3.1691 \(\int \sqrt{a+\frac{b}{x}} x^3 \, dx\)

Optimal. Leaf size=117 \[ -\frac{5 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^2}+\frac{5 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^3}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{7/2}}+\frac{b x^3 \sqrt{a+\frac{b}{x}}}{24 a}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x}} \]

[Out]

(5*b^3*Sqrt[a + b/x]*x)/(64*a^3) - (5*b^2*Sqrt[a + b/x]*x^2)/(96*a^2) + (b*Sqrt[a + b/x]*x^3)/(24*a) + (Sqrt[a
+ b/x]*x^4)/4 - (5*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(64*a^(7/2))

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Rubi [A]  time = 0.0549252, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ -\frac{5 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^2}+\frac{5 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^3}-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{7/2}}+\frac{b x^3 \sqrt{a+\frac{b}{x}}}{24 a}+\frac{1}{4} x^4 \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]*x^3,x]

[Out]

(5*b^3*Sqrt[a + b/x]*x)/(64*a^3) - (5*b^2*Sqrt[a + b/x]*x^2)/(96*a^2) + (b*Sqrt[a + b/x]*x^3)/(24*a) + (Sqrt[a
+ b/x]*x^4)/4 - (5*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(64*a^(7/2))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 \sqrt{a+\frac{b}{x}} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} \sqrt{a+\frac{b}{x}} x^4-\frac{1}{8} b \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b \sqrt{a+\frac{b}{x}} x^3}{24 a}+\frac{1}{4} \sqrt{a+\frac{b}{x}} x^4+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{48 a}\\ &=-\frac{5 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^2}+\frac{b \sqrt{a+\frac{b}{x}} x^3}{24 a}+\frac{1}{4} \sqrt{a+\frac{b}{x}} x^4-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{64 a^2}\\ &=\frac{5 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^3}-\frac{5 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^2}+\frac{b \sqrt{a+\frac{b}{x}} x^3}{24 a}+\frac{1}{4} \sqrt{a+\frac{b}{x}} x^4+\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{128 a^3}\\ &=\frac{5 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^3}-\frac{5 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^2}+\frac{b \sqrt{a+\frac{b}{x}} x^3}{24 a}+\frac{1}{4} \sqrt{a+\frac{b}{x}} x^4+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{64 a^3}\\ &=\frac{5 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^3}-\frac{5 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^2}+\frac{b \sqrt{a+\frac{b}{x}} x^3}{24 a}+\frac{1}{4} \sqrt{a+\frac{b}{x}} x^4-\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0148337, size = 39, normalized size = 0.33 \[ \frac{2 b^4 \left (a+\frac{b}{x}\right )^{3/2} \, _2F_1\left (\frac{3}{2},5;\frac{5}{2};\frac{b}{a x}+1\right )}{3 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]*x^3,x]

[Out]

(2*b^4*(a + b/x)^(3/2)*Hypergeometric2F1[3/2, 5, 5/2, 1 + b/(a*x)])/(3*a^5)

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Maple [A]  time = 0.011, size = 135, normalized size = 1.2 \begin{align*} -{\frac{x}{384}\sqrt{{\frac{ax+b}{x}}} \left ( -96\,x \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}+80\,{a}^{5/2} \left ( a{x}^{2}+bx \right ) ^{3/2}b-60\,{a}^{5/2}\sqrt{a{x}^{2}+bx}x{b}^{2}-30\,{a}^{3/2}\sqrt{a{x}^{2}+bx}{b}^{3}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{4} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*((a*x+b)/x)^(1/2)*x*(-96*x*(a*x^2+b*x)^(3/2)*a^(7/2)+80*a^(5/2)*(a*x^2+b*x)^(3/2)*b-60*a^(5/2)*(a*x^2+b
*x)^(1/2)*x*b^2-30*a^(3/2)*(a*x^2+b*x)^(1/2)*b^3+15*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^
4)/((a*x+b)*x)^(1/2)/a^(9/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78313, size = 406, normalized size = 3.47 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{4} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (48 \, a^{4} x^{4} + 8 \, a^{3} b x^{3} - 10 \, a^{2} b^{2} x^{2} + 15 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{384 \, a^{4}}, \frac{15 \, \sqrt{-a} b^{4} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (48 \, a^{4} x^{4} + 8 \, a^{3} b x^{3} - 10 \, a^{2} b^{2} x^{2} + 15 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{192 \, a^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/384*(15*sqrt(a)*b^4*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(48*a^4*x^4 + 8*a^3*b*x^3 - 10*a^2*b
^2*x^2 + 15*a*b^3*x)*sqrt((a*x + b)/x))/a^4, 1/192*(15*sqrt(-a)*b^4*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (48
*a^4*x^4 + 8*a^3*b*x^3 - 10*a^2*b^2*x^2 + 15*a*b^3*x)*sqrt((a*x + b)/x))/a^4]

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Sympy [A]  time = 10.4164, size = 153, normalized size = 1.31 \begin{align*} \frac{a x^{\frac{9}{2}}}{4 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{7 \sqrt{b} x^{\frac{7}{2}}}{24 \sqrt{\frac{a x}{b} + 1}} - \frac{b^{\frac{3}{2}} x^{\frac{5}{2}}}{96 a \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{5}{2}} x^{\frac{3}{2}}}{192 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{7}{2}} \sqrt{x}}{64 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{5 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{64 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**(9/2)/(4*sqrt(b)*sqrt(a*x/b + 1)) + 7*sqrt(b)*x**(7/2)/(24*sqrt(a*x/b + 1)) - b**(3/2)*x**(5/2)/(96*a*sqr
t(a*x/b + 1)) + 5*b**(5/2)*x**(3/2)/(192*a**2*sqrt(a*x/b + 1)) + 5*b**(7/2)*sqrt(x)/(64*a**3*sqrt(a*x/b + 1))
- 5*b**4*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(64*a**(7/2))

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Giac [A]  time = 1.1395, size = 146, normalized size = 1.25 \begin{align*} \frac{5 \, b^{4} \log \left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ) \mathrm{sgn}\left (x\right )}{128 \, a^{\frac{7}{2}}} - \frac{5 \, b^{4} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{128 \, a^{\frac{7}{2}}} + \frac{1}{192} \, \sqrt{a x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x \mathrm{sgn}\left (x\right ) + \frac{b \mathrm{sgn}\left (x\right )}{a}\right )} x - \frac{5 \, b^{2} \mathrm{sgn}\left (x\right )}{a^{2}}\right )} x + \frac{15 \, b^{3} \mathrm{sgn}\left (x\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/128*b^4*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sgn(x)/a^(7/2) - 5/128*b^4*log(abs(b))*sgn(
x)/a^(7/2) + 1/192*sqrt(a*x^2 + b*x)*(2*(4*(6*x*sgn(x) + b*sgn(x)/a)*x - 5*b^2*sgn(x)/a^2)*x + 15*b^3*sgn(x)/a
^3)