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

Optimal. Leaf size=132 \[ -\frac {3 c^2 (b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {(b c-2 a d) \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )-\frac {a b d^2 (2 a d+b c)}{x}}{a^2 b^2 \sqrt {a+\frac {b}{x}}}+\frac {c x \left (c+\frac {d}{x}\right )^2}{a \sqrt {a+\frac {b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 98, 146, 63, 208} \[ \frac {(b c-2 a d) \left (2 a^2 d^2-2 a b c d+3 b^2 c^2\right )-\frac {a b d^2 (2 a d+b c)}{x}}{a^2 b^2 \sqrt {a+\frac {b}{x}}}-\frac {3 c^2 (b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {c x \left (c+\frac {d}{x}\right )^2}{a \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d/x)^3/(a + b/x)^(3/2),x]

[Out]

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

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 98

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

Rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

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 375

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

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Mathematica [C]  time = 0.07, size = 92, normalized size = 0.70 \[ \frac {a \left (-4 a^2 d^3 x-2 a b d^2 (d-3 c x)+b^2 c^3 x^2\right )+3 b^2 c^2 x (b c-2 a d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b}{a x}+1\right )}{a^2 b^2 x \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.98, size = 336, normalized size = 2.55 \[ \left [-\frac {3 \, {\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d\right )} x\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (a^{2} b^{2} c^{3} x^{2} - 2 \, a^{3} b d^{3} + {\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} b^{2} x + a^{3} b^{3}\right )}}, \frac {3 \, {\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} b^{2} c^{3} x^{2} - 2 \, a^{3} b d^{3} + {\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} b^{2} x + a^{3} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 222, normalized size = 1.68 \[ -\frac {\frac {2 \, d^{3} \sqrt {\frac {a x + b}{x}}}{b} - \frac {3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {2 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3} - \frac {3 \, {\left (a x + b\right )} b^{3} c^{3}}{x} + \frac {6 \, {\left (a x + b\right )} a b^{2} c^{2} d}{x} - \frac {6 \, {\left (a x + b\right )} a^{2} b c d^{2}}{x} + \frac {2 \, {\left (a x + b\right )} a^{3} d^{3}}{x}}{{\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )} a^{2} b}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*d^3*sqrt((a*x + b)/x)/b - 3*(b^2*c^3 - 2*a*b*c^2*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2) - (2
*a*b^3*c^3 - 6*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - 2*a^4*d^3 - 3*(a*x + b)*b^3*c^3/x + 6*(a*x + b)*a*b^2*c^2*d/x -
6*(a*x + b)*a^2*b*c*d^2/x + 2*(a*x + b)*a^3*d^3/x)/((a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x + b)/x)/x)*a^2
*b))/b

