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

Optimal. Leaf size=164 \[ -\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{35 b^2}-3 c^2 \sqrt {a+\frac {b}{x}} (2 a d+b c)+3 \sqrt {a} c^2 (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \]

[Out]

-9/7*d*(a+b/x)^(3/2)*(c+d/x)^2-1/35*d*(a+b/x)^(3/2)*(2*(-a*d+13*b*c)*(2*a*d+5*b*c)+3*b*d*(2*a*d+19*b*c)/x)/b^2
+(a+b/x)^(3/2)*(c+d/x)^3*x+3*c^2*(2*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))*a^(1/2)-3*c^2*(2*a*d+b*c)*(a+b/x)^
(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {375, 97, 153, 147, 50, 63, 208} \[ -\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (\frac {3 b d (2 a d+19 b c)}{x}+2 (13 b c-a d) (2 a d+5 b c)\right )}{35 b^2}-3 c^2 \sqrt {a+\frac {b}{x}} (2 a d+b c)+3 \sqrt {a} c^2 (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+x \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*(c + d/x)^3,x]

[Out]

-3*c^2*(b*c + 2*a*d)*Sqrt[a + b/x] - (9*d*(a + b/x)^(3/2)*(c + d/x)^2)/7 - (d*(a + b/x)^(3/2)*(2*(13*b*c - a*d
)*(5*b*c + 2*a*d) + (3*b*d*(19*b*c + 2*a*d))/x))/(35*b^2) + (a + b/x)^(3/2)*(c + d/x)^3*x + 3*Sqrt[a]*c^2*(b*c
+ 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 97

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

Rule 147

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2} (c+d x)^3}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (c+d x)^2 \left (\frac {3}{2} (b c+2 a d)+\frac {9 b d x}{2}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x} (c+d x) \left (\frac {21}{4} b c (b c+2 a d)+\frac {3}{4} b d (19 b c+2 a d) x\right )}{x} \, dx,x,\frac {1}{x}\right )}{7 b}\\ &=-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {1}{2} \left (3 c^2 (b c+2 a d)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {1}{2} \left (3 a 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 )\\ &=-3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x-\frac {\left (3 a 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 )}{b}\\ &=-3 c^2 (b c+2 a d) \sqrt {a+\frac {b}{x}}-\frac {9}{7} d \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^2-\frac {d \left (a+\frac {b}{x}\right )^{3/2} \left (2 (13 b c-a d) (5 b c+2 a d)+\frac {3 b d (19 b c+2 a d)}{x}\right )}{35 b^2}+\left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )^3 x+3 \sqrt {a} c^2 (b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 159, normalized size = 0.97 \[ \frac {\sqrt {a+\frac {b}{x}} \left (4 a^3 d^3 x^3-2 a^2 b d^2 x^2 (21 c x+d)+a b^2 x \left (35 c^3 x^3-280 c^2 d x^2-84 c d^2 x-16 d^3\right )-2 b^3 \left (35 c^3 x^3+35 c^2 d x^2+21 c d^2 x+5 d^3\right )\right )}{35 b^2 x^3}+3 \sqrt {a} c^2 (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b/x]*(4*a^3*d^3*x^3 - 2*a^2*b*d^2*x^2*(d + 21*c*x) + a*b^2*x*(-16*d^3 - 84*c*d^2*x - 280*c^2*d*x^2 +
35*c^3*x^3) - 2*b^3*(5*d^3 + 21*c*d^2*x + 35*c^2*d*x^2 + 35*c^3*x^3)))/(35*b^2*x^3) + 3*Sqrt[a]*c^2*(b*c + 2*
a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]

