\(\int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx\) [405]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\frac {2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.12 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.90 \[ \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx=\frac {2 (b c-a d) \sqrt {c+d x} \left (3 a^2 d+3 b^2 c x+4 a b (c+d x)\right )}{3 a^2 b^2 (a+b x)^{3/2}}+\frac {2 i c^{3/2} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{a^{5/2} b}-\frac {2 i c^{3/2} \left (-i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{a^{5/2} b}-\frac {4 d^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b^{5/2}} \] Input:

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

Output:

(2*(b*c - a*d)*Sqrt[c + d*x]*(3*a^2*d + 3*b^2*c*x + 4*a*b*(c + d*x)))/(3*a 
^2*b^2*(a + b*x)^(3/2)) + ((2*I)*c^(3/2)*(I*Sqrt[a]*Sqrt[d] + Sqrt[b*c - a 
*d])*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqr 
t[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqr 
t[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(a^(5/2)*b) - ((2*I)*c 
^(3/2)*((-I)*Sqrt[a]*Sqrt[d] + Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d + (2*I)*S 
qrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*S 
qrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] 
 - Sqrt[a + b*x]))])/(a^(5/2)*b) - (4*d^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d* 
x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/b^(5/2)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {109, 27, 167, 27, 175, 66, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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 167

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[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] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

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

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. \(565\) vs. \(2(123)=246\).

Time = 0.25 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.61

method	result	size
default	\(-\frac {\sqrt {x d +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) \sqrt {d b}\, b^{4} c^{3} x^{2}-3 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b^{2} d^{3} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) \sqrt {d b}\, a \,b^{3} c^{3} x -6 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{3} b \,d^{3} x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) \sqrt {d b}\, a^{2} b^{2} c^{3}-3 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{4} d^{3}+8 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b \,d^{2} x -2 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a \,b^{2} c d x -6 b^{3} c^{2} x \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+6 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} d^{2}+2 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b c d -8 a \,b^{2} c^{2} \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\right )}{3 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}} b^{2} a^{2}}\)	\(566\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (123) = 246\).

Time = 0.92 (sec) , antiderivative size = 1361, normalized size of antiderivative = 8.67 \[ \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6*(3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*sqrt(d/b)*log(8*b^2*d^ 
2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt 
(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + 3*(b^4*c^2* 
x^2 + 2*a*b^3*c^2*x + a^2*b^2*c^2)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6 
*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sq 
rt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(4*a*b^2*c^2 - a 
^2*b*c*d - 3*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(b*x + 
 a)*sqrt(d*x + c))/(a^2*b^4*x^2 + 2*a^3*b^3*x + a^4*b^2), -1/6*(6*(a^2*b^2 
*d^2*x^2 + 2*a^3*b*d^2*x + a^4*d^2)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + 
 a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + 
 a*d^2)*x)) - 3*(b^4*c^2*x^2 + 2*a*b^3*c^2*x + a^2*b^2*c^2)*sqrt(c/a)*log( 
(8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a 
^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/ 
x^2) - 4*(4*a*b^2*c^2 - a^2*b*c*d - 3*a^3*d^2 + (3*b^3*c^2 + a*b^2*c*d - 4 
*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^4*x^2 + 2*a^3*b^3*x + a 
^4*b^2), 1/6*(6*(b^4*c^2*x^2 + 2*a*b^3*c^2*x + a^2*b^2*c^2)*sqrt(-c/a)*arc 
tan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-c/a)/(b* 
c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + 3*(a^2*b^2*d^2*x^2 + 2*a^3*b*d^2*x 
 + a^4*d^2)*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 
4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 645, normalized size of antiderivative = 4.11 \[ \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx=\frac {3 \sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,b^{3} c^{2}+3 \sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{4} c^{2} x +3 \sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,b^{3} c^{2}+3 \sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{4} c^{2} x -3 \sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a \,b^{3} c^{2}-3 \sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b^{4} c^{2} x +6 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} d^{2}+6 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b \,d^{2} x +2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{3} b c d -2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{2} c^{2}+2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{2} c d x -2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{3} c^{2} x -6 \sqrt {d x +c}\, a^{4} b \,d^{2}-2 \sqrt {d x +c}\, a^{3} b^{2} c d -8 \sqrt {d x +c}\, a^{3} b^{2} d^{2} x +8 \sqrt {d x +c}\, a^{2} b^{3} c^{2}+2 \sqrt {d x +c}\, a^{2} b^{3} c d x +6 \sqrt {d x +c}\, a \,b^{4} c^{2} x}{3 \sqrt {b x +a}\, a^{3} b^{3} \left (b x +a \right )} \] Input:

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

Output:

(3*sqrt(c)*sqrt(a)*sqrt(a + b*x)*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**3* 
c**2 + 3*sqrt(c)*sqrt(a)*sqrt(a + b*x)*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 
**4*c**2*x + 3*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2*sqrt(d)*sqrt(c)*sq 
rt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 
)*a*b**3*c**2 + 3*sqrt(c)*sqrt(a)*sqrt(a + b*x)*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**4*c**2*x - 3*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(2*sqrt(d)*sqrt(b)*s 
qrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)* 
a*b**3*c**2 - 3*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(2*sqrt(d)*sqrt(b)*sqrt(c 
 + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*b**4* 
c**2*x + 6*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt 
(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d**2 + 6*sqrt(d)*sqrt(b)*sqrt(a + 
 b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c)) 
*a**3*b*d**2*x + 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b*c*d - 2*sqrt(d)*sq 
rt(b)*sqrt(a + b*x)*a**2*b**2*c**2 + 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2* 
b**2*c*d*x - 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a*b**3*c**2*x - 6*sqrt(c + d* 
x)*a**4*b*d**2 - 2*sqrt(c + d*x)*a**3*b**2*c*d - 8*sqrt(c + d*x)*a**3*b**2 
*d**2*x + 8*sqrt(c + d*x)*a**2*b**3*c**2 + 2*sqrt(c + d*x)*a**2*b**3*c*...