3.1732 \(\int \frac {x^2}{(a+\frac {b}{x})^{3/2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {35 b^3}{8 a^4 \sqrt {a+\frac {b}{x}}}+\frac {35 b^2 x}{24 a^3 \sqrt {a+\frac {b}{x}}}-\frac {7 b x^2}{12 a^2 \sqrt {a+\frac {b}{x}}}+\frac {x^3}{3 a \sqrt {a+\frac {b}{x}}} \]

[Out]

-35/8*b^3*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(9/2)+35/8*b^3/a^4/(a+b/x)^(1/2)+35/24*b^2*x/a^3/(a+b/x)^(1/2)-7/12
*b*x^2/a^2/(a+b/x)^(1/2)+1/3*x^3/a/(a+b/x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 115, normalized size of antiderivative = 0.98, 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} \[ \frac {35 b^2 x \sqrt {a+\frac {b}{x}}}{8 a^4}-\frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{9/2}}-\frac {35 b x^2 \sqrt {a+\frac {b}{x}}}{12 a^3}+\frac {7 x^3 \sqrt {a+\frac {b}{x}}}{3 a^2}-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x)^(3/2),x]

[Out]

(35*b^2*Sqrt[a + b/x]*x)/(8*a^4) - (35*b*Sqrt[a + b/x]*x^2)/(12*a^3) - (2*x^3)/(a*Sqrt[a + b/x]) + (7*Sqrt[a +
b/x]*x^3)/(3*a^2) - (35*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(9/2))

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]

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]]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{3 a^2}+\frac {(35 b) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{6 a^2}\\ &=-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{12 a^3}-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{3 a^2}-\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{8 a^3}\\ &=\frac {35 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^4}-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{12 a^3}-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{3 a^2}+\frac {\left (35 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{16 a^4}\\ &=\frac {35 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^4}-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{12 a^3}-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{3 a^2}+\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{8 a^4}\\ &=\frac {35 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^4}-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{12 a^3}-\frac {2 x^3}{a \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{3 a^2}-\frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 37, normalized size = 0.32 \[ \frac {2 b^3 \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {b}{a x}+1\right )}{a^4 \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x)^(3/2),x]

[Out]

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

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fricas [A]  time = 1.06, size = 207, normalized size = 1.77 \[ \left [\frac {105 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{4} x^{4} - 14 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} + 105 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a^{6} x + a^{5} b\right )}}, \frac {105 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (8 \, a^{4} x^{4} - 14 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} + 105 \, a b^{3} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{6} x + a^{5} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(105*(a*b^3*x + b^4)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(8*a^4*x^4 - 14*a^3*b*x^
3 + 35*a^2*b^2*x^2 + 105*a*b^3*x)*sqrt((a*x + b)/x))/(a^6*x + a^5*b), 1/24*(105*(a*b^3*x + b^4)*sqrt(-a)*arcta
n(sqrt(-a)*sqrt((a*x + b)/x)/a) + (8*a^4*x^4 - 14*a^3*b*x^3 + 35*a^2*b^2*x^2 + 105*a*b^3*x)*sqrt((a*x + b)/x))
/(a^6*x + a^5*b)]

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giac [A]  time = 0.19, size = 131, normalized size = 1.12 \[ \frac {1}{24} \, b^{3} {\left (\frac {105 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {48}{a^{4} \sqrt {\frac {a x + b}{x}}} - \frac {87 \, a^{2} \sqrt {\frac {a x + b}{x}} - \frac {136 \, {\left (a x + b\right )} a \sqrt {\frac {a x + b}{x}}}{x} + \frac {57 \, {\left (a x + b\right )}^{2} \sqrt {\frac {a x + b}{x}}}{x^{2}}}{{\left (a - \frac {a x + b}{x}\right )}^{3} a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b^3*(105*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^4) + 48/(a^4*sqrt((a*x + b)/x)) - (87*a^2*sqrt((a
*x + b)/x) - 136*(a*x + b)*a*sqrt((a*x + b)/x)/x + 57*(a*x + b)^2*sqrt((a*x + b)/x)/x^2)/((a - (a*x + b)/x)^3*
a^4))

