\(\int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx\) [408]

Optimal result

Integrand size = 22, antiderivative size = 278 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx=-\frac {7 b (15 b c-7 a d) (b c-a d) \sqrt {c+d x}}{24 a^4 (a+b x)^{3/2}}+\frac {3 c (b c-a d) \sqrt {c+d x}}{4 a^2 x^2 (a+b x)^{3/2}}-\frac {(21 b c-11 a d) (b c-a d) \sqrt {c+d x}}{8 a^3 x (a+b x)^{3/2}}-\frac {b \left (315 b^2 c^2-420 a b c d+113 a^2 d^2\right ) \sqrt {c+d x}}{24 a^5 \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}+\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{11/2} \sqrt {c}} \] Output:

-7/24*b*(-7*a*d+15*b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/a^4/(b*x+a)^(3/2)+3/4*c*( 
-a*d+b*c)*(d*x+c)^(1/2)/a^2/x^2/(b*x+a)^(3/2)-1/8*(-11*a*d+21*b*c)*(-a*d+b 
*c)*(d*x+c)^(1/2)/a^3/x/(b*x+a)^(3/2)-1/24*b*(113*a^2*d^2-420*a*b*c*d+315* 
b^2*c^2)*(d*x+c)^(1/2)/a^5/(b*x+a)^(1/2)-1/3*c*(d*x+c)^(3/2)/a/x^3/(b*x+a) 
^(3/2)+5/8*(-a*d+b*c)*(a^2*d^2-14*a*b*c*d+21*b^2*c^2)*arctanh(c^(1/2)*(b*x 
+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(11/2)/c^(1/2)
 

Mathematica [A] (verified)

Time = 10.25 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.72 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx=\frac {-8 a^{9/2} c (c+d x)^{7/2}+2 a^{7/2} (9 b c-a d) x (c+d x)^{7/2}-\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) x^2 \left (3 a^{5/2} (c+d x)^{5/2}+5 (b c-a d) x \left (\sqrt {a} \sqrt {c+d x} (4 a c+3 b c x+a d x)-3 c^{3/2} (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{24 a^{11/2} c^2 x^3 (a+b x)^{3/2}} \] Input:

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

Output:

(-8*a^(9/2)*c*(c + d*x)^(7/2) + 2*a^(7/2)*(9*b*c - a*d)*x*(c + d*x)^(7/2) 
- (21*b^2*c^2 - 14*a*b*c*d + a^2*d^2)*x^2*(3*a^(5/2)*(c + d*x)^(5/2) + 5*( 
b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*(4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2)* 
(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])) 
)/(24*a^(11/2)*c^2*x^3*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {109, 27, 166, 27, 168, 27, 169, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle -\frac {(b c-a d) \left (\frac {\int -\frac {c (21 b c-11 a d)+2 d (9 b c-4 a d) x}{2 x^2 (a+b x)^{5/2} \sqrt {c+d x}}dx}{2 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {\int \frac {c (21 b c-11 a d)+2 d (9 b c-4 a d) x}{x^2 (a+b x)^{5/2} \sqrt {c+d x}}dx}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\int \frac {c \left (5 \left (21 b^2 c^2-14 a b d c+a^2 d^2\right )+4 b d (21 b c-11 a d) x\right )}{2 x (a+b x)^{5/2} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\int \frac {5 \left (21 b^2 c^2-14 a b d c+a^2 d^2\right )+4 b d (21 b c-11 a d) x}{x (a+b x)^{5/2} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {2 \int \frac {(b c-a d) \left (15 \left (21 b^2 c^2-14 a b d c+a^2 d^2\right )+14 b d (15 b c-7 a d) x\right )}{2 x (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a (b c-a d)}+\frac {14 b \sqrt {c+d x} (15 b c-7 a d)}{3 a (a+b x)^{3/2}}}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {\int \frac {15 \left (21 b^2 c^2-14 a b d c+a^2 d^2\right )+14 b d (15 b c-7 a d) x}{x (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a}+\frac {14 b \sqrt {c+d x} (15 b c-7 a d)}{3 a (a+b x)^{3/2}}}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {\frac {2 \int \frac {15 (b c-a d) \left (21 b^2 c^2-14 a b d c+a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}}{3 a}+\frac {14 b \sqrt {c+d x} (15 b c-7 a d)}{3 a (a+b x)^{3/2}}}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {\frac {15 \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}+\frac {2 b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}}{3 a}+\frac {14 b \sqrt {c+d x} (15 b c-7 a d)}{3 a (a+b x)^{3/2}}}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {\frac {30 \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}+\frac {2 b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}}{3 a}+\frac {14 b \sqrt {c+d x} (15 b c-7 a d)}{3 a (a+b x)^{3/2}}}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {(b c-a d) \left (-\frac {-\frac {\frac {\frac {2 b \sqrt {c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {30 \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}}{3 a}+\frac {14 b \sqrt {c+d x} (15 b c-7 a d)}{3 a (a+b x)^{3/2}}}{2 a}-\frac {\sqrt {c+d x} (21 b c-11 a d)}{a x (a+b x)^{3/2}}}{4 a}-\frac {3 c \sqrt {c+d x}}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}}\)

