\(\int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx\) [139]

Optimal result

Integrand size = 20, antiderivative size = 284 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=-\frac {b \left (32 b^2 c^2-52 a b c d+19 a^2 d^2\right ) \sqrt {c+d x}}{8 a^4 (a+b x)}+\frac {c (8 b c-9 a d) \sqrt {c+d x}}{12 a^2 x^2 (a+b x)}-\frac {\left (48 b^2 c^2-82 a b c d+33 a^2 d^2\right ) \sqrt {c+d x}}{24 a^3 x (a+b x)}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}+\frac {\left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{8 a^5 \sqrt {c}}-\frac {\sqrt {b} (8 b c-3 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^5} \] Output:

-1/8*b*(19*a^2*d^2-52*a*b*c*d+32*b^2*c^2)*(d*x+c)^(1/2)/a^4/(b*x+a)+1/12*c 
*(-9*a*d+8*b*c)*(d*x+c)^(1/2)/a^2/x^2/(b*x+a)-1/24*(33*a^2*d^2-82*a*b*c*d+ 
48*b^2*c^2)*(d*x+c)^(1/2)/a^3/x/(b*x+a)-1/3*c*(d*x+c)^(3/2)/a/x^3/(b*x+a)+ 
1/8*(-5*a^3*d^3+60*a^2*b*c*d^2-120*a*b^2*c^2*d+64*b^3*c^3)*arctanh((d*x+c) 
^(1/2)/c^(1/2))/a^5/c^(1/2)-b^(1/2)*(-3*a*d+8*b*c)*(-a*d+b*c)^(3/2)*arctan 
h(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/a^5
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=\frac {-\frac {a \sqrt {c+d x} \left (96 b^3 c^2 x^3+12 a b^2 c x^2 (4 c-13 d x)+a^3 \left (8 c^2+26 c d x+33 d^2 x^2\right )+a^2 b x \left (-16 c^2-82 c d x+57 d^2 x^2\right )\right )}{x^3 (a+b x)}+24 \sqrt {b} (8 b c-3 a d) (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )+\frac {3 \left (64 b^3 c^3-120 a b^2 c^2 d+60 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{24 a^5} \] Input:

Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]
 

Output:

(-((a*Sqrt[c + d*x]*(96*b^3*c^2*x^3 + 12*a*b^2*c*x^2*(4*c - 13*d*x) + a^3* 
(8*c^2 + 26*c*d*x + 33*d^2*x^2) + a^2*b*x*(-16*c^2 - 82*c*d*x + 57*d^2*x^2 
)))/(x^3*(a + b*x))) + 24*Sqrt[b]*(8*b*c - 3*a*d)*(-(b*c) + a*d)^(3/2)*Arc 
Tan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]] + (3*(64*b^3*c^3 - 120*a*b 
^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/Sqr 
t[c])/(24*a^5)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 174, 73, 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.

