\(\int \frac {c+\frac {d}{x}}{(a+\frac {b}{x})^{3/2}} \, dx\) [254]

Optimal result

Integrand size = 19, antiderivative size = 76 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

-(-2*a*d+3*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(5/2)+(-2*a*d+3*b*c)/a^2/(a+b/x)^(1/2)+c*x/a/(a+b/x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {382, 79, 53, 65, 214} \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (3 b c-2 a d)}{a^{5/2}}+\frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}} \]

[In]

Int[(c + d/x)/(a + b/x)^(3/2),x]

[Out]

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

Rule 53

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 65

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[(a + b/x^n)^p*((c +
d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {c+d x}{x^2 (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {\left (-\frac {3 b c}{2}+a d\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}+\frac {(3 b c-2 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}+\frac {(3 b c-2 a d) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a^2 b} \\ & = \frac {3 b c-2 a d}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \sqrt {a+\frac {b}{x}}}-\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x (3 b c-2 a d+a c x)}{a^2 (b+a x)}+\frac {(-3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]

[In]

Integrate[(c + d/x)/(a + b/x)^(3/2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(66)=132\).

Time = 0.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.09

method	result	size
risch	\(\frac {c \left (a x +b \right )}{a^{2} \sqrt {\frac {a x +b}{x}}}+\frac {\left (2 \sqrt {a}\, d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )-\frac {3 b c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}-\frac {4 \left (a d -b c \right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{2} x \sqrt {\frac {a x +b}{x}}}\)	\(159\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, d \,x^{2}-6 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b c \,x^{2}-4 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} d +8 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b d x +4 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b c -12 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c x -2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b d \,x^{2}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} c \,x^{2}+4 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} d -4 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} d x +6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} c x -6 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{3} c -2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} d +3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4} c \right )}{2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{2}}\)	\(387\)

[In]

int((c+d/x)/(a+b/x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

c/a^2*(a*x+b)/((a*x+b)/x)^(1/2)+1/2/a^2*(2*a^(1/2)*d*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))-3*b*c*ln((1/2*b
+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)-4*(a*d-b*c)/a/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2))/x/((a*x+b)/x)^(1
/2)*(x*(a*x+b))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.76 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\left [-\frac {{\left (3 \, b^{2} c - 2 \, a b d + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{2} c x^{2} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {{\left (3 \, b^{2} c - 2 \, a b d + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} c x^{2} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (65) = 130\).

Time = 14.55 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.95 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=c \left (\frac {x^{\frac {3}{2}}}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} \sqrt {x}}{a^{2} \sqrt {\frac {a x}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {5}{2}}}\right ) + d \left (- \frac {2 a^{3} x \sqrt {1 + \frac {b}{a x}}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{3} x \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{3} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b}\right ) \]

[In]

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

[Out]

c*(x**(3/2)/(a*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*sqrt(x)/(a**2*sqrt(a*x/b + 1)) - 3*b*asinh(sqrt(a)*sqrt(x)
/sqrt(b))/a**(5/2)) + d*(-2*a**3*x*sqrt(1 + b/(a*x))/(a**(9/2)*x + a**(7/2)*b) - a**3*x*log(b/(a*x))/(a**(9/2)
*x + a**(7/2)*b) + 2*a**3*x*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b) - a**2*b*log(b/(a*x))/(a**(9/
2)*x + a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (66) = 132\).

Time = 0.35 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.89 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {1}{2} \, c {\left (\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} - d {\left (\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} a}\right )} \]

[In]

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

[Out]

1/2*c*(2*(3*(a + b/x)*b - 2*a*b)/((a + b/x)^(3/2)*a^2 - sqrt(a + b/x)*a^3) + 3*b*log((sqrt(a + b/x) - sqrt(a))
/(sqrt(a + b/x) + sqrt(a)))/a^(5/2)) - d*(log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(3/2) + 2
/(sqrt(a + b/x)*a))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (66) = 132\).

Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.96 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=-\frac {{\left (3 \, b c \log \left ({\left | b \right |}\right ) - 2 \, a d \log \left ({\left | b \right |}\right ) + 4 \, b c - 4 \, a d\right )} \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {5}{2}}} + \frac {\sqrt {a x^{2} + b x} c}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {{\left (3 \, b c - 2 \, a d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (\sqrt {a} b^{2} c - a^{\frac {3}{2}} b d\right )}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )} a^{3} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-1/2*(3*b*c*log(abs(b)) - 2*a*d*log(abs(b)) + 4*b*c - 4*a*d)*sgn(x)/a^(5/2) + sqrt(a*x^2 + b*x)*c/(a^2*sgn(x))
+ 1/2*(3*b*c - 2*a*d)*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(5/2)*sgn(x)) + 2*(sqrt(a)*b
^2*c - a^(3/2)*b*d)/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)*a^3*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2\,d}{a\,\sqrt {a+\frac {b}{x}}}+\frac {2\,c\,x\,{\left (\frac {a\,x}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,x}{b}\right )}{5\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]

[In]

int((c + d/x)/(a + b/x)^(3/2),x)

[Out]

(2*d*atanh((a + b/x)^(1/2)/a^(1/2)))/a^(3/2) - (2*d)/(a*(a + b/x)^(1/2)) + (2*c*x*((a*x)/b + 1)^(3/2)*hypergeo
m([3/2, 5/2], 7/2, -(a*x)/b))/(5*(a + b/x)^(3/2))