Input:

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

Output:

-1/3*(c*(c + d*x)^(3/2))/(a*x^3*(a + b*x)^(3/2)) - ((b*c - a*d)*((-3*c*Sqr 
t[c + d*x])/(2*a*x^2*(a + b*x)^(3/2)) - (-(((21*b*c - 11*a*d)*Sqrt[c + d*x 
])/(a*x*(a + b*x)^(3/2))) - ((14*b*(15*b*c - 7*a*d)*Sqrt[c + d*x])/(3*a*(a 
 + b*x)^(3/2)) + ((2*b*(315*b^2*c^2 - 420*a*b*c*d + 113*a^2*d^2)*Sqrt[c + 
d*x])/(a*(b*c - a*d)*Sqrt[a + b*x]) - (30*(21*b^2*c^2 - 14*a*b*c*d + a^2*d 
^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sqr 
t[c]))/(3*a))/(2*a))/(4*a)))/(2*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 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 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 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. \(1008\) vs. \(2(234)=468\).

Time = 0.28 (sec) , antiderivative size = 1009, normalized size of antiderivative = 3.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1009\)

Input:

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

Output:

-1/48*(d*x+c)^(1/2)*(15*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^3*b^2*d^3*x^5-225*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*b^3*c*d^2*x^5+525*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^4*c^2*d*x^5-315*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^5*c^3*x^5+30*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^4*b*d^3*x^4-450*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^3*b^2*c*d^2*x 
^4+1050*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*b^3*c^2*d*x^4-630*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^4*c^3*x^4+15*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^5*d^3*x^3-225*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^4*b*c*d^2*x^3+525*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^3*b^2*c^2*d*x^3-315*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*b^3*c^3*x^3+226*(a*c)^(1/ 
2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^2*d^2*x^4-840*(a*c)^(1/2)*((b*x+a)*(d*x+c 
))^(1/2)*a*b^3*c*d*x^4+630*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^4*c^2*x^4 
+324*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*d^2*x^3-1148*(a*c)^(1/2)*(( 
b*x+a)*(d*x+c))^(1/2)*a^2*b^2*c*d*x^3+840*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 
/2)*a*b^3*c^2*x^3+66*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*d^2*x^2-192*( 
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c*d*x^2+126*(a*c)^(1/2)*((b*x+...
 

Fricas [A] (verification not implemented)