\(\displaystyle \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (8 b c-9 a d)+d (5 b c-6 a d) x)}{2 x^3 (a+b x)^2}dx}{3 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sqrt {c+d x} (c (8 b c-9 a d)+d (5 b c-6 a d) x)}{x^3 (a+b x)^2}dx}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {\frac {\int -\frac {c \left (48 b^2 c^2-82 a b d c+33 a^2 d^2\right )+d \left (40 b^2 c^2-65 a b d c+24 a^2 d^2\right ) x}{2 x^2 (a+b x)^2 \sqrt {c+d x}}dx}{2 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {c \left (48 b^2 c^2-82 a b d c+33 a^2 d^2\right )+d \left (40 b^2 c^2-65 a b d c+24 a^2 d^2\right ) x}{x^2 (a+b x)^2 \sqrt {c+d x}}dx}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {3 c \left (64 b^3 c^3-120 a b^2 d c^2+60 a^2 b d^2 c-5 a^3 d^3+b d \left (48 b^2 c^2-82 a b d c+33 a^2 d^2\right ) x\right )}{2 x (a+b x)^2 \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {3 \int \frac {64 b^3 c^3-120 a b^2 d c^2+60 a^2 b d^2 c-5 a^3 d^3+b d \left (48 b^2 c^2-82 a b d c+33 a^2 d^2\right ) x}{x (a+b x)^2 \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {-\frac {3 \left (\frac {\int \frac {(b c-a d) \left (64 b^3 c^3-120 a b^2 d c^2+60 a^2 b d^2 c-5 a^3 d^3+b d \left (32 b^2 c^2-52 a b d c+19 a^2 d^2\right ) x\right )}{x (a+b x) \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{a (a+b x)}\right )}{2 a}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {3 \left (\frac {\int \frac {64 b^3 c^3-120 a b^2 d c^2+60 a^2 b d^2 c-5 a^3 d^3+b d \left (32 b^2 c^2-52 a b d c+19 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}}dx}{a}+\frac {2 b \sqrt {c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{a (a+b x)}\right )}{2 a}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {-\frac {-\frac {3 \left (\frac {\frac {\left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \int \frac {1}{x \sqrt {c+d x}}dx}{a}-\frac {8 b (8 b c-3 a d) (b c-a d)^2 \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{a}}{a}+\frac {2 b \sqrt {c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{a (a+b x)}\right )}{2 a}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {-\frac {-\frac {3 \left (\frac {\frac {2 \left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \int \frac {1}{\frac {c+d x}{d}-\frac {c}{d}}d\sqrt {c+d x}}{a d}-\frac {16 b (8 b c-3 a d) (b c-a d)^2 \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{a d}}{a}+\frac {2 b \sqrt {c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{a (a+b x)}\right )}{2 a}-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {-\frac {\sqrt {c+d x} \left (33 a^2 d^2-82 a b c d+48 b^2 c^2\right )}{a x (a+b x)}-\frac {3 \left (\frac {2 b \sqrt {c+d x} \left (19 a^2 d^2-52 a b c d+32 b^2 c^2\right )}{a (a+b x)}+\frac {\frac {16 \sqrt {b} (8 b c-3 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a}-\frac {2 \left (-5 a^3 d^3+60 a^2 b c d^2-120 a b^2 c^2 d+64 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}}{a}\right )}{2 a}}{4 a}-\frac {c \sqrt {c+d x} (8 b c-9 a d)}{2 a x^2 (a+b x)}}{6 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)}\)

Input:

Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^2),x]
 

Output:

-1/3*(c*(c + d*x)^(3/2))/(a*x^3*(a + b*x)) - (-1/2*(c*(8*b*c - 9*a*d)*Sqrt 
[c + d*x])/(a*x^2*(a + b*x)) - (-(((48*b^2*c^2 - 82*a*b*c*d + 33*a^2*d^2)* 
Sqrt[c + d*x])/(a*x*(a + b*x))) - (3*((2*b*(32*b^2*c^2 - 52*a*b*c*d + 19*a 
^2*d^2)*Sqrt[c + d*x])/(a*(a + b*x)) + ((-2*(64*b^3*c^3 - 120*a*b^2*c^2*d 
+ 60*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a*Sqrt[c]) 
+ (16*Sqrt[b]*(8*b*c - 3*a*d)*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + 
d*x])/Sqrt[b*c - a*d]])/a)/a))/(2*a))/(4*a))/(6*a)
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 166

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] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168

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] && ILtQ[m, -1]
 

