\(\int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx\) [361]

Optimal result

Integrand size = 22, antiderivative size = 124 \[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 d \sqrt {a+b x}}{3 c (b c-a d) (c+d x)^{3/2}}-\frac {2 d (5 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 d \sqrt {a+b x} \left (-6 b c+3 a d+\frac {c d (a+b x)}{c+d x}\right )}{3 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{5/2}} \] Input:

Integrate[1/(x*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
 

Output:

(2*d*Sqrt[a + b*x]*(-6*b*c + 3*a*d + (c*d*(a + b*x))/(c + d*x)))/(3*c^2*(b 
*c - a*d)^2*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S 
qrt[c + d*x])])/(Sqrt[a]*c^(5/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {115, 27, 169, 27, 104, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 115

\(\displaystyle -\frac {2 \int -\frac {3 (b c-a d)-2 b d x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 (b c-a d)-2 b d x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {-\frac {2 \int -\frac {3 (b c-a d)^2}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}-\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 (b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}-\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {6 (b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}-\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {6 (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}-\frac {2 d \sqrt {a+b x} (5 b c-3 a d)}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}-\frac {2 d \sqrt {a+b x}}{3 c (c+d x)^{3/2} (b c-a d)}\)

Input:

Int[1/(x*Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
 

Output:

(-2*d*Sqrt[a + b*x])/(3*c*(b*c - a*d)*(c + d*x)^(3/2)) + ((-2*d*(5*b*c - 3 
*a*d)*Sqrt[a + b*x])/(c*(b*c - a*d)*Sqrt[c + d*x]) - (6*(b*c - a*d)*ArcTan 
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(3/2)))/(3* 
c*(b*c - a*d))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 221

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs. \(2(100)=200\).

Time = 0.28 (sec) , antiderivative size = 586, normalized size of antiderivative = 4.73

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} d^{4} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b c \,d^{3} x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{2} c^{2} d^{2} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{3} x -12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b \,c^{2} d^{2} x +6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{2} c^{3} d x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} c^{2} d^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b \,c^{3} d +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{2} c^{4}-6 a \,d^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b c \,d^{2} x -8 a c \,d^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b \,c^{2} d \right )}{3 \left (a d -b c \right )^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \left (x d +c \right )^{\frac {3}{2}} c^{2}}\)	\(586\)

Input:

int(1/x/(b*x+a)^(1/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(b*x+a)^(1/2)*(3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2 
)+2*a*c)/x)*a^2*d^4*x^2-6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^ 
(1/2)+2*a*c)/x)*a*b*c*d^3*x^2+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d* 
x+c))^(1/2)+2*a*c)/x)*b^2*c^2*d^2*x^2+6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b* 
x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*c*d^3*x-12*ln((a*d*x+b*c*x+2*(a*c)^(1/2) 
*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b*c^2*d^2*x+6*ln((a*d*x+b*c*x+2*(a*c) 
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^2*c^3*d*x+3*ln((a*d*x+b*c*x+2*( 
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*c^2*d^2-6*ln((a*d*x+b*c*x 
+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*b*c^3*d+3*ln((a*d*x+b*c 
*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^2*c^4-6*a*d^3*x*(a*c) 
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c* 
d^2*x-8*a*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*(a*c)^(1/2)*((b*x+a 
)*(d*x+c))^(1/2)*b*c^2*d)/(a*d-b*c)^2/(a*c)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/ 
(d*x+c)^(3/2)/c^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (100) = 200\).

