3.244 \(\int \frac {(a+\frac {b}{x})^{5/2}}{(c+\frac {d}{x})^3} \, dx\)

Optimal. Leaf size=237 \[ \frac {a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^4}-\frac {\sqrt {b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 d^{3/2}}-\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{4 c^3 d \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left (c+\frac {d}{x}\right )^2}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {375, 98, 149, 151, 156, 63, 208, 205} \[ -\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{4 c^3 d \left (c+\frac {d}{x}\right )}-\frac {\sqrt {b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 d^{3/2}}+\frac {a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^4}+\frac {\sqrt {a+\frac {b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left (c+\frac {d}{x}\right )^2}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)/(c + d/x)^3,x]

[Out]

((b*c - 3*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(2*c^2*d*(c + d/x)^2) - ((b^2*c^2 + 7*a*b*c*d - 12*a^2*d^2)*Sqrt[a +
b/x])/(4*c^3*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)/(c*(c + d/x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 2
4*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*d^(3/2)) + (a^(3/2)*(5*b*c - 6*a*d)*ArcTanh
[Sqrt[a + b/x]/Sqrt[a]])/c^4

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 149

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

Rule 151

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

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 205

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

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 (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (5 b c-6 a d)-\frac {1}{2} b (2 b c-3 a d) x\right )}{x (c+d x)^3} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {(b c-3 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{2 c^2 d \left (c+\frac {d}{x}\right )^2}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {a^2 d (5 b c-6 a d)+\frac {1}{2} b \left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{2 c^2 d}\\ &=\frac {(b c-3 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{2 c^2 d \left (c+\frac {d}{x}\right )^2}-\frac {\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt {a+\frac {b}{x}}}{4 c^3 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {-a^2 d (5 b c-6 a d) (b c-a d)-\frac {1}{4} b (b c-a d) \left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 c^3 d (b c-a d)}\\ &=\frac {(b c-3 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{2 c^2 d \left (c+\frac {d}{x}\right )^2}-\frac {\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt {a+\frac {b}{x}}}{4 c^3 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\left (a^2 (5 b c-6 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 c^4}-\frac {\left ((b c-a d) \left (b^2 c^2+8 a b c d-24 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{8 c^4 d}\\ &=\frac {(b c-3 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{2 c^2 d \left (c+\frac {d}{x}\right )^2}-\frac {\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt {a+\frac {b}{x}}}{4 c^3 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\left (a^2 (5 b c-6 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 c^4}-\frac {\left ((b c-a d) \left (b^2 c^2+8 a b c d-24 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{4 b c^4 d}\\ &=\frac {(b c-3 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{2 c^2 d \left (c+\frac {d}{x}\right )^2}-\frac {\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt {a+\frac {b}{x}}}{4 c^3 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\sqrt {b c-a d} \left (b^2 c^2+8 a b c d-24 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 d^{3/2}}+\frac {a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^4}\\ \end {align*}

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Mathematica [A]  time = 0.85, size = 191, normalized size = 0.81 \[ \frac {-4 a^{3/2} (6 a d-5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+\frac {c x \sqrt {a+\frac {b}{x}} \left (2 a^2 d \left (2 c^2 x^2+9 c d x+6 d^2\right )-a b c d (11 c x+7 d)+b^2 c^2 (c x-d)\right )}{d (c x+d)^2}-\frac {\sqrt {b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{d^{3/2}}}{4 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)/(c + d/x)^3,x]

[Out]

((c*Sqrt[a + b/x]*x*(b^2*c^2*(-d + c*x) - a*b*c*d*(7*d + 11*c*x) + 2*a^2*d*(6*d^2 + 9*c*d*x + 2*c^2*x^2)))/(d*
(d + c*x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a
*d]])/d^(3/2) - 4*a^(3/2)*(-5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*c^4)

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fricas [A]  time = 1.16, size = 1445, normalized size = 6.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="fricas")

[Out]