rule 174

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

Time = 0.62 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {8 \left (b x +a \right ) \left (-a d +b c \right )^{2} \sqrt {c}\, b \left (b c -\frac {3 a d}{8}\right ) x^{3} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\left (\frac {15 \left (b x +a \right ) \left (a^{3} d^{3}-12 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -\frac {64}{5} b^{3} c^{3}\right ) x^{3} \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{8}+a \sqrt {c}\, \left (12 x^{3} b^{3} c^{2}+6 a c \left (-\frac {13 x d}{4}+c \right ) x^{2} b^{2}-2 a^{2} \left (-\frac {57}{16} d^{2} x^{2}+\frac {41}{8} c d x +c^{2}\right ) x b +a^{3} \left (c^{2}+\frac {13}{4} c d x +\frac {33}{8} d^{2} x^{2}\right )\right ) \sqrt {x d +c}\right ) \sqrt {\left (a d -b c \right ) b}}{3}}{a^{5} \left (b x +a \right ) \sqrt {\left (a d -b c \right ) b}\, x^{3} \sqrt {c}}\)	\(243\)
risch	\(-\frac {\sqrt {x d +c}\, \left (33 a^{2} d^{2} x^{2}-108 a b c d \,x^{2}+72 b^{2} c^{2} x^{2}+26 a^{2} c d x -24 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 a^{4} x^{3}}-\frac {d \left (\frac {16 b \left (\frac {\left (\frac {1}{2} a^{3} d^{3}-a^{2} b c \,d^{2}+\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {\left (3 a^{3} d^{3}-14 a^{2} b c \,d^{2}+19 a \,b^{2} c^{2} d -8 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a d}-\frac {\left (-5 a^{3} d^{3}+60 a^{2} b c \,d^{2}-120 a \,b^{2} c^{2} d +64 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}\right )}{8 a^{4}}\)	\(277\)
derivativedivides	\(2 d^{5} \left (-\frac {\left (a d -b c \right )^{2} b \left (\frac {\sqrt {x d +c}\, a d}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -8 b c \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{5} d^{5}}+\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{4} a^{2} b c \,d^{2}+\frac {3}{2} a \,b^{2} c^{2} d \right ) \left (x d +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+4 a^{2} d^{2} b \,c^{2}-3 a \,b^{2} c^{3} d \right ) \left (x d +c \right )^{\frac {3}{2}}+\left (-\frac {7}{4} a^{2} b \,c^{3} d^{2}+\frac {3}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {x d +c}}{x^{3} d^{3}}-\frac {\left (5 a^{3} d^{3}-60 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{5} d^{5}}\right )\)	\(290\)
default	\(2 d^{5} \left (-\frac {\left (a d -b c \right )^{2} b \left (\frac {\sqrt {x d +c}\, a d}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -8 b c \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{5} d^{5}}+\frac {-\frac {\left (\frac {11}{16} a^{3} d^{3}-\frac {9}{4} a^{2} b c \,d^{2}+\frac {3}{2} a \,b^{2} c^{2} d \right ) \left (x d +c \right )^{\frac {5}{2}}+\left (-\frac {5}{6} a^{3} c \,d^{3}+4 a^{2} d^{2} b \,c^{2}-3 a \,b^{2} c^{3} d \right ) \left (x d +c \right )^{\frac {3}{2}}+\left (-\frac {7}{4} a^{2} b \,c^{3} d^{2}+\frac {3}{2} a \,b^{2} c^{4} d +\frac {5}{16} a^{3} c^{2} d^{3}\right ) \sqrt {x d +c}}{x^{3} d^{3}}-\frac {\left (5 a^{3} d^{3}-60 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{a^{5} d^{5}}\right )\)	\(290\)

Input:

int((d*x+c)^(5/2)/x^4/(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

8/((a*d-b*c)*b)^(1/2)*((b*x+a)*(-a*d+b*c)^2*c^(1/2)*b*(b*c-3/8*a*d)*x^3*ar 
ctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-1/24*(15/8*(b*x+a)*(a^3*d^3-12*a 
^2*b*c*d^2+24*a*b^2*c^2*d-64/5*b^3*c^3)*x^3*arctanh((d*x+c)^(1/2)/c^(1/2)) 
+a*c^(1/2)*(12*x^3*b^3*c^2+6*a*c*(-13/4*x*d+c)*x^2*b^2-2*a^2*(-57/16*d^2*x 
^2+41/8*c*d*x+c^2)*x*b+a^3*(c^2+13/4*c*d*x+33/8*d^2*x^2))*(d*x+c)^(1/2))*( 
(a*d-b*c)*b)^(1/2))/c^(1/2)/a^5/(b*x+a)/x^3
 

Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1476, normalized size of antiderivative = 5.20 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="fricas")
 

Output:

