\(\int \frac {c+\frac {d}{x}}{(a+\frac {b}{x})^{5/2}} \, dx\) [261]

Optimal result

Integrand size = 19, antiderivative size = 103 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {(5 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]

[Out]

1/3*(-2*a*d+5*b*c)/a^2/(a+b/x)^(3/2)+c*x/a/(a+b/x)^(3/2)-(-2*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(7/2)
+(-2*a*d+5*b*c)/a^3/(a+b/x)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {382, 79, 53, 65, 214} \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (5 b c-2 a d)}{a^{7/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}} \]

[In]

Int[(c + d/x)/(a + b/x)^(5/2),x]

[Out]

(5*b*c - 2*a*d)/(3*a^2*(a + b/x)^(3/2)) + (5*b*c - 2*a*d)/(a^3*Sqrt[a + b/x]) + (c*x)/(a*(a + b/x)^(3/2)) - ((
5*b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)

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)^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {\left (-\frac {5 b c}{2}+a d\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b c-2 a d) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 b c-2 a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^3} \\ & = \frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(5 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^3 b} \\ & = \frac {5 b c-2 a d}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {5 b c-2 a d}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {(5 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^2 c+a^2 x (-8 d+3 c x)+a b (-6 d+20 c x)\right )}{3 a^3 (b+a x)^2}+\frac {(-5 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]

[In]

Integrate[(c + d/x)/(a + b/x)^(5/2),x]

[Out]

(Sqrt[a + b/x]*x*(15*b^2*c + a^2*x*(-8*d + 3*c*x) + a*b*(-6*d + 20*c*x)))/(3*a^3*(b + a*x)^2) + ((-5*b*c + 2*a
*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(253\) vs. \(2(89)=178\).

Time = 0.18 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.47

method	result	size
risch	\(\frac {c \left (a x +b \right )}{a^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (2 \sqrt {a}\, d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )-\frac {5 b c \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}+\frac {2 \left (a d -b c \right ) b^{2} \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{a^{2}}-\frac {4 \left (2 a d -3 b c \right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{3} x \sqrt {\frac {a x +b}{x}}}\)	\(254\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (12 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, d \,x^{3}-30 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b c \,x^{3}-12 a^{\frac {7}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} d x +36 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b d \,x^{2}+24 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b c x -90 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} c \,x^{2}-8 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b d +36 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} d x -6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b d \,x^{3}+15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} c \,x^{3}+20 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b^{2} c -90 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} c x -18 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} d \,x^{2}+45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} c \,x^{2}+12 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} d -18 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} d x +45 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} c x -30 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{4} c -6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{4} d +15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{5} c \right )}{6 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{3}}\)	\(541\)

[In]

int((c+d/x)/(a+b/x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/a^3*c*(a*x+b)/((a*x+b)/x)^(1/2)+1/2/a^3*(2*a^(1/2)*d*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))-5*b*c*ln((1/2
*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)+2*(a*d-b*c)*b^2/a^2*(2/3/b/(x+b/a)^2*(a*(x+b/a)^2-b*(x+b/a))^(1/2)+
4/3*a/b^2/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2))-4*(2*a*d-3*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 = 0.27 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.21 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c - 2 \, a b^{2} d + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a^{3} c x^{3} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}, \frac {3 \, {\left (5 \, b^{3} c - 2 \, a b^{2} d + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a^{3} c x^{3} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{2} + 3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}}\right ] \]

[In]

integrate((c+d/x)/(a+b/x)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(5*b^3*c - 2*a*b^2*d + (5*a^2*b*c - 2*a^3*d)*x^2 + 2*(5*a*b^2*c - 2*a^2*b*d)*x)*sqrt(a)*log(2*a*x + 2
*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(3*a^3*c*x^3 + 4*(5*a^2*b*c - 2*a^3*d)*x^2 + 3*(5*a*b^2*c - 2*a^2*b*d)*x
)*sqrt((a*x + b)/x))/(a^6*x^2 + 2*a^5*b*x + a^4*b^2), 1/3*(3*(5*b^3*c - 2*a*b^2*d + (5*a^2*b*c - 2*a^3*d)*x^2
+ 2*(5*a*b^2*c - 2*a^2*b*d)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (3*a^3*c*x^3 + 4*(5*a^2*b*c - 2
*a^3*d)*x^2 + 3*(5*a*b^2*c - 2*a^2*b*d)*x)*sqrt((a*x + b)/x))/(a^6*x^2 + 2*a^5*b*x + a^4*b^2)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (90) = 180\).