[-1/8*(4*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^2*d^2)*x^2 + 2*(5*a*b*c^2*d^2 - 6*a^2*c*d^3)*x)*sqr
t(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^2*d^4 + (b^2*c^4 + 8*a
*b*c^3*d - 24*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt(-(b*c - a*d)/d)*log((2*d
*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) - 2*(4*a^2*c^3*d*x^3 + (b^2*c^4
- 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(c^6*d
*x^2 + 2*c^5*d^2*x + c^4*d^3), 1/4*((b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^2*d^4 + (b^2*c^4 + 8*a*b*c^3*d - 24*a^2*
c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/
d)*sqrt((a*x + b)/x)/(b*c - a*d)) - 2*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^2*d^2)*x^2 + 2*(5*a*b*
c^2*d^2 - 6*a^2*c*d^3)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (4*a^2*c^3*d*x^3 + (b^2*c^4
- 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(c^6*
d*x^2 + 2*c^5*d^2*x + c^4*d^3), -1/8*(8*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^2*d^2)*x^2 + 2*(5*a*
b*c^2*d^2 - 6*a^2*c*d^3)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^
2*d^4 + (b^2*c^4 + 8*a*b*c^3*d - 24*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt(-(
b*c - a*d)/d)*log((2*d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) - 2*(4*a^2
*c^3*d*x^3 + (b^2*c^4 - 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqr
t((a*x + b)/x))/(c^6*d*x^2 + 2*c^5*d^2*x + c^4*d^3), 1/4*((b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^2*d^4 + (b^2*c^4 +
8*a*b*c^3*d - 24*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt((b*c - a*d)/d)*arcta
n(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a*d)) - 4*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^
2*d^2)*x^2 + 2*(5*a*b*c^2*d^2 - 6*a^2*c*d^3)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (4*a^2*c^3*d*x
^3 + (b^2*c^4 - 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x +
b)/x))/(c^6*d*x^2 + 2*c^5*d^2*x + c^4*d^3)]

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giac [B]  time = 0.49, size = 945, normalized size = 3.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="giac")

[Out]

sqrt(a*x^2 + b*x)*a^2*sgn(x)/c^3 - 1/2*(5*a^2*b*c*sgn(x) - 6*a^3*d*sgn(x))*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 +
b*x))*sqrt(a) + b))/(sqrt(a)*c^4) + 1/4*(b^3*c^3*sgn(x) + 7*a*b^2*c^2*d*sgn(x) - 32*a^2*b*c*d^2*sgn(x) + 24*a
^3*d^3*sgn(x))*arctan(-((sqrt(a)*x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^
2)*c^4*d) + 1/4*(sqrt(a)*b^3*c^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 7*a^(3/2)*b^2*c^2*d*arctan(sqrt(a)*d/
sqrt(b*c*d - a*d^2)) - 32*a^(5/2)*b*c*d^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(7/2)*d^3*arctan(sqrt(a
)*d/sqrt(b*c*d - a*d^2)) + 10*sqrt(b*c*d - a*d^2)*a^2*b*c*d*log(abs(b)) - 12*sqrt(b*c*d - a*d^2)*a^3*d^2*log(a
bs(b)) - sqrt(b*c*d - a*d^2)*a*b^2*c^2 + 11*sqrt(b*c*d - a*d^2)*a^2*b*c*d - 10*sqrt(b*c*d - a*d^2)*a^3*d^2)*sg
n(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*c^4*d) - 1/4*((sqrt(a)*x - sqrt(a*x^2 + b*x))^3*sqrt(a)*b^3*c^4*sgn(x) - 17*
(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^2*c^3*d*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(5/2)*b*c^
2*d^2*sgn(x) - 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(7/2)*c*d^3*sgn(x) - 5*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2
*a*b^3*c^3*d*sgn(x) - 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^2*b^2*c^2*d^2*sgn(x) + 48*(sqrt(a)*x - sqrt(a*x^2
+ b*x))^2*a^3*b*c*d^3*sgn(x) - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^4*d^4*sgn(x) - (sqrt(a)*x - sqrt(a*x^2 +
b*x))*sqrt(a)*b^4*c^3*d*sgn(x) - 11*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(3/2)*b^3*c^2*d^2*sgn(x) + 52*(sqrt(a)*
x - sqrt(a*x^2 + b*x))*a^(5/2)*b^2*c*d^3*sgn(x) - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(7/2)*b*d^4*sgn(x) - a*
b^4*c^2*d^2*sgn(x) + 11*a^2*b^3*c*d^3*sgn(x) - 10*a^3*b^2*d^4*sgn(x))/(((sqrt(a)*x - sqrt(a*x^2 + b*x))^2*c +
2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*d + b*d)^2*sqrt(a)*c^4*d)

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maple [B]  time = 0.07, size = 1638, normalized size = 6.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)/(c+d/x)^3,x)

[Out]

