3.350 \(\int \frac{1}{x^3 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ \frac{15 b^2}{4 a^3 \sqrt{a+b x}}-\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 b}{4 a^2 x \sqrt{a+b x}}-\frac{1}{2 a x^2 \sqrt{a+b x}} \]

[Out]

(15*b^2)/(4*a^3*Sqrt[a + b*x]) - 1/(2*a*x^2*Sqrt[a + b*x]) + (5*b)/(4*a^2*x*Sqrt[a + b*x]) - (15*b^2*ArcTanh[S
qrt[a + b*x]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.0231892, antiderivative size = 85, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}+\frac{2}{a x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)^(3/2)),x]

[Out]

2/(a*x^2*Sqrt[a + b*x]) - (5*Sqrt[a + b*x])/(2*a^2*x^2) + (15*b*Sqrt[a + b*x])/(4*a^3*x) - (15*b^2*ArcTanh[Sqr
t[a + b*x]/Sqrt[a]])/(4*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^3 (a+b x)^{3/2}} \, dx &=\frac{2}{a x^2 \sqrt{a+b x}}+\frac{5 \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}-\frac{(15 b) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{4 a^2}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}+\frac{\left (15 b^2\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{8 a^3}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}+\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{4 a^3}\\ &=\frac{2}{a x^2 \sqrt{a+b x}}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}-\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0093782, size = 33, normalized size = 0.38 \[ \frac{2 b^2 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{b x}{a}+1\right )}{a^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 67, normalized size = 0.8 \begin{align*} 2\,{b}^{2} \left ({\frac{1}{{a}^{3}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ({\frac{7\, \left ( bx+a \right ) ^{3/2}}{8}}-{\frac{9\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{15}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }+{\frac{1}{\sqrt{bx+a}{a}^{3}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x+a)^(3/2),x)

[Out]

2*b^2*(1/a^3*((7/8*(b*x+a)^(3/2)-9/8*a*(b*x+a)^(1/2))/b^2/x^2-15/8*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))+1/a
^3/(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^3/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57, size = 420, normalized size = 4.83 \begin{align*} \left [\frac{15 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\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} + 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt{b x + a}}{8 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}, \frac{15 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt{b x + a}}{4 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(15*(b^3*x^3 + a*b^2*x^2)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(15*a*b^2*x^2 + 5*a^2*
b*x - 2*a^3)*sqrt(b*x + a))/(a^4*b*x^3 + a^5*x^2), 1/4*(15*(b^3*x^3 + a*b^2*x^2)*sqrt(-a)*arctan(sqrt(b*x + a)
*sqrt(-a)/a) + (15*a*b^2*x^2 + 5*a^2*b*x - 2*a^3)*sqrt(b*x + a))/(a^4*b*x^3 + a^5*x^2)]

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Sympy [A]  time = 7.50486, size = 107, normalized size = 1.23 \begin{align*} - \frac{1}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{5 \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{15 b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x+a)**(3/2),x)

[Out]

-1/(2*a*sqrt(b)*x**(5/2)*sqrt(a/(b*x) + 1)) + 5*sqrt(b)/(4*a**2*x**(3/2)*sqrt(a/(b*x) + 1)) + 15*b**(3/2)/(4*a
**3*sqrt(x)*sqrt(a/(b*x) + 1)) - 15*b**2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*a**(7/2))

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Giac [A]  time = 1.17616, size = 108, normalized size = 1.24 \begin{align*} \frac{15 \, b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{3}} + \frac{2 \, b^{2}}{\sqrt{b x + a} a^{3}} + \frac{7 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2} - 9 \, \sqrt{b x + a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

15/4*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + 2*b^2/(sqrt(b*x + a)*a^3) + 1/4*(7*(b*x + a)^(3/2)*b^
2 - 9*sqrt(b*x + a)*a*b^2)/(a^3*b^2*x^2)