\(\int \frac {(c+\frac {d}{x})^2}{(a+\frac {b}{x})^{5/2}} \, dx\) [260]

Optimal result

Integrand size = 21, antiderivative size = 122 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 a^2 d^2+b c (5 b c-4 a d)}{3 a^2 b \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c (5 b c-4 a d)}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {c (5 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]

[Out]

1/3*(2*a^2*d^2+b*c*(-4*a*d+5*b*c))/a^2/b/(a+b/x)^(3/2)+c^2*x/a/(a+b/x)^(3/2)-c*(-4*a*d+5*b*c)*arctanh((a+b/x)^
(1/2)/a^(1/2))/a^(7/2)+c*(-4*a*d+5*b*c)/a^3/(a+b/x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {382, 91, 79, 53, 65, 214} \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {c \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (5 b c-4 a d)}{a^{7/2}}+\frac {c (5 b c-4 a d)}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {\frac {5 b c^2}{a}+\frac {2 a d^2}{b}-4 c d}{3 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}} \]

[In]

Int[(c + d/x)^2/(a + b/x)^(5/2),x]

[Out]

((5*b*c^2)/a - 4*c*d + (2*a*d^2)/b)/(3*a*(a + b/x)^(3/2)) + (c*(5*b*c - 4*a*d))/(a^3*Sqrt[a + b/x]) + (c^2*x)/
(a*(a + b/x)^(3/2)) - (c*(5*b*c - 4*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 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2}{x^2 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} c (5 b c-4 a d)+a d^2 x}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {\frac {5 b c^2}{a}-4 c d+\frac {2 a d^2}{b}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(c (5 b c-4 a d)) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a^2} \\ & = \frac {\frac {5 b c^2}{a}-4 c d+\frac {2 a d^2}{b}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c (5 b c-4 a d)}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(c (5 b c-4 a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^3} \\ & = \frac {\frac {5 b c^2}{a}-4 c d+\frac {2 a d^2}{b}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c (5 b c-4 a d)}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {(c (5 b c-4 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 {\frac {5 b c^2}{a}-4 c d+\frac {2 a d^2}{b}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {c (5 b c-4 a d)}{a^3 \sqrt {a+\frac {b}{x}}}+\frac {c^2 x}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {c (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x \left (15 b^3 c^2+2 a^3 d^2 x+a^2 b c x (-16 d+3 c x)+4 a b^2 c (-3 d+5 c x)\right )}{3 a^3 b (b+a x)^2}+\frac {c (-5 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2}} \]

[In]

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

[Out]

(Sqrt[a + b/x]*x*(15*b^3*c^2 + 2*a^3*d^2*x + a^2*b*c*x*(-16*d + 3*c*x) + 4*a*b^2*c*(-3*d + 5*c*x)))/(3*a^3*b*(
b + a*x)^2) + (c*(-5*b*c + 4*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. \(286\) vs. \(2(108)=216\).

Time = 0.19 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.35

method	result	size
risch	\(\frac {c^{2} \left (a x +b \right )}{a^{3} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {5 b \,c^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{\sqrt {a}}+4 \sqrt {a}\, c d \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )+\frac {2 \left (2 a^{2} d^{2}-8 a b c d +6 b^{2} c^{2}\right ) \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a b \left (x +\frac {b}{a}\right )}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \left (\frac {2 \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b \left (x +\frac {b}{a}\right )^{2}}+\frac {4 a \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{3 b^{2} \left (x +\frac {b}{a}\right )}\right )}{a^{2}}\right ) \sqrt {x \left (a x +b \right )}}{2 a^{3} x \sqrt {\frac {a x +b}{x}}}\)	\(287\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (24 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, c d \,x^{3}-30 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b \,c^{2} x^{3}-24 a^{\frac {7}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} c d x +72 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} b c d \,x^{2}-4 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {7}{2}} d^{2}+24 a^{\frac {5}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b \,c^{2} x -90 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b^{2} c^{2} x^{2}-16 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} b c d +72 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b^{2} c d x -12 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b c 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^{2} x^{3}+20 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} c^{2}-90 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{3} c^{2} x -36 \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 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^{2} x^{2}+24 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{3} c d -36 \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 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^{2} x -30 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{4} c^{2}-12 \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 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^{2}\right )}{6 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{3}}\)	\(588\)

[In]

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

[Out]

c^2/a^3*(a*x+b)/((a*x+b)/x)^(1/2)+1/2/a^3*(-5*b*c^2*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))/a^(1/2)+4*a^(1/2
)*c*d*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x)^(1/2))+2*(2*a^2*d^2-8*a*b*c*d+6*b^2*c^2)/a/b/(x+b/a)*(a*(x+b/a)^2-b*(
x+b/a))^(1/2)-2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*b/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)))/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 = 407, normalized size of antiderivative = 3.34 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{4} c^{2} - 4 \, a b^{3} c d + {\left (5 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (3 \, a^{3} b c^{2} x^{3} + 2 \, {\left (10 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{6} b x^{2} + 2 \, a^{5} b^{2} x + a^{4} b^{3}\right )}}, \frac {3 \, {\left (5 \, b^{4} c^{2} - 4 \, a b^{3} c d + {\left (5 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (3 \, a^{3} b c^{2} x^{3} + 2 \, {\left (10 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} + 3 \, {\left (5 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} b x^{2} + 2 \, a^{5} b^{2} x + a^{4} b^{3}\right )}}\right ] \]