Time = 33.93 (sec) , antiderivative size = 1479, normalized size of antiderivative = 14.36 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c+d/x)/(a+b/x)**(5/2),x)

[Out]

c*(6*a**17*x**4*sqrt(1 + b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*
b**3) + 46*a**16*b*x**3*sqrt(1 + b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a*
*(33/2)*b**3) + 15*a**16*b*x**3*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6
*a**(33/2)*b**3) - 30*a**16*b*x**3*log(sqrt(1 + b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**
(35/2)*b**2*x + 6*a**(33/2)*b**3) + 70*a**15*b**2*x**2*sqrt(1 + b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x*
*2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 45*a**15*b**2*x**2*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2
)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 90*a**15*b**2*x**2*log(sqrt(1 + b/(a*x)) + 1)/(6*a**(39/2
)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 30*a**14*b**3*x*sqrt(1 + b/(a*x))/(6*
a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 45*a**14*b**3*x*log(b/(a*x))/
(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 90*a**14*b**3*x*log(sqrt(1
+ b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) + 15*a**13*
b**4*log(b/(a*x))/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b**3) - 30*a**13
*b**4*log(sqrt(1 + b/(a*x)) + 1)/(6*a**(39/2)*x**3 + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x + 6*a**(33/2)*b
**3)) + d*(-8*a**7*x**3*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(
13/2)*b**3) - 3*a**7*x**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/
2)*b**3) + 6*a**7*x**3*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x
+ 3*a**(13/2)*b**3) - 14*a**6*b*x**2*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b*
*2*x + 3*a**(13/2)*b**3) - 9*a**6*b*x**2*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**
2*x + 3*a**(13/2)*b**3) + 18*a**6*b*x**2*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9
*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 6*a**5*b**2*x*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2
+ 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 9*a**5*b**2*x*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2
+ 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) + 18*a**5*b**2*x*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**
(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 3*a**4*b**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17
/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3
+ 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.65 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{6} \, c {\left (\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - \frac {1}{3} \, d {\left (\frac {3 \, \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}\right )} \]

[In]

integrate((c+d/x)/(a+b/x)^(5/2),x, algorithm="maxima")

[Out]

1/6*c*(2*(15*(a + b/x)^2*b - 10*(a + b/x)*a*b - 2*a^2*b)/((a + b/x)^(5/2)*a^3 - (a + b/x)^(3/2)*a^4) + 15*b*lo
g((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(7/2)) - 1/3*d*(3*log((sqrt(a + b/x) - sqrt(a))/(sqrt
(a + b/x) + sqrt(a)))/a^(5/2) + 2*(4*a + 3*b/x)/((a + b/x)^(3/2)*a^2))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (89) = 178\).

Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.51 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {{\left (15 \, b c \log \left ({\left | b \right |}\right ) - 6 \, a d \log \left ({\left | b \right |}\right ) + 28 \, b c - 16 \, a d\right )} \mathrm {sgn}\left (x\right )}{6 \, a^{\frac {7}{2}}} + \frac {\sqrt {a x^{2} + b x} c}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (5 \, b c - 2 \, a d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{2} b d + 15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c - 9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} d + 7 \, b^{4} c - 4 \, a b^{3} d\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

integrate((c+d/x)/(a+b/x)^(5/2),x, algorithm="giac")

[Out]

-1/6*(15*b*c*log(abs(b)) - 6*a*d*log(abs(b)) + 28*b*c - 16*a*d)*sgn(x)/a^(7/2) + sqrt(a*x^2 + b*x)*c/(a^3*sgn(
x)) + 1/2*(5*b*c - 2*a*d)*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(7/2)*sgn(x)) + 2/3*(9*(s
qrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2*c - 6*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^2*b*d + 15*(sqrt(a)*x - sqrt(a
*x^2 + b*x))*sqrt(a)*b^3*c - 9*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(3/2)*b^2*d + 7*b^4*c - 4*a*b^3*d)/(((sqrt(a)
*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(7/2)*sgn(x))

Mupad [B] (verification not implemented)

Time = 7.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {c+\frac {d}{x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,d}{3\,a}+\frac {2\,d\,\left (a+\frac {b}{x}\right )}{a^2}}{{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {2\,c\,x\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {7}{2};\ \frac {9}{2};\ -\frac {a\,x}{b}\right )}{7\,{\left (a+\frac {b}{x}\right )}^{5/2}} \]

[In]

int((c + d/x)/(a + b/x)^(5/2),x)

[Out]

(2*d*atanh((a + b/x)^(1/2)/a^(1/2)))/a^(5/2) - ((2*d)/(3*a) + (2*d*(a + b/x))/a^2)/(a + b/x)^(3/2) + (2*c*x*((
a*x)/b + 1)^(5/2)*hypergeom([5/2, 7/2], 9/2, -(a*x)/b))/(7*(a + b/x)^(5/2))