\(\int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx\) [307]

Optimal result

Integrand size = 22, antiderivative size = 317 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}} \] Output:

-1/192*d*(-945*a^3*d^3+1785*a^2*b*c*d^2-839*a*b^2*c^2*d+15*b^3*c^3)*(b*x+a 
)^(1/2)/a/c^5/(d*x+c)^(1/2)-1/24*a*(-9*a*d+11*b*c)*(b*x+a)^(1/2)/c^2/x^3/( 
d*x+c)^(1/2)-1/96*(-63*a*d+59*b*c)*(-a*d+b*c)*(b*x+a)^(1/2)/c^3/x^2/(d*x+c 
)^(1/2)-1/192*(-a*d+b*c)*(315*a^2*d^2-322*a*b*c*d+15*b^2*c^2)*(b*x+a)^(1/2 
)/a/c^4/x/(d*x+c)^(1/2)-1/4*a*(b*x+a)^(3/2)/c/x^4/(d*x+c)^(1/2)+5/64*(-a*d 
+b*c)^2*(-63*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^(3/2)/c^(11/2)
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (-15 b^3 c^3 x^3 (c+d x)+a b^2 c^2 x^2 \left (-118 c^2+337 c d x+839 d^2 x^2\right )-a^2 b c x \left (136 c^3-244 c^2 d x+637 c d^2 x^2+1785 d^3 x^3\right )+a^3 \left (-48 c^4+72 c^3 d x-126 c^2 d^2 x^2+315 c d^3 x^3+945 d^4 x^4\right )\right )}{192 a c^5 x^4 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}} \] Input:

Integrate[(a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x]
 

Output:

(Sqrt[a + b*x]*(-15*b^3*c^3*x^3*(c + d*x) + a*b^2*c^2*x^2*(-118*c^2 + 337* 
c*d*x + 839*d^2*x^2) - a^2*b*c*x*(136*c^3 - 244*c^2*d*x + 637*c*d^2*x^2 + 
1785*d^3*x^3) + a^3*(-48*c^4 + 72*c^3*d*x - 126*c^2*d^2*x^2 + 315*c*d^3*x^ 
3 + 945*d^4*x^4)))/(192*a*c^5*x^4*Sqrt[c + d*x]) + (5*(b*c - a*d)^2*(b^2*c 
^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqr 
t[c + d*x])])/(64*a^(3/2)*c^(11/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.09, 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, 168, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {a+b x} (a (11 b c-9 a d)+2 b (4 b c-3 a d) x)}{x^4 (c+d x)^{3/2}}dx}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int \frac {a (59 b c-63 a d) (b c-a d)+6 b (8 b c-9 a d) x (b c-a d)}{2 x^3 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {a (59 b c-63 a d) (b c-a d)+6 b (8 b c-9 a d) x (b c-a d)}{x^3 \sqrt {a+b x} (c+d x)^{3/2}}dx}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {-\frac {\int -\frac {a (b c-a d) \left (15 b^2 c^2-322 a b d c+315 a^2 d^2-4 b d (59 b c-63 a d) x\right )}{2 x^2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{2 a c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {(b c-a d) \int \frac {15 b^2 c^2-322 a b d c+315 a^2 d^2-4 b d (59 b c-63 a d) x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {(b c-a d) \left (-\frac {\int \frac {15 (b c-a d) \left (b^2 c^2+14 a b d c-63 a^2 d^2\right )+2 b d \left (15 b^2 c^2-322 a b d c+315 a^2 d^2\right ) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {315 a d^2}{c}-322 b d\right )}{x \sqrt {c+d x}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {(b c-a d) \left (-\frac {\int \frac {15 (b c-a d) \left (b^2 c^2+14 a b d c-63 a^2 d^2\right )+2 b d \left (15 b^2 c^2-322 a b d c+315 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {315 a d^2}{c}-322 b d\right )}{x \sqrt {c+d x}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {(b c-a d) \left (-\frac {\frac {2 d \sqrt {a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {15 (b c-a d)^2 \left (b^2 c^2+14 a b d c-63 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {315 a d^2}{c}-322 b d\right )}{x \sqrt {c+d x}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {(b c-a d) \left (-\frac {\frac {15 (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {315 a d^2}{c}-322 b d\right )}{x \sqrt {c+d x}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {\frac {(b c-a d) \left (-\frac {\frac {30 (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {315 a d^2}{c}-322 b d\right )}{x \sqrt {c+d x}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {(b c-a d) \left (-\frac {\frac {2 d \sqrt {a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {30 (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{2 a c}-\frac {\sqrt {a+b x} \left (\frac {15 b^2 c}{a}+\frac {315 a d^2}{c}-322 b d\right )}{x \sqrt {c+d x}}\right )}{4 c}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{2 c x^2 \sqrt {c+d x}}}{6 c}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{3 c x^3 \sqrt {c+d x}}}{8 c}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}\)

Input:

Int[(a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x]
 

