3.366 \(\int \frac{1}{x^2 (-a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=81 \[ \frac{5 b}{a^3 \sqrt{b x-a}}-\frac{5 b}{3 a^2 (b x-a)^{3/2}}+\frac{5 b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{1}{a x (b x-a)^{3/2}} \]
[Out]
(-5*b)/(3*a^2*(-a + b*x)^(3/2)) + 1/(a*x*(-a + b*x)^(3/2)) + (5*b)/(a^3*Sqrt[-a + b*x]) + (5*b*ArcTan[Sqrt[-a
+ b*x]/Sqrt[a]])/a^(7/2)
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Rubi [A] time = 0.0234259, antiderivative size = 88, normalized size of antiderivative =
1.09, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}
Mathematica [C] time = 0.0124766, 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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Maple [A] time = 0.013, size = 68, normalized size = 0.8 \begin{align*} -{\frac{2\,b}{3\,{a}^{2}} \left ( bx-a \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{b}{{a}^{3}\sqrt{bx-a}}}+{\frac{1}{{a}^{3}x}\sqrt{bx-a}}+5\,{\frac{b}{{a}^{7/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) } \end{align*}
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)
________________________________________________________________________________________
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^2/(b*x-a)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.89575, size = 491, normalized size = 6.06 \begin{align*} \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 ] \end{align*}
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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Sympy [C] time = 7.7245, size = 2236, normalized size = 27.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**2/(b*x-a)**(5/2),x)
[Out]
Piecewise((-6*a**17*sqrt(-1 + b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33
/2)*b**3*x**4) + 46*a**16*b*x*sqrt(-1 + b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3
+ 6*a**(33/2)*b**3*x**4) + 15*I*a**16*b*x*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2
*x**3 + 6*a**(33/2)*b**3*x**4) - 30*I*a**16*b*x*log(sqrt(b)*sqrt(x)/sqrt(a))/(-6*a**(39/2)*x + 18*a**(37/2)*b*
x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*a**16*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-6*a**(
39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 70*a**15*b**2*x**2*sqrt(-1 +
b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 45*I*a**15*b
**2*x**2*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) +
90*I*a**15*b**2*x**2*log(sqrt(b)*sqrt(x)/sqrt(a))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x*
*3 + 6*a**(33/2)*b**3*x**4) - 90*a**15*b**2*x**2*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-6*a**(39/2)*x + 18*a**(37/2
)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*a**14*b**3*x**3*sqrt(-1 + b*x/a)/(-6*a**(39/2)
*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 45*I*a**14*b**3*x**3*log(b*x/a)/(
-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 90*I*a**14*b**3*x**3*
log(sqrt(b)*sqrt(x)/sqrt(a))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3
*x**4) + 90*a**14*b**3*x**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/
2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 15*I*a**13*b**4*x**4*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2
- 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*I*a**13*b**4*x**4*log(sqrt(b)*sqrt(x)/sqrt(a))/(-6*a**(
39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*a**13*b**4*x**4*asin(sqrt
(a)/(sqrt(b)*sqrt(x)))/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4)
, Abs(b*x)/Abs(a) > 1), (-6*I*a**17*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*
x**3 + 6*a**(33/2)*b**3*x**4) + 46*I*a**16*b*x*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(
35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 15*I*a**16*b*x*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 1
8*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*I*a**16*b*x*log(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*
a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 15*pi*a**16*b*x/(-6*a**(39/2)*x + 18*a**(
37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 70*I*a**15*b**2*x**2*sqrt(1 - b*x/a)/(-6*a**(
39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 45*I*a**15*b**2*x**2*log(b*x
/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*I*a**15*b**2*
x**2*log(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**
3*x**4) - 45*pi*a**15*b**2*x**2/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b
**3*x**4) + 30*I*a**14*b**3*x**3*sqrt(1 - b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**
3 + 6*a**(33/2)*b**3*x**4) + 45*I*a**14*b**3*x**3*log(b*x/a)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35
/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 90*I*a**14*b**3*x**3*log(sqrt(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a*
*(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 45*pi*a**14*b**3*x**3/(-6*a**(39/2)*x + 18*
a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 15*I*a**13*b**4*x**4*log(b*x/a)/(-6*a**(3
9/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*I*a**13*b**4*x**4*log(sqrt
(1 - b*x/a) + 1)/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 15*
pi*a**13*b**4*x**4/(-6*a**(39/2)*x + 18*a**(37/2)*b*x**2 - 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4), Tr
ue))
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Giac [A] time = 1.17017, size = 89, normalized size = 1.1 \begin{align*} \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} \end{align*}
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)