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maple [B]  time = 0.07, size = 969, normalized size = 7.34 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 a^{4} b^{2} c \,d^{2} x^{4} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 a^{4} b^{2} c \,d^{2} x^{4} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 a^{3} b^{3} c^{2} d \,x^{4} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+3 a^{2} b^{4} c^{3} x^{4} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 a^{3} b^{3} c \,d^{2} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 a^{3} b^{3} c \,d^{2} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-12 a^{2} b^{4} c^{2} d \,x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 a \,b^{5} c^{3} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b c \,d^{2} x^{4}-6 \sqrt {\left (a x +b \right ) x}\, a^{\frac {9}{2}} b c \,d^{2} x^{4}+12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b^{2} c^{2} d \,x^{4}-6 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{3} c^{3} x^{4}+3 a^{2} b^{4} c \,d^{2} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 a^{2} b^{4} c \,d^{2} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 a \,b^{5} c^{2} d \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+3 b^{6} c^{3} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-12 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{2} c \,d^{2} x^{3}-12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b^{2} c \,d^{2} x^{3}+24 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{3} c^{2} d \,x^{3}-12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{4} c^{3} x^{3}-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{3} c \,d^{2} x^{2}-6 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{3} c \,d^{2} x^{2}+12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{4} c^{2} d \,x^{2}-6 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b^{5} c^{3} x^{2}+4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} d^{3} x^{2}-4 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {9}{2}} d^{3} x^{2}+12 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b c \,d^{2} x^{2}-12 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} c^{2} d \,x^{2}+4 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{3} c^{3} x^{2}+8 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b \,d^{3} x +4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2} d^{3}\right )}{2 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{2} a^{\frac {5}{2}} b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a*x+b)/x)^(1/2)/x/a^(5/2)*(3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*b^6*c^3+4*(a*x^2
+b*x)^(3/2)*a^(9/2)*x^2*d^3-4*a^(9/2)*((a*x+b)*x)^(3/2)*x^2*d^3+4*(a*x^2+b*x)^(3/2)*a^(5/2)*b^2*d^3+6*ln(1/2*(
2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^3*a^3*b^3*c*d^2-12*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2)
)/a^(1/2))*x^3*a^2*b^4*c^2*d-6*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^4*a^3*b^3*c^2*d-3*ln(1/
2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x^4*a^4*b^2*c*d^2+3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/
2))/a^(1/2))*x^4*a^4*b^2*c*d^2-6*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x^3*a^3*b^3*c*d^2+12*a^
(3/2)*((a*x+b)*x)^(1/2)*x^2*b^4*c^2*d-6*(a*x^2+b*x)^(1/2)*a^(9/2)*x^4*b*c*d^2-6*a^(9/2)*((a*x+b)*x)^(1/2)*x^4*
b*c*d^2+12*a^(7/2)*((a*x+b)*x)^(1/2)*x^4*b^2*c^2*d-12*(a*x^2+b*x)^(1/2)*a^(7/2)*x^3*b^2*c*d^2+12*a^(7/2)*((a*x
+b)*x)^(3/2)*x^2*b*c*d^2-12*a^(5/2)*((a*x+b)*x)^(3/2)*x^2*b^2*c^2*d-12*a^(7/2)*((a*x+b)*x)^(1/2)*x^3*b^2*c*d^2
+24*a^(5/2)*((a*x+b)*x)^(1/2)*x^3*b^3*c^2*d-6*(a*x^2+b*x)^(1/2)*a^(5/2)*x^2*b^3*c*d^2-6*ln(1/2*(2*a*x+b+2*((a*
x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*a*b^5*c^2*d-3*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*a^
2*b^4*c*d^2+3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*a^2*b^4*c*d^2-6*a^(5/2)*((a*x+b)*x)^(1
/2)*x^2*b^3*c*d^2+8*(a*x^2+b*x)^(3/2)*a^(7/2)*x*b*d^3-12*a^(3/2)*((a*x+b)*x)^(1/2)*x^3*b^4*c^3+4*a^(3/2)*((a*x
+b)*x)^(3/2)*x^2*b^3*c^3-6*a^(5/2)*((a*x+b)*x)^(1/2)*x^4*b^3*c^3+3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2)
)/a^(1/2))*x^4*a^2*b^4*c^3+6*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^3*a*b^5*c^3-6*a^(1/2)*((a
*x+b)*x)^(1/2)*x^2*b^5*c^3)/((a*x+b)*x)^(1/2)/b^3/(a*x+b)^2

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maxima [A]  time = 1.26, size = 200, normalized size = 1.52 \[ \frac {1}{2} \, c^{3} {\left (\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b\right )}}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} - 3 \, c^{2} d {\left (\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} a}\right )} - 2 \, d^{3} {\left (\frac {\sqrt {a + \frac {b}{x}}}{b^{2}} + \frac {a}{\sqrt {a + \frac {b}{x}} b^{2}}\right )} + \frac {6 \, c d^{2}}{\sqrt {a + \frac {b}{x}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.91, size = 172, normalized size = 1.30 \[ \frac {\frac {2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a}-\frac {\left (a+\frac {b}{x}\right )\,\left (2\,a^3\,d^3-6\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{a^2}}{b^2\,{\left (a+\frac {b}{x}\right )}^{3/2}-a\,b^2\,\sqrt {a+\frac {b}{x}}}-\frac {2\,d^3\,\sqrt {a+\frac {b}{x}}}{b^2}+\frac {3\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\,\left (2\,a\,d-b\,c\right )}{a^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d/x)^3/(a + b/x)^(3/2),x)

[Out]

((2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/a - ((a + b/x)*(2*a^3*d^3 - 3*b^3*c^3 + 6*a*b^2*c^2*d
- 6*a^2*b*c*d^2))/a^2)/(b^2*(a + b/x)^(3/2) - a*b^2*(a + b/x)^(1/2)) - (2*d^3*(a + b/x)^(1/2))/b^2 + (3*c^2*a
tanh((a + b/x)^(1/2)/a^(1/2))*(2*a*d - b*c))/a^(5/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x + d\right )^{3}}{x^{3} \left (a + \frac {b}{x}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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