[In]

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

[Out]

[-1/6*(3*(5*b^4*c^2 - 4*a*b^3*c*d + (5*a^2*b^2*c^2 - 4*a^3*b*c*d)*x^2 + 2*(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqr
t(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(3*a^3*b*c^2*x^3 + 2*(10*a^2*b^2*c^2 - 8*a^3*b*c*d + a
^4*d^2)*x^2 + 3*(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqrt((a*x + b)/x))/(a^6*b*x^2 + 2*a^5*b^2*x + a^4*b^3), 1/3*(
3*(5*b^4*c^2 - 4*a*b^3*c*d + (5*a^2*b^2*c^2 - 4*a^3*b*c*d)*x^2 + 2*(5*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqrt(-a)*a
rctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (3*a^3*b*c^2*x^3 + 2*(10*a^2*b^2*c^2 - 8*a^3*b*c*d + a^4*d^2)*x^2 + 3*(5
*a*b^3*c^2 - 4*a^2*b^2*c*d)*x)*sqrt((a*x + b)/x))/(a^6*b*x^2 + 2*a^5*b^2*x + a^4*b^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.56 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {1}{6} \, c^{2} {\left (\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{2} b - 10 \, {\left (a + \frac {b}{x}\right )} a b - 2 \, a^{2} b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} - \frac {2}{3} \, c d {\left (\frac {3 \, \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}\right )} + \frac {2 \, d^{2}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b} \]

[In]

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

[Out]

1/6*c^2*(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*
log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(7/2)) - 2/3*c*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)) + 2/3*d^2/((a + b/x)^(3/2)*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (108) = 216\).

Time = 0.31 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.98 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\sqrt {a x^{2} + b x} c^{2}}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (5 \, b c^{2} - 4 \, a c d\right )} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} - \frac {{\left (15 \, b^{2} c^{2} \log \left ({\left | b \right |}\right ) - 12 \, a b c d \log \left ({\left | b \right |}\right ) + 28 \, b^{2} c^{2} - 32 \, a b c d + 4 \, a^{2} d^{2}\right )} \mathrm {sgn}\left (x\right )}{6 \, a^{\frac {7}{2}} b} + \frac {2 \, {\left (9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} c^{2} - 12 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{2} b c d + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a^{3} d^{2} + 15 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} c^{2} - 18 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {3}{2}} b^{2} c d + 3 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} a^{\frac {5}{2}} b d^{2} + 7 \, b^{4} c^{2} - 8 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

sqrt(a*x^2 + b*x)*c^2/(a^3*sgn(x)) + 1/2*(5*b*c^2 - 4*a*c*d)*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)
+ b))/(a^(7/2)*sgn(x)) - 1/6*(15*b^2*c^2*log(abs(b)) - 12*a*b*c*d*log(abs(b)) + 28*b^2*c^2 - 32*a*b*c*d + 4*a
^2*d^2)*sgn(x)/(a^(7/2)*b) + 2/3*(9*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2*c^2 - 12*(sqrt(a)*x - sqrt(a*x^2 +
b*x))^2*a^2*b*c*d + 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^3*d^2 + 15*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*
b^3*c^2 - 18*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(3/2)*b^2*c*d + 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(5/2)*b*d^2
+ 7*b^4*c^2 - 8*a*b^3*c*d + a^2*b^2*d^2)/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(7/2)*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+\frac {d}{x}\right )^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\frac {2\,\left (a+\frac {b}{x}\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d-5\,b^2\,c^2\right )}{3\,a^2}-\frac {2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a}+\frac {b\,{\left (a+\frac {b}{x}\right )}^2\,\left (5\,b\,c^2-4\,a\,c\,d\right )}{a^3}}{b\,{\left (a+\frac {b}{x}\right )}^{5/2}-a\,b\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {c\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\,\left (4\,a\,d-5\,b\,c\right )}{a^{7/2}} \]

[In]

int((c + d/x)^2/(a + b/x)^(5/2),x)

[Out]

((2*(a + b/x)*(a^2*d^2 - 5*b^2*c^2 + 4*a*b*c*d))/(3*a^2) - (2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(3*a) + (b*(a +
b/x)^2*(5*b*c^2 - 4*a*c*d))/a^3)/(b*(a + b/x)^(5/2) - a*b*(a + b/x)^(3/2)) + (c*atanh((a + b/x)^(1/2)/a^(1/2)
)*(4*a*d - 5*b*c))/a^(7/2)