\(\int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx\) [315]

Optimal result

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

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

Mathematica [A] (verified)

Time = 10.25 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\frac {-8 a c^{9/2} (a+b x)^{7/2}-2 c^{7/2} (b c-9 a d) x (a+b x)^{7/2}-\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) x^2 \left (3 c^{5/2} (a+b x)^{5/2}-5 (b c-a d) x \left (\sqrt {c} \sqrt {a+b x} (4 a c+b c x+3 a d x)-3 a^{3/2} (c+d x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{24 a^2 c^{11/2} x^3 (c+d x)^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {109, 27, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 = 0.25 (sec) , antiderivative size = 1009, normalized size of antiderivative = 3.63

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

Input:

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

Output:

1/48*(b*x+a)^(1/2)*(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^3*d^5*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^2*b*c*d^4*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*b^2*c^2*d^3*x^5-15*ln((a*d*x+b*c*x+2*(a*c) 
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*d^2*x^5+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^3*c*d^4*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*c^2*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* 
b^2*c^3*d^2*x^4-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2 
*a*c)/x)*b^3*c^4*d*x^4+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^3*c^2*d^3*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^2*b*c^3*d^2*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*b^2*c^4*d*x^3-15*ln((a*d*x+b*c*x+ 
2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^5*x^3-630*(a*c)^(1/2 
)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^4*x^4+840*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 
/2)*a*b*c*d^3*x^4-226*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*d^2*x^4- 
840*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d^3*x^3+1148*(a*c)^(1/2)*((b 
*x+a)*(d*x+c))^(1/2)*a*b*c^2*d^2*x^3-324*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/ 
2)*b^2*c^3*d*x^3-126*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*d^2*x^2+1 
92*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^3*d*x^2-66*(a*c)^(1/2)*((b...
 

Fricas [A] (verification not implemented)

Time = 3.85 (sec) , antiderivative size = 842, normalized size of antiderivative = 3.03 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(5/2)/x^4/(d*x+c)^(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. 2354 vs. \(2 (234) = 468\).

Time = 3.65 (sec) , antiderivative size = 2354, normalized size of antiderivative = 8.47 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*sqrt(b*x + a)*((5*b^6*c^8*d^3*abs(b) - 22*a*b^5*c^7*d^4*abs(b) + 29*a 
^2*b^4*c^6*d^5*abs(b) - 12*a^3*b^3*c^5*d^6*abs(b))*(b*x + a)/(b^3*c^11*d - 
 a*b^2*c^10*d^2) + 6*(b^7*c^9*d^2*abs(b) - 5*a*b^6*c^8*d^3*abs(b) + 9*a^2* 
b^5*c^7*d^4*abs(b) - 7*a^3*b^4*c^6*d^5*abs(b) + 2*a^4*b^3*c^5*d^6*abs(b))/ 
(b^3*c^11*d - a*b^2*c^10*d^2))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5/8 
*(sqrt(b*d)*b^5*c^3 - 15*sqrt(b*d)*a*b^4*c^2*d + 35*sqrt(b*d)*a^2*b^3*c*d^ 
2 - 21*sqrt(b*d)*a^3*b^2*d^3)*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))/(s 
qrt(-a*b*c*d)*b*c^5*abs(b)) - 1/12*(33*sqrt(b*d)*b^15*c^8 - 346*sqrt(b*d)* 
a*b^14*c^7*d + 1506*sqrt(b*d)*a^2*b^13*c^6*d^2 - 3618*sqrt(b*d)*a^3*b^12*c 
^5*d^3 + 5300*sqrt(b*d)*a^4*b^11*c^4*d^4 - 4878*sqrt(b*d)*a^5*b^10*c^3*d^5 
 + 2766*sqrt(b*d)*a^6*b^9*c^2*d^6 - 886*sqrt(b*d)*a^7*b^8*c*d^7 + 123*sqrt 
(b*d)*a^8*b^7*d^8 - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d))^2*b^13*c^7 + 1287*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + 
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^12*c^6*d - 3441*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^11 
*c^5*d^2 + 3579*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^10*c^4*d^3 + 129*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^9*c^3*d^4 - 3291*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...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 5.73 (sec) , antiderivative size = 3222, normalized size of antiderivative = 11.59 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 48*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**5*d + 108*sqrt(c + d*x)*sqrt(a 
+ b*x)*a**4*c**4*d**2*x - 378*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**3*d**3*x 
**2 - 2520*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**2*d**4*x**3 - 1890*sqrt(c + 
 d*x)*sqrt(a + b*x)*a**4*c*d**5*x**4 - 16*sqrt(c + d*x)*sqrt(a + b*x)*a**3 
*b*c**6 - 120*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*c**5*d*x + 450*sqrt(c + d 
*x)*sqrt(a + b*x)*a**3*b*c**4*d**2*x**2 + 2604*sqrt(c + d*x)*sqrt(a + b*x) 
*a**3*b*c**3*d**3*x**3 + 1890*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*c**2*d**4 
*x**4 - 52*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*c**6*x - 6*sqrt(c + d*x)* 
sqrt(a + b*x)*a**2*b**2*c**5*d*x**2 + 176*sqrt(c + d*x)*sqrt(a + b*x)*a**2 
*b**2*c**4*d**2*x**3 + 162*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*c**3*d**3 
*x**4 - 66*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c**6*x**2 - 324*sqrt(c + d*x 
)*sqrt(a + b*x)*a*b**3*c**5*d*x**3 - 226*sqrt(c + d*x)*sqrt(a + b*x)*a*b** 
3*c**4*d**2*x**4 - 945*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 
**4*c**2*d**4*x**3 - 1890*sqrt(c)*sqrt(a)*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**4*c*d**5*x**4 - 945*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**4*d**6*x**5 + 1260*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)...