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fricas [A]  time = 1.24, size = 380, normalized size = 2.32 \[ \left [\frac {105 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt {a} x^{3} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \, {\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \, {\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{70 \, b^{2} x^{3}}, -\frac {105 \, {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (35 \, a b^{2} c^{3} x^{4} - 10 \, b^{3} d^{3} - 2 \, {\left (35 \, b^{3} c^{3} + 140 \, a b^{2} c^{2} d + 21 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} d + 42 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 2 \, {\left (21 \, b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{35 \, b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/70*(105*(b^3*c^3 + 2*a*b^2*c^2*d)*sqrt(a)*x^3*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(35*a*b^2*
c^3*x^4 - 10*b^3*d^3 - 2*(35*b^3*c^3 + 140*a*b^2*c^2*d + 21*a^2*b*c*d^2 - 2*a^3*d^3)*x^3 - 2*(35*b^3*c^2*d + 4
2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 - 2*(21*b^3*c*d^2 + 8*a*b^2*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^3), -1/35*(105*(b
^3*c^3 + 2*a*b^2*c^2*d)*sqrt(-a)*x^3*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (35*a*b^2*c^3*x^4 - 10*b^3*d^3 - 2
*(35*b^3*c^3 + 140*a*b^2*c^2*d + 21*a^2*b*c*d^2 - 2*a^3*d^3)*x^3 - 2*(35*b^3*c^2*d + 42*a*b^2*c*d^2 + a^2*b*d^
3)*x^2 - 2*(21*b^3*c*d^2 + 8*a*b^2*d^3)*x)*sqrt((a*x + b)/x))/(b^2*x^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, choosing root of [1,0,%%%{-2,[1,2,0]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%%%{1,[2,4,0]%%
%}+%%%{-2,[2,2,0]%%%}+%%%{1,[2,0,0]%%%}+%%%{2,[1,3,1]%%%}+%%%{-2,[1,1,1]%%%}+%%%{1,[0,2,2]%%%}] at parameters
values [86,-97,-82]Warning, choosing root of [1,0,%%%{-2,[1,2,0]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%
%%{1,[2,4,0]%%%}+%%%{-2,[2,2,0]%%%}+%%%{1,[2,0,0]%%%}+%%%{2,[1,3,1]%%%}+%%%{-2,[1,1,1]%%%}+%%%{1,[0,2,2]%%%}]
at parameters values [7,-27,26]Warning, choosing root of [1,0,%%%{-4,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%%%{1,[0
,2,2]%%%}] at parameters values [18.6420984049,-49,-86]Warning, choosing root of [1,0,%%%{-4,[1,0,0]%%%}+%%%{-
2,[0,1,1]%%%},0,%%%{1,[0,2,2]%%%}] at parameters values [78.6493344628,22,42]Warning, choosing root of [1,0,%%
%{-2,[1,0,1]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,2]%%%}] at parameters values [-13,74.7709350525,24]Sign error
(%%%{-b,0%%%}+%%%{2*sqrt(a)*sqrt(b),1/2%%%}+%%%{-2*a,1%%%}+%%%{a*sqrt(a)*sqrt(b)/b,3/2%%%}+%%%{-a^2*sqrt(a)*s
qrt(b)/(4*b^2),5/2%%%}+%%%{undef,7/2%%%})Evaluation time: 0.8Limit: Max order reached or unable to make series
expansion Error: Bad Argument Value

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maple [B]  time = 0.06, size = 353, normalized size = 2.15 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (210 a^{2} b^{2} c^{2} d \,x^{5} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+105 a \,b^{3} c^{3} x^{5} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+420 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b \,c^{2} d \,x^{5}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2} c^{3} x^{5}-420 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,c^{2} d \,x^{3}-140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c^{3} x^{3}+8 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d^{3} x^{2}-84 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c \,d^{2} x^{2}-140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c^{2} d \,x^{2}-12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,d^{3} x -84 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c \,d^{2} x -20 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} d^{3}\right )}{70 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/70*((a*x+b)/x)^(1/2)*(420*(a*x^2+b*x)^(1/2)*a^(5/2)*x^5*b*c^2*d+210*(a*x^2+b*x)^(1/2)*a^(3/2)*x^5*b^2*c^3+8*
(a*x^2+b*x)^(3/2)*a^(5/2)*x^2*d^3-420*(a*x^2+b*x)^(3/2)*a^(3/2)*x^3*b*c^2*d+210*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^
(1/2)*a^(1/2))/a^(1/2))*x^5*a^2*b^2*c^2*d+105*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x^5*a*b^3*
c^3-84*(a*x^2+b*x)^(3/2)*a^(3/2)*x^2*b*c*d^2-140*(a*x^2+b*x)^(3/2)*a^(1/2)*x^3*b^2*c^3-12*(a*x^2+b*x)^(3/2)*a^
(3/2)*x*b*d^3-140*(a*x^2+b*x)^(3/2)*a^(1/2)*x^2*b^2*c^2*d-84*(a*x^2+b*x)^(3/2)*a^(1/2)*x*b^2*c*d^2-20*(a*x^2+b
*x)^(3/2)*a^(1/2)*b^2*d^3)/x^4/b^2/((a*x+b)*x)^(1/2)/a^(1/2)