-1/8*(7*a^(5/2)*b^2*c^4*d^2*x^2*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))
+24*((a*d-b*c)/c^2*d)^(1/2)*a^4*c^3*d^3*x^2*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))-36*((a*x+b)*
x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(7/2)*c^3*d^3*x-64*a^(7/2)*b*c^2*d^4*x*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/
c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))+14*a^(5/2)*b^2*c^3*d^3*x*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)
^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))+48*((a*d-b*c)/c^2*d)^(1/2)*a^4*c^2*d^4*x*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1
/2)*a^(1/2))/a^(1/2))+14*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*b*c^3*d^3-20*((a*d-b*c)/c^2*d)^(1/2
)*a^3*b*c^2*d^4*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))+2*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1
/2)*a^(3/2)*b^2*c^4*d^2-2*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(3/2)*b*c^6*x+2*((a*x+b)*x)^(1/2)*((a*d-
b*c)/c^2*d)^(1/2)*a^(3/2)*b^2*c^6*x^2+a^(3/2)*b^3*c^5*d*x^2*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*(
(a*x+b)*x)^(1/2)*c)/(c*x+d))-6*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(3/2)*b*c^5*d+12*((a*x+b)*x)^(1/2)*
((a*d-b*c)/c^2*d)^(1/2)*a^(7/2)*c^5*d*x^3+2*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*b*c^6*x^3-12*((a
*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*c^5*d*x-32*a^(7/2)*b*c^3*d^3*x^2*ln((-2*a*d*x+b*c*x-b*d+2*((a*d
-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))+2*a^(3/2)*b^3*c^4*d^2*x*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c
^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))+24*a^(9/2)*d^6*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*
x+b)*x)^(1/2)*c)/(c*x+d))+30*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*b*c^4*d^2*x-40*((a*d-b*c)/c^2*d
)^(1/2)*a^3*b*c^3*d^3*x*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))+18*((a*x+b)*x)^(1/2)*((a*d-b*c)/
c^2*d)^(1/2)*a^(5/2)*b*c^5*d*x^2-20*((a*d-b*c)/c^2*d)^(1/2)*a^3*b*c^4*d^2*x^2*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1
/2)*a^(1/2))/a^(1/2))+4*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(3/2)*b^2*c^5*d*x+48*a^(9/2)*c*d^5*x*ln((-
2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))-24*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*
d)^(1/2)*a^(7/2)*c^2*d^4-32*a^(7/2)*b*c*d^5*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)
*c)/(c*x+d))+7*a^(5/2)*b^2*c^2*d^4*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+
d))+24*((a*d-b*c)/c^2*d)^(1/2)*a^4*c*d^5*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))+24*a^(9/2)*c^2*
d^4*x^2*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))-8*((a*x+b)*x)^(3/2)*((a
*d-b*c)/c^2*d)^(1/2)*a^(5/2)*c^4*d^2+a^(3/2)*b^3*c^3*d^3*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*
x+b)*x)^(1/2)*c)/(c*x+d)))*x*((a*x+b)/x)^(1/2)/c^5/((a*d-b*c)/c^2*d)^(1/2)/a^(3/2)/(c*x+d)^2/d^2/((a*x+b)*x)^(
1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{{\left (c + \frac {d}{x}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="maxima")

[Out]

integrate((a + b/x)^(5/2)/(c + d/x)^3, x)

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mupad [B]  time = 3.44, size = 1476, normalized size = 6.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(5/2)/(c + d/x)^3,x)

[Out]