Output:

-1/4*(a*(a + b*x)^(3/2))/(c*x^4*Sqrt[c + d*x]) + (-1/3*(a*(11*b*c - 9*a*d) 
*Sqrt[a + b*x])/(c*x^3*Sqrt[c + d*x]) + (-1/2*((59*b*c - 63*a*d)*(b*c - a* 
d)*Sqrt[a + b*x])/(c*x^2*Sqrt[c + d*x]) + ((b*c - a*d)*(-((((15*b^2*c)/a - 
 322*b*d + (315*a*d^2)/c)*Sqrt[a + b*x])/(x*Sqrt[c + d*x])) - ((2*d*(15*b^ 
3*c^3 - 839*a*b^2*c^2*d + 1785*a^2*b*c*d^2 - 945*a^3*d^3)*Sqrt[a + b*x])/( 
c*(b*c - a*d)*Sqrt[c + d*x]) - (30*(b*c - a*d)*(b^2*c^2 + 14*a*b*c*d - 63* 
a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a 
]*c^(3/2)))/(2*a*c)))/(4*c))/(6*c))/(8*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. \(981\) vs. \(2(273)=546\).

Time = 0.25 (sec) , antiderivative size = 982, normalized size of antiderivative = 3.10

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \left (945 \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^{4} d^{5} x^{5}-2100 \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^{3} b c \,d^{4} x^{5}+1350 \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} b^{2} c^{2} d^{3} x^{5}-180 \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^{3} c^{3} d^{2} x^{5}-15 \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^{4} c^{4} d \,x^{5}+945 \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^{4} c \,d^{4} x^{4}-2100 \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^{3} b \,c^{2} d^{3} x^{4}+1350 \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} b^{2} c^{3} d^{2} x^{4}-180 \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^{3} c^{4} d \,x^{4}-15 \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^{4} c^{5} x^{4}-1890 a^{3} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+3570 a^{2} b c \,d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-1678 a \,b^{2} c^{2} d^{2} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+30 b^{3} c^{3} d \,x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-630 a^{3} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+1274 a^{2} b \,c^{2} d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-674 a \,b^{2} c^{3} d \,x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+30 b^{3} c^{4} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+252 a^{3} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-488 a^{2} b \,c^{3} d \,x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+236 a \,b^{2} c^{4} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}-144 a^{3} c^{3} d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+272 a^{2} b \,c^{4} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}+96 a^{3} c^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\right )}{384 a \,c^{5} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{4} \sqrt {a c}\, \sqrt {x d +c}}\)	\(982\)

Input:

int((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/384*(b*x+a)^(1/2)*(945*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*d^5*x^5-2100*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*c*d^4*x^5+1350*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^2*c^2*d^3*x^5-180*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^3*c^3*d^2*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^4*c^4*d*x^5+9 
45*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*c*d 
^4*x^4-2100*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*c^2*d^3*x^4+1350*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^2*c^3*d^2*x^4-180*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^3*c^4*d*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)*b^4*c^5*x^4-1890*a^3*d^4*x^4*((b*x+a 
)*(d*x+c))^(1/2)*(a*c)^(1/2)+3570*a^2*b*c*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)* 
(a*c)^(1/2)-1678*a*b^2*c^2*d^2*x^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30* 
b^3*c^3*d*x^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-630*a^3*c*d^3*x^3*((b*x+ 
a)*(d*x+c))^(1/2)*(a*c)^(1/2)+1274*a^2*b*c^2*d^2*x^3*((b*x+a)*(d*x+c))^(1/ 
2)*(a*c)^(1/2)-674*a*b^2*c^3*d*x^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30* 
b^3*c^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+252*a^3*c^2*d^2*x^2*((b*x+ 
a)*(d*x+c))^(1/2)*(a*c)^(1/2)-488*a^2*b*c^3*d*x^2*((b*x+a)*(d*x+c))^(1/2)* 
(a*c)^(1/2)+236*a*b^2*c^4*x^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-144*a...
 

Fricas [A] (verification not implemented)

Time = 4.11 (sec) , antiderivative size = 828, normalized size of antiderivative = 2.61 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