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maple [B]  time = 0.02, size = 458, normalized size = 3.91 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (-120 a^{3} b^{3} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+15 a^{3} b^{3} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-60 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b \,x^{3}-240 a^{2} b^{4} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+30 a^{2} b^{4} x \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-150 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{2} x^{2}+240 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b^{2} x^{2}-120 a \,b^{5} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+15 a \,b^{5} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{2}-120 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{3} x +480 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{3} x +32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b x -30 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{4}+240 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{4}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}-96 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}\right ) x}{48 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{2} a^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((a*x+b)/x)^(1/2)*x/a^(11/2)*(16*(a*x^2+b*x)^(3/2)*a^(9/2)*x^2-60*(a*x^2+b*x)^(1/2)*a^(9/2)*x^3*b+32*(a*x
^2+b*x)^(3/2)*a^(7/2)*x*b-150*(a*x^2+b*x)^(1/2)*a^(7/2)*x^2*b^2+240*a^(7/2)*((a*x+b)*x)^(1/2)*x^2*b^2-120*a^3*
ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*b^3+16*(a*x^2+b*x)^(3/2)*a^(5/2)*b^2-120*(a*x^2+b*x)
^(1/2)*a^(5/2)*x*b^3-96*a^(5/2)*((a*x+b)*x)^(3/2)*b^2+480*a^(5/2)*((a*x+b)*x)^(1/2)*x*b^3-240*a^2*ln(1/2*(2*a*
x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x*b^4+15*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*a
^3*b^3-30*(a*x^2+b*x)^(1/2)*a^(3/2)*b^4+240*a^(3/2)*((a*x+b)*x)^(1/2)*b^4-120*a*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^
(1/2)*a^(1/2))/a^(1/2))*b^5+30*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x*a^2*b^4+15*ln(1/2*(2*a*
x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*a*b^5)/((a*x+b)*x)^(1/2)/(a*x+b)^2

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maxima [A]  time = 2.44, size = 154, normalized size = 1.32 \[ \frac {105 \, {\left (a + \frac {b}{x}\right )}^{3} b^{3} - 280 \, {\left (a + \frac {b}{x}\right )}^{2} a b^{3} + 231 \, {\left (a + \frac {b}{x}\right )} a^{2} b^{3} - 48 \, a^{3} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{4} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{5} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{6} - \sqrt {a + \frac {b}{x}} a^{7}\right )}} + \frac {35 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(105*(a + b/x)^3*b^3 - 280*(a + b/x)^2*a*b^3 + 231*(a + b/x)*a^2*b^3 - 48*a^3*b^3)/((a + b/x)^(7/2)*a^4 -
3*(a + b/x)^(5/2)*a^5 + 3*(a + b/x)^(3/2)*a^6 - sqrt(a + b/x)*a^7) + 35/16*b^3*log((sqrt(a + b/x) - sqrt(a))/
(sqrt(a + b/x) + sqrt(a)))/a^(9/2)

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mupad [B]  time = 1.57, size = 93, normalized size = 0.79 \[ \frac {35\,b^3}{8\,a^4\,\sqrt {a+\frac {b}{x}}}-\frac {35\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8\,a^{9/2}}+\frac {x^3}{3\,a\,\sqrt {a+\frac {b}{x}}}-\frac {7\,b\,x^2}{12\,a^2\,\sqrt {a+\frac {b}{x}}}+\frac {35\,b^2\,x}{24\,a^3\,\sqrt {a+\frac {b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x)^(3/2),x)

[Out]

(35*b^3)/(8*a^4*(a + b/x)^(1/2)) - (35*b^3*atanh((a + b/x)^(1/2)/a^(1/2)))/(8*a^(9/2)) + x^3/(3*a*(a + b/x)^(1
/2)) - (7*b*x^2)/(12*a^2*(a + b/x)^(1/2)) + (35*b^2*x)/(24*a^3*(a + b/x)^(1/2))

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sympy [A]  time = 9.82, size = 133, normalized size = 1.14 \[ \frac {x^{\frac {7}{2}}}{3 a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} - \frac {7 \sqrt {b} x^{\frac {5}{2}}}{12 a^{2} \sqrt {\frac {a x}{b} + 1}} + \frac {35 b^{\frac {3}{2}} x^{\frac {3}{2}}}{24 a^{3} \sqrt {\frac {a x}{b} + 1}} + \frac {35 b^{\frac {5}{2}} \sqrt {x}}{8 a^{4} \sqrt {\frac {a x}{b} + 1}} - \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{8 a^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(7/2)/(3*a*sqrt(b)*sqrt(a*x/b + 1)) - 7*sqrt(b)*x**(5/2)/(12*a**2*sqrt(a*x/b + 1)) + 35*b**(3/2)*x**(3/2)/(
24*a**3*sqrt(a*x/b + 1)) + 35*b**(5/2)*sqrt(x)/(8*a**4*sqrt(a*x/b + 1)) - 35*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(b
))/(8*a**(9/2))

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