Time = 1.93 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.05 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/96*(15*((21*b^5*c^3 - 35*a*b^4*c^2*d + 15*a^2*b^3*c*d^2 - a^3*b^2*d^3) 
*x^5 + 2*(21*a*b^4*c^3 - 35*a^2*b^3*c^2*d + 15*a^3*b^2*c*d^2 - a^4*b*d^3)* 
x^4 + (21*a^2*b^3*c^3 - 35*a^3*b^2*c^2*d + 15*a^4*b*c*d^2 - a^5*d^3)*x^3)* 
sqrt(a*c)*log((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*(8*a^5*c^3 + (315*a*b^4*c^3 - 420*a^2*b^3*c^2*d + 113*a^3 
*b^2*c*d^2)*x^4 + 2*(210*a^2*b^3*c^3 - 287*a^3*b^2*c^2*d + 81*a^4*b*c*d^2) 
*x^3 + 3*(21*a^3*b^2*c^3 - 32*a^4*b*c^2*d + 11*a^5*c*d^2)*x^2 - 2*(9*a^4*b 
*c^3 - 13*a^5*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b^2*c*x^5 + 2*a^ 
7*b*c*x^4 + a^8*c*x^3), -1/48*(15*((21*b^5*c^3 - 35*a*b^4*c^2*d + 15*a^2*b 
^3*c*d^2 - a^3*b^2*d^3)*x^5 + 2*(21*a*b^4*c^3 - 35*a^2*b^3*c^2*d + 15*a^3* 
b^2*c*d^2 - a^4*b*d^3)*x^4 + (21*a^2*b^3*c^3 - 35*a^3*b^2*c^2*d + 15*a^4*b 
*c*d^2 - a^5*d^3)*x^3)*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*(8*a^5*c^3 + (315*a*b^4*c^3 - 420*a^2*b^3*c^2*d + 113*a^3*b^2 
*c*d^2)*x^4 + 2*(210*a^2*b^3*c^3 - 287*a^3*b^2*c^2*d + 81*a^4*b*c*d^2)*x^3 
 + 3*(21*a^3*b^2*c^3 - 32*a^4*b*c^2*d + 11*a^5*c*d^2)*x^2 - 2*(9*a^4*b*c^3 
 - 13*a^5*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^6*b^2*c*x^5 + 2*a^7*b* 
c*x^4 + a^8*c*x^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Leaf count of result is larger than twice the leaf count of optimal. 4178 vs. \(2 (234) = 468\).

Time = 8.28 (sec) , antiderivative size = 4178, normalized size of antiderivative = 15.03 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

5/8*(21*sqrt(b*d)*b^3*c^3*abs(b) - 35*sqrt(b*d)*a*b^2*c^2*d*abs(b) + 15*sq 
rt(b*d)*a^2*b*c*d^2*abs(b) - sqrt(b*d)*a^3*d^3*abs(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)*a^5*b) - 1/12*(315*sqrt(b*d)*b^19*c 
^11*abs(b) - 3255*sqrt(b*d)*a*b^18*c^10*d*abs(b) + 15233*sqrt(b*d)*a^2*b^1 
7*c^9*d^2*abs(b) - 42597*sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b) + 79038*sqrt(b* 
d)*a^4*b^15*c^7*d^4*abs(b) - 102102*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) + 93 
618*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) - 60858*sqrt(b*d)*a^7*b^12*c^4*d^7*a 
bs(b) + 27447*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) - 8163*sqrt(b*d)*a^9*b^10* 
c^2*d^9*abs(b) + 1437*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 113*sqrt(b*d)*a^1 
1*b^8*d^11*abs(b) - 2520*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
 (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) + 19950*sqrt(b*d)*(sqrt(b*d)*s 
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^16*c^9*d*abs(b) 
- 68622*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - 
a*b*d))^2*a^2*b^15*c^8*d^2*abs(b) + 133080*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + 
 a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^14*c^7*d^3*abs(b) - 156 
744*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* 
d))^2*a^4*b^13*c^6*d^4*abs(b) + 109956*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) 
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^12*c^5*d^5*abs(b) - 37380*s 
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 15.13 (sec) , antiderivative size = 2405, normalized size of antiderivative = 8.65 \[ \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sq 
rt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**5*d 
**4*x**3 - 180*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))*a**4*b*c*d**3*x**3 + 15*sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*s 
qrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt 
(b)*sqrt(c + d*x))*a**4*b*d**4*x**4 - 150*sqrt(c)*sqrt(a)*sqrt(a + b*x)*lo 
g( - 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**3*b**2*c**2*d**2*x**3 - 180*sqrt(c)*sqr 
t(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))*a**3*b**2*c*d**3*x**4 
 + 1260*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))*a* 
*2*b**3*c**3*d*x**3 - 150*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))*a**2*b**3*c**2*d**2*x**4 - 945*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)*sq 
rt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*b**4*c**4*x**3 + 1260*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))*a*b**4*c**3*d*x**4 -...