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maxima [A]  time = 1.49, size = 190, normalized size = 1.16 \[ -\frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} c d^{2}}{5 \, b} + \frac {1}{2} \, {\left (2 \, \sqrt {a + \frac {b}{x}} a x - 3 \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, \sqrt {a + \frac {b}{x}} b\right )} c^{3} - {\left (3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {b}{x}} a\right )} c^{2} d - \frac {2}{35} \, {\left (\frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{b^{2}} - \frac {7 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{b^{2}}\right )} d^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/5*(a + b/x)^(5/2)*c*d^2/b + 1/2*(2*sqrt(a + b/x)*a*x - 3*sqrt(a)*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a +
b/x) + sqrt(a))) - 4*sqrt(a + b/x)*b)*c^3 - (3*a^(3/2)*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a))
) + 2*(a + b/x)^(3/2) + 6*sqrt(a + b/x)*a)*c^2*d - 2/35*(5*(a + b/x)^(7/2)/b^2 - 7*(a + b/x)^(5/2)*a/b^2)*d^3

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mupad [B]  time = 3.88, size = 327, normalized size = 1.99 \[ {\left (a+\frac {b}{x}\right )}^{5/2}\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{5\,b^2}-\frac {4\,a\,d^3}{5\,b^2}\right )+\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,{\left (a\,d-b\,c\right )}^3}{b^2}+2\,a\,\left (2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )-\frac {6\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{b^2}\right )-a^2\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )\right )+{\left (a+\frac {b}{x}\right )}^{3/2}\,\left (\frac {2\,a\,\left (\frac {6\,a\,d^3-6\,b\,c\,d^2}{b^2}-\frac {4\,a\,d^3}{b^2}\right )}{3}-\frac {2\,d\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,a^2\,d^3}{3\,b^2}\right )-\frac {2\,d^3\,{\left (a+\frac {b}{x}\right )}^{7/2}}{7\,b^2}+a\,c^3\,x\,\sqrt {a+\frac {b}{x}}-2\,c^2\,\mathrm {atan}\left (\frac {2\,c^2\,\sqrt {a+\frac {b}{x}}\,\left (2\,a\,d+b\,c\right )\,\sqrt {-\frac {9\,a}{4}}}{6\,d\,a^2\,c^2+3\,b\,a\,c^3}\right )\,\left (2\,a\,d+b\,c\right )\,\sqrt {-\frac {9\,a}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(3/2)*(c + d/x)^3,x)

[Out]

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

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sympy [A]  time = 123.48, size = 1817, normalized size = 11.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a**(19/2)*b**(11/2)*d**3*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2)
+ 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 40*a**(17/2)*b**(13/2)*d**3*x**5*sqrt(a*x/b + 1
)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**
10*x**(7/2)) - 30*a**(15/2)*b**(15/2)*d**3*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*
b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 40*a**(13/2)*b**(17/2)*d**3*x**3*
sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 10
5*a**(7/2)*b**10*x**(7/2)) + 4*a**(13/2)*b**(3/2)*d**3*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a*
*(5/2)*b**4*x**(5/2)) - 100*a**(11/2)*b**(19/2)*d**3*x**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*
a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 12*a**(11/2)*b**(5/2)*c
*d**2*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 2*a**(11/2)*b**(5/2)*d**3
*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 96*a**(9/2)*b**(21/2)*d**3*x*s
qrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105
*a**(7/2)*b**10*x**(7/2)) + 6*a**(9/2)*b**(7/2)*c*d**2*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a*
*(5/2)*b**4*x**(5/2)) - 8*a**(9/2)*b**(7/2)*d**3*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b*
*4*x**(5/2)) - 30*a**(7/2)*b**(23/2)*d**3*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x
**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 24*a**(7/2)*b**(9/2)*c*d**2*x*sqrt(a*x/
b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**(7/2)*b**(9/2)*d**3*sqrt(a*x/b + 1)/(15*
a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 18*a**(5/2)*b**(11/2)*c*d**2*sqrt(a*x/b + 1)/(15*a**(7/2
)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + sqrt(a)*b*c**3*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 16*a**10*b**5*d
**3*x**(13/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*
a**(7/2)*b**10*x**(7/2)) + 48*a**9*b**6*d**3*x**(11/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(
11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 48*a**8*b**7*d**3*x**(9/2)/(105*a**(13/2)*
b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 16
*a**7*b**8*d**3*x**(7/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(
9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 4*a**7*b*d**3*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**
(5/2)) - 12*a**6*b**2*c*d**2*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**6*b**2*d*
*3*x**(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 12*a**5*b**3*c*d**2*x**(5/2)/(15*a**(7/2
)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**2*c**2*d*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + a*sqrt(b)
*c**3*sqrt(x)*sqrt(a*x/b + 1) - 2*a*b*c**3*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) - 6*a*c**2*d*sqrt(a + b/x) +
3*a*c*d**2*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) - 2*b*c**3*sqrt(a + b/x) + 3*b
*c**2*d*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True))

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