3.1722 \(\int \frac{x^3}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=120 \[ \frac{35 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^3}-\frac{35 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^4}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{9/2}}-\frac{7 b x^3 \sqrt{a+\frac{b}{x}}}{24 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x}}}{4 a} \]

[Out]

(-35*b^3*Sqrt[a + b/x]*x)/(64*a^4) + (35*b^2*Sqrt[a + b/x]*x^2)/(96*a^3) - (7*b*Sqrt[a + b/x]*x^3)/(24*a^2) +
(Sqrt[a + b/x]*x^4)/(4*a) + (35*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(64*a^(9/2))

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Rubi [A]  time = 0.0507232, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{35 b^2 x^2 \sqrt{a+\frac{b}{x}}}{96 a^3}-\frac{35 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^4}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{9/2}}-\frac{7 b x^3 \sqrt{a+\frac{b}{x}}}{24 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x}}}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a + b/x],x]

[Out]

(-35*b^3*Sqrt[a + b/x]*x)/(64*a^4) + (35*b^2*Sqrt[a + b/x]*x^2)/(96*a^3) - (7*b*Sqrt[a + b/x]*x^3)/(24*a^2) +
(Sqrt[a + b/x]*x^4)/(4*a) + (35*b^4*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(64*a^(9/2))

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 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{x^3}{\sqrt{a+\frac{b}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x}} x^4}{4 a}+\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=-\frac{7 b \sqrt{a+\frac{b}{x}} x^3}{24 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^4}{4 a}-\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{48 a^2}\\ &=\frac{35 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^3}-\frac{7 b \sqrt{a+\frac{b}{x}} x^3}{24 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^4}{4 a}+\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{64 a^3}\\ &=-\frac{35 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^4}+\frac{35 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^3}-\frac{7 b \sqrt{a+\frac{b}{x}} x^3}{24 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^4}{4 a}-\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{128 a^4}\\ &=-\frac{35 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^4}+\frac{35 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^3}-\frac{7 b \sqrt{a+\frac{b}{x}} x^3}{24 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^4}{4 a}-\frac{\left (35 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^4}\\ &=-\frac{35 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^4}+\frac{35 b^2 \sqrt{a+\frac{b}{x}} x^2}{96 a^3}-\frac{7 b \sqrt{a+\frac{b}{x}} x^3}{24 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^4}{4 a}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0110393, size = 37, normalized size = 0.31 \[ \frac{2 b^4 \sqrt{a+\frac{b}{x}} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{b}{a x}+1\right )}{a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a + b/x],x]

[Out]

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

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Maple [A]  time = 0.009, size = 184, normalized size = 1.5 \begin{align*} -{\frac{x}{384}\sqrt{{\frac{ax+b}{x}}} \left ( -96\,x \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}+208\,{a}^{5/2} \left ( a{x}^{2}+bx \right ) ^{3/2}b-348\,{a}^{5/2}\sqrt{a{x}^{2}+bx}x{b}^{2}+384\,\sqrt{ \left ( ax+b \right ) x}{a}^{3/2}{b}^{3}-174\,{a}^{3/2}\sqrt{a{x}^{2}+bx}{b}^{3}-192\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{4}+87\,\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{11}{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)+208*a^(5/2)*(a*x^2+b*x)^(3/2)*b-348*a^(5/2)*(a*x^2
+b*x)^(1/2)*x*b^2+384*((a*x+b)*x)^(1/2)*a^(3/2)*b^3-174*a^(3/2)*(a*x^2+b*x)^(1/2)*b^3-192*ln(1/2*(2*((a*x+b)*x
)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^4+87*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^(11/2)

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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 = 1.8154, size = 416, normalized size = 3.47 \begin{align*} \left [\frac{105 \, \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} - 56 \, a^{3} b x^{3} + 70 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{384 \, a^{5}}, -\frac{105 \, \sqrt{-a} b^{4} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (48 \, a^{4} x^{4} - 56 \, a^{3} b x^{3} + 70 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{192 \, a^{5}}\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*(105*sqrt(a)*b^4*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(48*a^4*x^4 - 56*a^3*b*x^3 + 70*a^2
*b^2*x^2 - 105*a*b^3*x)*sqrt((a*x + b)/x))/a^5, -1/192*(105*sqrt(-a)*b^4*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a)
- (48*a^4*x^4 - 56*a^3*b*x^3 + 70*a^2*b^2*x^2 - 105*a*b^3*x)*sqrt((a*x + b)/x))/a^5]

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Sympy [A]  time = 8.83759, size = 155, normalized size = 1.29 \begin{align*} \frac{x^{\frac{9}{2}}}{4 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{7}{2}}}{24 a \sqrt{\frac{a x}{b} + 1}} + \frac{7 b^{\frac{3}{2}} x^{\frac{5}{2}}}{96 a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{35 b^{\frac{5}{2}} x^{\frac{3}{2}}}{192 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{35 b^{\frac{7}{2}} \sqrt{x}}{64 a^{4} \sqrt{\frac{a x}{b} + 1}} + \frac{35 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{64 a^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(9/2)/(4*sqrt(b)*sqrt(a*x/b + 1)) - sqrt(b)*x**(7/2)/(24*a*sqrt(a*x/b + 1)) + 7*b**(3/2)*x**(5/2)/(96*a**2*
sqrt(a*x/b + 1)) - 35*b**(5/2)*x**(3/2)/(192*a**3*sqrt(a*x/b + 1)) - 35*b**(7/2)*sqrt(x)/(64*a**4*sqrt(a*x/b +
1)) + 35*b**4*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(64*a**(9/2))

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Giac [A]  time = 1.15931, size = 208, normalized size = 1.73 \begin{align*} -\frac{1}{192} \, b{\left (\frac{105 \, b^{3} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{279 \, a^{3} b^{3} \sqrt{\frac{a x + b}{x}} - \frac{511 \,{\left (a x + b\right )} a^{2} b^{3} \sqrt{\frac{a x + b}{x}}}{x} + \frac{385 \,{\left (a x + b\right )}^{2} a b^{3} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{105 \,{\left (a x + b\right )}^{3} b^{3} \sqrt{\frac{a x + b}{x}}}{x^{3}}}{{\left (a - \frac{a x + b}{x}\right )}^{4} 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="giac")

[Out]

-1/192*b*(105*b^3*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^4) - (279*a^3*b^3*sqrt((a*x + b)/x) - 511*(a*
x + b)*a^2*b^3*sqrt((a*x + b)/x)/x + 385*(a*x + b)^2*a*b^3*sqrt((a*x + b)/x)/x^2 - 105*(a*x + b)^3*b^3*sqrt((a
*x + b)/x)/x^3)/((a - (a*x + b)/x)^4*a^4))