Time = 0.33 (sec) , antiderivative size = 678, normalized size of antiderivative = 5.47 \[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + {\left (5 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + {\left (5 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{2} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x\right )}}\right ] \] Input:

integrate(1/x/(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 
+ a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x)*sqrt(a*c)*lo 
g((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d 
)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) 
 - 4*(6*a*b*c^3*d - 4*a^2*c^2*d^2 + (5*a*b*c^2*d^2 - 3*a^2*c*d^3)*x)*sqrt( 
b*x + a)*sqrt(d*x + c))/(a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2 + (a*b^2* 
c^5*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^2 + 2*(a*b^2*c^6*d - 2*a^2*b*c^ 
5*d^2 + a^3*c^4*d^3)*x), 1/3*(3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^ 
2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^ 
2*c*d^3)*x)*sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt( 
b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 
2*(6*a*b*c^3*d - 4*a^2*c^2*d^2 + (5*a*b*c^2*d^2 - 3*a^2*c*d^3)*x)*sqrt(b*x 
 + a)*sqrt(d*x + c))/(a*b^2*c^7 - 2*a^2*b*c^6*d + a^3*c^5*d^2 + (a*b^2*c^5 
*d^2 - 2*a^2*b*c^4*d^3 + a^3*c^3*d^4)*x^2 + 2*(a*b^2*c^6*d - 2*a^2*b*c^5*d 
^2 + a^3*c^4*d^3)*x)]
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {1}{x \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/x/(b*x+a)**(1/2)/(d*x+c)**(5/2),x)
 

Output:

Integral(1/(x*sqrt(a + b*x)*(c + d*x)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate(1/x/(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (100) = 200\).

Time = 0.16 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.19 \[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (5 \, b^{4} c^{3} d^{3} {\left | b \right |} - 3 \, a b^{3} c^{2} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + a^{2} b^{2} c^{4} d^{3}} + \frac {3 \, {\left (2 \, b^{5} c^{4} d^{2} {\left | b \right |} - 3 \, a b^{4} c^{3} d^{3} {\left | b \right |} + a^{2} b^{3} c^{2} d^{4} {\left | b \right |}\right )}}{b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + a^{2} b^{2} c^{4} d^{3}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, \sqrt {b d} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} \] Input:

integrate(1/x/(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

-2/3*sqrt(b*x + a)*((5*b^4*c^3*d^3*abs(b) - 3*a*b^3*c^2*d^4*abs(b))*(b*x + 
 a)/(b^4*c^6*d - 2*a*b^3*c^5*d^2 + a^2*b^2*c^4*d^3) + 3*(2*b^5*c^4*d^2*abs 
(b) - 3*a*b^4*c^3*d^3*abs(b) + a^2*b^3*c^2*d^4*abs(b))/(b^4*c^6*d - 2*a*b^ 
3*c^5*d^2 + a^2*b^2*c^4*d^3))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 2*sq 
rt(b*d)*b*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2 
*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*c^2*ab 
s(b))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(1/(x*(a + b*x)^(1/2)*(c + d*x)^(5/2)),x)
 

Output:

int(1/(x*(a + b*x)^(1/2)*(c + d*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 1683, normalized size of antiderivative = 13.57 \[ \int \frac {1}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/x/(b*x+a)^(1/2)/(d*x+c)^(5/2),x)
 

Output:

(8*sqrt(c + d*x)*sqrt(a + b*x)*a**2*c**2*d**2 + 6*sqrt(c + d*x)*sqrt(a + b 
*x)*a**2*c*d**3*x - 12*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c**3*d - 10*sqrt(c 
+ d*x)*sqrt(a + b*x)*a*b*c**2*d**2*x + 3*sqrt(c)*sqrt(a)*log( - sqrt(2*sqr 
t(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b 
)*sqrt(c + d*x))*a**2*c**2*d**2 + 6*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)* 
sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqr 
t(c + d*x))*a**2*c*d**3*x + 3*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c 
)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + 
d*x))*a**2*d**4*x**2 - 6*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqr 
t(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x)) 
*a*b*c**3*d - 12*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqr 
t(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b*c** 
2*d**2*x - 6*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) 
 + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b*c*d**3* 
x**2 + 3*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a 
*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**2*c**4 + 6*s 
qrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) 
+ sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**2*c**3*d*x + 3*sqrt(c) 
*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt 
(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**2*c**2*d**2*x**2 + 3*sqrt...