[1/48*(24*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)*x^4 + (8*a*b^2*c^3 
 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b 
*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 3*((64*b^4*c^ 
3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 
- 120*a^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(c)*log((d*x - 
2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(8*a^4*c^3 + 3*(32*a*b^3*c^3 - 52*a^ 
2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33* 
a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b*c 
*x^4 + a^6*c*x^3), 1/48*(48*((8*b^3*c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)* 
x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2*d + 3*a^3*c*d^2)*x^3)*sqrt(-b^2*c + a*b* 
d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) - 3*((64*b^4*c 
^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 
 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d^2 - 5*a^4*d^3)*x^3)*sqrt(c)*log((d*x - 
 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(8*a^4*c^3 + 3*(32*a*b^3*c^3 - 52*a 
^2*b^2*c^2*d + 19*a^3*b*c*d^2)*x^3 + (48*a^2*b^2*c^3 - 82*a^3*b*c^2*d + 33 
*a^4*c*d^2)*x^2 - 2*(8*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(d*x + c))/(a^5*b* 
c*x^4 + a^6*c*x^3), -1/24*(3*((64*b^4*c^3 - 120*a*b^3*c^2*d + 60*a^2*b^2*c 
*d^2 - 5*a^3*b*d^3)*x^4 + (64*a*b^3*c^3 - 120*a^2*b^2*c^2*d + 60*a^3*b*c*d 
^2 - 5*a^4*d^3)*x^3)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x + c)) - 12*((8*b^3* 
c^3 - 11*a*b^2*c^2*d + 3*a^2*b*c*d^2)*x^4 + (8*a*b^2*c^3 - 11*a^2*b*c^2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(5/2)/x**4/(b*x+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^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 [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.30 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=\frac {{\left (8 \, b^{4} c^{3} - 19 \, a b^{3} c^{2} d + 14 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{5}} - \frac {{\left (64 \, b^{3} c^{3} - 120 \, a b^{2} c^{2} d + 60 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{8 \, a^{5} \sqrt {-c}} - \frac {\sqrt {d x + c} b^{3} c^{2} d - 2 \, \sqrt {d x + c} a b^{2} c d^{2} + \sqrt {d x + c} a^{2} b d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a^{4}} - \frac {72 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c^{2} d - 144 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{3} d + 72 \, \sqrt {d x + c} b^{2} c^{4} d - 108 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c d^{2} + 192 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{2} d^{2} - 84 \, \sqrt {d x + c} a b c^{3} d^{2} + 33 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} d^{3} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} c d^{3} + 15 \, \sqrt {d x + c} a^{2} c^{2} d^{3}}{24 \, a^{4} d^{3} x^{3}} \] Input:

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^2,x, algorithm="giac")
 

Output:

(8*b^4*c^3 - 19*a*b^3*c^2*d + 14*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*arctan(sqrt( 
d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^5) - 1/8*(64*b^3* 
c^3 - 120*a*b^2*c^2*d + 60*a^2*b*c*d^2 - 5*a^3*d^3)*arctan(sqrt(d*x + c)/s 
qrt(-c))/(a^5*sqrt(-c)) - (sqrt(d*x + c)*b^3*c^2*d - 2*sqrt(d*x + c)*a*b^2 
*c*d^2 + sqrt(d*x + c)*a^2*b*d^3)/(((d*x + c)*b - b*c + a*d)*a^4) - 1/24*( 
72*(d*x + c)^(5/2)*b^2*c^2*d - 144*(d*x + c)^(3/2)*b^2*c^3*d + 72*sqrt(d*x 
 + c)*b^2*c^4*d - 108*(d*x + c)^(5/2)*a*b*c*d^2 + 192*(d*x + c)^(3/2)*a*b* 
c^2*d^2 - 84*sqrt(d*x + c)*a*b*c^3*d^2 + 33*(d*x + c)^(5/2)*a^2*d^3 - 40*( 
d*x + c)^(3/2)*a^2*c*d^3 + 15*sqrt(d*x + c)*a^2*c^2*d^3)/(a^4*d^3*x^3)
 

Mupad [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 2151, normalized size of antiderivative = 7.57 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^(5/2)/(x^4*(a + b*x)^2),x)
                                                                                    
                                                                                    
 

Output:

(((c + d*x)^(3/2)*(40*a^3*c*d^4 - 288*b^3*c^4*d + 564*a*b^2*c^3*d^2 - 319* 
a^2*b*c^2*d^3))/(24*a^4) - ((c + d*x)^(5/2)*(33*a^3*d^4 - 288*b^3*c^3*d + 
516*a*b^2*c^2*d^2 - 253*a^2*b*c*d^3))/(24*a^4) + ((c + d*x)^(1/2)*(32*b^3* 
c^5*d - 5*a^3*c^2*d^4 - 68*a*b^2*c^4*d^2 + 41*a^2*b*c^3*d^3))/(8*a^4) - (b 
*d*(c + d*x)^(7/2)*(19*a^2*d^2 + 32*b^2*c^2 - 52*a*b*c*d))/(8*a^4))/(b*(c 
+ d*x)^4 - (4*b*c^3 - 3*a*c^2*d)*(c + d*x) + (6*b*c^2 - 3*a*c*d)*(c + d*x) 
^2 + (a*d - 4*b*c)*(c + d*x)^3 + b*c^4 - a*c^3*d) + (atan((((((c + d*x)^(1 
/2)*(601*a^6*b^3*d^8 + 8192*b^9*c^6*d^2 - 34816*a*b^8*c^5*d^3 - 5976*a^5*b 
^4*c*d^7 + 59520*a^2*b^7*c^4*d^4 - 52160*a^3*b^6*c^3*d^5 + 24640*a^4*b^5*c 
^2*d^6))/(32*a^8) - ((((5*a^13*b^2*d^6)/2 - (41*a^12*b^3*c*d^5)/2 - 16*a^1 
0*b^5*c^3*d^3 + 34*a^11*b^4*c^2*d^4)/a^12 - ((256*a^11*b^2*d^3 - 512*a^10* 
b^3*c*d^2)*(c + d*x)^(1/2)*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60* 
a^2*b*c*d^2))/(512*a^13*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2* 
d - 60*a^2*b*c*d^2))/(16*a^5*c^(1/2)))*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2 
*c^2*d - 60*a^2*b*c*d^2)*1i)/(16*a^5*c^(1/2)) + ((((c + d*x)^(1/2)*(601*a^ 
6*b^3*d^8 + 8192*b^9*c^6*d^2 - 34816*a*b^8*c^5*d^3 - 5976*a^5*b^4*c*d^7 + 
59520*a^2*b^7*c^4*d^4 - 52160*a^3*b^6*c^3*d^5 + 24640*a^4*b^5*c^2*d^6))/(3 
2*a^8) + ((((5*a^13*b^2*d^6)/2 - (41*a^12*b^3*c*d^5)/2 - 16*a^10*b^5*c^3*d 
^3 + 34*a^11*b^4*c^2*d^4)/a^12 + ((256*a^11*b^2*d^3 - 512*a^10*b^3*c*d^2)* 
(c + d*x)^(1/2)*(5*a^3*d^3 - 64*b^3*c^3 + 120*a*b^2*c^2*d - 60*a^2*b*c*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 876, normalized size of antiderivative = 3.08 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^2} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(5/2)/x^4/(b*x+a)^2,x)
 

Output:

( - 144*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - 
 b*c)))*a**3*c*d**2*x**3 + 528*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x) 
*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b*c**2*d*x**3 - 144*sqrt(b)*sqrt(a*d - 
 b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b*c*d**2*x**4 
 - 384*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - 
b*c)))*a*b**2*c**3*x**3 + 528*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)* 
b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**2*c**2*d*x**4 - 384*sqrt(b)*sqrt(a*d - 
b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**3*c**3*x**4 - 16 
*sqrt(c + d*x)*a**4*c**3 - 52*sqrt(c + d*x)*a**4*c**2*d*x - 66*sqrt(c + d* 
x)*a**4*c*d**2*x**2 + 32*sqrt(c + d*x)*a**3*b*c**3*x + 164*sqrt(c + d*x)*a 
**3*b*c**2*d*x**2 - 114*sqrt(c + d*x)*a**3*b*c*d**2*x**3 - 96*sqrt(c + d*x 
)*a**2*b**2*c**3*x**2 + 312*sqrt(c + d*x)*a**2*b**2*c**2*d*x**3 - 192*sqrt 
(c + d*x)*a*b**3*c**3*x**3 + 15*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a**4* 
d**3*x**3 - 180*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a**3*b*c*d**2*x**3 + 
15*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a**3*b*d**3*x**4 + 360*sqrt(c)*log 
(sqrt(c + d*x) - sqrt(c))*a**2*b**2*c**2*d*x**3 - 180*sqrt(c)*log(sqrt(c + 
 d*x) - sqrt(c))*a**2*b**2*c*d**2*x**4 - 192*sqrt(c)*log(sqrt(c + d*x) - s 
qrt(c))*a*b**3*c**3*x**3 + 360*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*b**3 
*c**2*d*x**4 - 192*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*b**4*c**3*x**4 - 1 
5*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a**4*d**3*x**3 + 180*sqrt(c)*log...