3*x**3) + 18*I*a**6*b*x*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2
)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*a**6*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)
*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*a**5*b**2*x**2*sqrt(-1 + b*x/a)/(-3*a**(19/2) + 9*a*
*(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*I*a**5*b**2*x**2*log(b*x/a)/(-3*a**(19/2) + 9
*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*I*a**5*b**2*x**2*log(sqrt(b)*sqrt(x)/sqrt
(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 18*a**5*b**2*x**2*asin
(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) -
3*I*a**4*b**3*x**3*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3
) + 6*I*a**4*b**3*x**3*log(sqrt(b)*sqrt(x)/sqrt(a))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 +
3*a**(13/2)*b**3*x**3) - 6*a**4*b**3*x**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*
a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3), Abs(b*x)/Abs(a) > 1), (8*I*a**7*sqrt(1 - b*x/a)/(-3*a**(19/2) +
9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*I*a**7*log(b*x/a)/(-3*a**(19/2) + 9*a**(1
7/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*I*a**7*log(sqrt(1 - b*x/a) + 1)/(-3*a**(19/2) +
9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*pi*a**7/(-3*a**(19/2) + 9*a**(17/2)*b*x -
9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 14*I*a**6*b*x*sqrt(1 - b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*
x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 9*I*a**6*b*x*log(b*x/a)/(-3*a**(19/2) + 9*a**(17/2)*b*x -
9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 18*I*a**6*b*x*log(sqrt(1 - b*x/a) + 1)/(-3*a**(19/2) + 9*a**
(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 9*pi*a**6*b*x/(-3*a**(19/2) + 9*a**(17/2)*b*x -
9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*I*a**5*b**2*x**2*sqrt(1 - b*x/a)/(-3*a**(19/2) + 9*a**(17/2
)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*I*a**5*b**2*x**2*log(b*x/a)/(-3*a**(19/2) + 9*a**(1
7/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*I*a**5*b**2*x**2*log(sqrt(1 - b*x/a) + 1)/(-3*a
**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*pi*a**5*b**2*x**2/(-3*a**(19/2
) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 3*I*a**4*b**3*x**3*log(b*x/a)/(-3*a**(1
9/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*I*a**4*b**3*x**3*log(sqrt(1 - b*x/
a) + 1)/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 3*pi*a**4*b**3*x**3
/(-3*a**(19/2) + 9*a**(17/2)*b*x - 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3), True))
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Giac [A] time = 1.19288, size = 57, normalized size = 0.95 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{5}{2}}} + \frac{2 \,{\left (3 \, b x - 4 \, a\right )}}{3 \,{\left (b x - a\right )}^{\frac{3}{2}} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b*x-a)^(5/2),x, algorithm="giac")
[Out]
2*arctan(sqrt(b*x - a)/sqrt(a))/a^(5/2) + 2/3*(3*b*x - 4*a)/((b*x - a)^(3/2)*a^2)