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

Optimal. Leaf size=54 \[ \frac{2}{a^2 \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{3 a (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0152049, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ \frac{2}{a^2 \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2}{3 a (a+b x)^{3/2}} \]

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*ArcTanh[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 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 (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 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0058095, size = 32, normalized size = 0.59 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b x}{a}+1\right )}{3 a (a+b x)^{3/2}} \]

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 = 43, normalized size = 0.8 \begin{align*}{\frac{2}{3\,a} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{1}{{a}^{5/2}}{\it Artanh} \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*arctanh((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 [B]  time = 1.69967, size = 409, normalized size = 7.57 \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}}{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)*s
qrt(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)/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 [B]  time = 3.71646, size = 697, normalized size = 12.91 \begin{align*} \frac{8 a^{7} \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{3 a^{7} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{6 a^{7} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{14 a^{6} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{9 a^{6} b x \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{18 a^{6} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{6 a^{5} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{9 a^{5} b^{2} x^{2} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{18 a^{5} b^{2} x^{2} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} + \frac{3 a^{4} b^{3} x^{3} \log{\left (\frac{b x}{a} \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} - \frac{6 a^{4} b^{3} x^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{3 a^{\frac{19}{2}} + 9 a^{\frac{17}{2}} b x + 9 a^{\frac{15}{2}} b^{2} x^{2} + 3 a^{\frac{13}{2}} b^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*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*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) + 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*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*a**6*b*x*log(s
qrt(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) + 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*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*
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) + 3*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*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)

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Giac [A]  time = 1.19167, size = 61, normalized size = 1.13 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{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))/(sqrt(-a)*a^2) + 2/3*(3*b*x + 4*a)/((b*x + a)^(3/2)*a^2)