[-1/768*(15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^2*c^2*d^3 + 140*a^3* 
b*c*d^4 - 63*a^4*d^5)*x^5 + (b^4*c^5 + 12*a*b^3*c^4*d - 90*a^2*b^2*c^3*d^2 
 + 140*a^3*b*c^2*d^3 - 63*a^4*c*d^4)*x^4)*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*(48*a^4*c^5 + ( 
15*a*b^3*c^4*d - 839*a^2*b^2*c^3*d^2 + 1785*a^3*b*c^2*d^3 - 945*a^4*c*d^4) 
*x^4 + (15*a*b^3*c^5 - 337*a^2*b^2*c^4*d + 637*a^3*b*c^3*d^2 - 315*a^4*c^2 
*d^3)*x^3 + 2*(59*a^2*b^2*c^5 - 122*a^3*b*c^4*d + 63*a^4*c^3*d^2)*x^2 + 8* 
(17*a^3*b*c^5 - 9*a^4*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^6*d*x^ 
5 + a^2*c^7*x^4), -1/384*(15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^2*c 
^2*d^3 + 140*a^3*b*c*d^4 - 63*a^4*d^5)*x^5 + (b^4*c^5 + 12*a*b^3*c^4*d - 9 
0*a^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*d^4)*x^4)*sqrt(-a*c)*arct 
an(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*(48*a^4*c^5 + (15*a*b^3*c 
^4*d - 839*a^2*b^2*c^3*d^2 + 1785*a^3*b*c^2*d^3 - 945*a^4*c*d^4)*x^4 + (15 
*a*b^3*c^5 - 337*a^2*b^2*c^4*d + 637*a^3*b*c^3*d^2 - 315*a^4*c^2*d^3)*x^3 
+ 2*(59*a^2*b^2*c^5 - 122*a^3*b*c^4*d + 63*a^4*c^3*d^2)*x^2 + 8*(17*a^3*b* 
c^5 - 9*a^4*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^6*d*x^5 + a^2*c^ 
7*x^4)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{5} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((b*x+a)**(5/2)/x**5/(d*x+c)**(3/2),x)
 

Output:

Integral((a + b*x)**(5/2)/(x**5*(c + d*x)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),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. 3762 vs. \(2 (273) = 546\).

Time = 7.14 (sec) , antiderivative size = 3762, normalized size of antiderivative = 11.87 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

2*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*sqrt(b*x + a)/(sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d)*c^5*abs(b)) + 5/64*(sqrt(b*d)*b^6*c^4 + 12*sqrt(b*d 
)*a*b^5*c^3*d - 90*sqrt(b*d)*a^2*b^4*c^2*d^2 + 140*sqrt(b*d)*a^3*b^3*c*d^3 
 - 63*sqrt(b*d)*a^4*b^2*d^4)*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))/(sq 
rt(-a*b*c*d)*a*b*c^5*abs(b)) - 1/96*(15*sqrt(b*d)*b^20*c^11 - 575*sqrt(b*d 
)*a*b^19*c^10*d + 5077*sqrt(b*d)*a^2*b^18*c^9*d^2 - 22277*sqrt(b*d)*a^3*b^ 
17*c^8*d^3 + 59494*sqrt(b*d)*a^4*b^16*c^7*d^4 - 105350*sqrt(b*d)*a^5*b^15* 
c^6*d^5 + 128506*sqrt(b*d)*a^6*b^14*c^5*d^6 - 109082*sqrt(b*d)*a^7*b^13*c^ 
4*d^7 + 63547*sqrt(b*d)*a^8*b^12*c^3*d^8 - 24299*sqrt(b*d)*a^9*b^11*c^2*d^ 
9 + 5505*sqrt(b*d)*a^10*b^10*c*d^10 - 561*sqrt(b*d)*a^11*b^9*d^11 - 105*sq 
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2* 
b^18*c^10 + 3946*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^17*c^9*d - 26165*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^16*c^8*d^2 + 77304*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 
^15*c^7*d^3 - 118178*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^14*c^6*d^4 + 80188*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^13*c^5*d^5 + 22494 
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^5\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 7.99 (sec) , antiderivative size = 2722, normalized size of antiderivative = 8.59 \[ \int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x)
 

Output:

( - 864*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**5*d + 1296*sqrt(c + d*x)*sqrt( 
a + b*x)*a**5*c**4*d**2*x - 2268*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**3*d** 
3*x**2 + 5670*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**2*d**4*x**3 + 17010*sqrt 
(c + d*x)*sqrt(a + b*x)*a**5*c*d**5*x**4 - 672*sqrt(c + d*x)*sqrt(a + b*x) 
*a**4*b*c**6 - 1440*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**5*d*x + 2628*sqr 
t(c + d*x)*sqrt(a + b*x)*a**4*b*c**4*d**2*x**2 - 7056*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**4*b*c**3*d**3*x**3 - 18900*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c 
**2*d**4*x**4 - 1904*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**6*x + 1292*s 
qrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**5*d*x**2 - 2852*sqrt(c + d*x)*sqrt 
(a + b*x)*a**3*b**2*c**4*d**2*x**3 - 9888*sqrt(c + d*x)*sqrt(a + b*x)*a**3 
*b**2*c**3*d**3*x**4 - 1652*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c**6*x** 
2 + 4448*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c**5*d*x**3 + 11476*sqrt(c 
+ d*x)*sqrt(a + b*x)*a**2*b**3*c**4*d**2*x**4 - 210*sqrt(c + d*x)*sqrt(a + 
 b*x)*a*b**4*c**6*x**3 - 210*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**5*d*x** 
4 + 8505*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**5*c*d**5*x** 
4 + 8505*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**5*d**6*x**5 
- 12285*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*b*c**2*d...