(atan((b^9*(a + b/x)^(1/2)*(a^3)^(1/2)*5i)/(8*((5*a^2*b^9)/8 + (8*a^3*b^8*d)/c - (159*a^4*b^7*d^2)/(8*c^2) + (
45*a^5*b^6*d^3)/(4*c^3))) + (a*b^8*(a + b/x)^(1/2)*(a^3)^(1/2)*8i)/(8*a^3*b^8 + (5*a^2*b^9*c)/(8*d) - (159*a^4
*b^7*d)/(8*c) + (45*a^5*b^6*d^2)/(4*c^2)) - (a^2*b^7*d*(a + b/x)^(1/2)*(a^3)^(1/2)*159i)/(8*(8*a^3*b^8*c - (15
9*a^4*b^7*d)/8 + (5*a^2*b^9*c^2)/(8*d) + (45*a^5*b^6*d^2)/(4*c))) + (a^3*b^6*d^2*(a + b/x)^(1/2)*(a^3)^(1/2)*4
5i)/(4*(8*a^3*b^8*c^2 + (45*a^5*b^6*d^2)/4 + (5*a^2*b^9*c^3)/(8*d) - (159*a^4*b^7*c*d)/8)))*(6*a*d - 5*b*c)*(a
^3)^(1/2)*1i)/c^4 - (((a + b/x)^(3/2)*(b^4*c^3 - 24*a^3*b*d^3 + 32*a^2*b^2*c*d^2 - 9*a*b^3*c^2*d))/(4*c^3*d) -
(b*(a + b/x)^(5/2)*(b^2*c^2 - 12*a^2*d^2 + 7*a*b*c*d))/(4*c^3) + (b*(a + b/x)^(1/2)*(12*a^4*d^3 - a*b^3*c^3 +
14*a^2*b^2*c^2*d - 25*a^3*b*c*d^2))/(4*c^3*d))/((a + b/x)^2*(3*a*d^2 - 2*b*c*d) - (a + b/x)*(3*a^2*d^2 + b^2*
c^2 - 4*a*b*c*d) - d^2*(a + b/x)^3 + a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) + (log(- (5*a^2*b^9*c^6 + 1728*a^8*b^3
*d^6 + 64*a^3*b^8*c^5*d - 4752*a^7*b^4*c*d^5 - 59*a^4*b^7*c^4*d^2 - 1450*a^5*b^6*c^3*d^3 + 4464*a^6*b^5*c^2*d^
4)/(16*c^9*d) - ((((a + b/x)^(1/2)*(b^8*c^6 + 1152*a^6*b^2*d^6 - 2496*a^5*b^3*c*d^5 - 15*a^2*b^6*c^4*d^2 - 400
*a^3*b^5*c^3*d^3 + 1760*a^4*b^4*c^2*d^4 + 14*a*b^7*c^5*d))/(8*c^6*d) - (((16*a*b^5*c^10*d^2 - 208*a^2*b^4*c^9*
d^3 + 192*a^3*b^3*c^8*d^4)/(16*c^9*d) - ((64*b^3*c^9*d^3 - 128*a*b^2*c^8*d^4)*(a + b/x)^(1/2)*(d^3*(a*d - b*c)
)^(1/2)*((b^2*c^2)/8 - 3*a^2*d^2 + a*b*c*d))/(8*c^10*d^4))*(d^3*(a*d - b*c))^(1/2)*((b^2*c^2)/8 - 3*a^2*d^2 +
a*b*c*d))/(c^4*d^3))*(d^3*(a*d - b*c))^(1/2)*((b^2*c^2)/8 - 3*a^2*d^2 + a*b*c*d))/(c^4*d^3))*(d^3*(a*d - b*c))
^(1/2)*((b^2*c^2)/8 - 3*a^2*d^2 + a*b*c*d))/(c^4*d^3) - (log(((((a + b/x)^(1/2)*(b^8*c^6 + 1152*a^6*b^2*d^6 -
2496*a^5*b^3*c*d^5 - 15*a^2*b^6*c^4*d^2 - 400*a^3*b^5*c^3*d^3 + 1760*a^4*b^4*c^2*d^4 + 14*a*b^7*c^5*d))/(8*c^6
*d) + (((16*a*b^5*c^10*d^2 - 208*a^2*b^4*c^9*d^3 + 192*a^3*b^3*c^8*d^4)/(16*c^9*d) + ((64*b^3*c^9*d^3 - 128*a*
b^2*c^8*d^4)*(a + b/x)^(1/2)*(d^3*(a*d - b*c))^(1/2)*(b^2*c^2 - 24*a^2*d^2 + 8*a*b*c*d))/(64*c^10*d^4))*(d^3*(
a*d - b*c))^(1/2)*(b^2*c^2 - 24*a^2*d^2 + 8*a*b*c*d))/(8*c^4*d^3))*(d^3*(a*d - b*c))^(1/2)*(b^2*c^2 - 24*a^2*d
^2 + 8*a*b*c*d))/(8*c^4*d^3) - (5*a^2*b^9*c^6 + 1728*a^8*b^3*d^6 + 64*a^3*b^8*c^5*d - 4752*a^7*b^4*c*d^5 - 59*
a^4*b^7*c^4*d^2 - 1450*a^5*b^6*c^3*d^3 + 4464*a^6*b^5*c^2*d^4)/(16*c^9*d))*(d^3*(a*d - b*c))^(1/2)*(b^2*c^2 -
24*a^2*d^2 + 8*a*b*c*d))/(8*c^4*d^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(5/2)/(c+d/x)**3,x)

[Out]

Timed out

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