\(\int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx\) [331]

Optimal result

Integrand size = 22, antiderivative size = 266 \[ \int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}+\frac {\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^4 c^2 x}-\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{5/2}} \] Output:

-1/4*c*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/x^4+1/24*(-9*a*d+7*b*c)*(b*x+a)^(1/2) 
*(d*x+c)^(1/2)/a^2/x^3-1/96*(3*a^2*d^2-46*a*b*c*d+35*b^2*c^2)*(b*x+a)^(1/2 
)*(d*x+c)^(1/2)/a^3/c/x^2+1/192*(9*a^3*d^3+15*a^2*b*c*d^2-145*a*b^2*c^2*d+ 
105*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2/x-1/64*(-a*d+b*c)^2*(3*a^ 
2*d^2+10*a*b*c*d+35*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c) 
^(1/2))/a^(9/2)/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3 x^3-5 a b^2 c^2 x^2 (14 c+29 d x)+a^2 b c x \left (56 c^2+92 c d x+15 d^2 x^2\right )-3 a^3 \left (16 c^3+24 c^2 d x+2 c d^2 x^2-3 d^3 x^3\right )\right )}{192 a^4 c^2 x^4}-\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{5/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^3*c^3*x^3 - 5*a*b^2*c^2*x^2*(14*c + 29 
*d*x) + a^2*b*c*x*(56*c^2 + 92*c*d*x + 15*d^2*x^2) - 3*a^3*(16*c^3 + 24*c^ 
2*d*x + 2*c*d^2*x^2 - 3*d^3*x^3)))/(192*a^4*c^2*x^4) - ((b*c - a*d)^2*(35* 
b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a] 
*Sqrt[c + d*x])])/(64*a^(9/2)*c^(5/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {109, 27, 168, 27, 168, 27, 168, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {\int \frac {c (7 b c-9 a d)+2 d (3 b c-4 a d) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {c (7 b c-9 a d)+2 d (3 b c-4 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {c \left (35 b^2 c^2-46 a b d c+3 a^2 d^2+4 b d (7 b c-9 a d) x\right )}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {35 b^2 c^2-46 a b d c+3 a^2 d^2+4 b d (7 b c-9 a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {105 b^3 c^3-145 a b^2 d c^2+15 a^2 b d^2 c+9 a^3 d^3+2 b d \left (35 b^2 c^2-46 a b d c+3 a^2 d^2\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {35 b^2 c}{a}+\frac {3 a d^2}{c}-46 b d\right )}{2 x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {105 b^3 c^3-145 a b^2 d c^2+15 a^2 b d^2 c+9 a^3 d^3+2 b d \left (35 b^2 c^2-46 a b d c+3 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {35 b^2 c}{a}+\frac {3 a d^2}{c}-46 b d\right )}{2 x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\int \frac {3 (b c-a d)^2 \left (35 b^2 c^2+10 a b d c+3 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {35 b^2 c}{a}+\frac {3 a d^2}{c}-46 b d\right )}{2 x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {35 b^2 c}{a}+\frac {3 a d^2}{c}-46 b d\right )}{2 x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {35 b^2 c}{a}+\frac {3 a d^2}{c}-46 b d\right )}{2 x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {-\frac {\frac {3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {35 b^2 c}{a}+\frac {3 a d^2}{c}-46 b d\right )}{2 x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{3 a x^3}}{8 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}\)

Input:

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

Output:

-1/4*(c*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x^4) - (-1/3*((7*b*c - 9*a*d)*Sqrt 
[a + b*x]*Sqrt[c + d*x])/(a*x^3) - (-1/2*(((35*b^2*c)/a - 46*b*d + (3*a*d^ 
2)/c)*Sqrt[a + b*x]*Sqrt[c + d*x])/x^2 - (-(((105*b^3*c^3 - 145*a*b^2*c^2* 
d + 15*a^2*b*c*d^2 + 9*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (3 
*(b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt 
[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(3/2)))/(4*a*c))/(6*a))/(8 
*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 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 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. \(592\) vs. \(2(228)=456\).

Time = 0.23 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.23

method	result	size
default	\(-\frac {\sqrt {x d +c}\, \sqrt {b x +a}\, \left (9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}+12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}+54 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-180 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-18 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} d^{3} x^{3}-30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}+290 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}-210 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, b^{3} c^{3} x^{3}+12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} c \,d^{2} x^{2}-184 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b \,c^{2} d \,x^{2}+140 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a \,b^{2} c^{3} x^{2}+144 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} c^{2} d x -112 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b \,c^{3} x +96 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{4} c^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{4} \sqrt {a c}}\)	\(593\)

Input:

int((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^4/c^2*(9*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^4*x^4+12*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^3*x^4+54*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^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^3* 
d*x^4+105*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*x^4-18*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*d^3*x^3-30*((b*x+a) 
*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c*d^2*x^3+290*((b*x+a)*(d*x+c))^(1/2)*(a 
*c)^(1/2)*a*b^2*c^2*d*x^3-210*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^3*c^3* 
x^3+12*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c*d^2*x^2-184*(a*c)^(1/2)*( 
(b*x+a)*(d*x+c))^(1/2)*a^2*b*c^2*d*x^2+140*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 
1/2)*a*b^2*c^3*x^2+144*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c^2*d*x-112 
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c^3*x+96*((b*x+a)*(d*x+c))^(1/2 
)*a^3*c^3*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.16 \[ \int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (48 \, a^{4} c^{4} - {\left (105 \, a b^{3} c^{4} - 145 \, a^{2} b^{2} c^{3} d + 15 \, a^{3} b c^{2} d^{2} + 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (35 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (7 \, a^{3} b c^{4} - 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{5} c^{3} x^{4}}, \frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (48 \, a^{4} c^{4} - {\left (105 \, a b^{3} c^{4} - 145 \, a^{2} b^{2} c^{3} d + 15 \, a^{3} b c^{2} d^{2} + 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (35 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (7 \, a^{3} b c^{4} - 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{5} c^{3} x^{4}}\right ] \] Input:

integrate((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/768*(3*(35*b^4*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^ 
3 + 3*a^4*d^4)*sqrt(a*c)*x^4*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^4 - (105*a*b^3*c^4 - 145*a^2 
*b^2*c^3*d + 15*a^3*b*c^2*d^2 + 9*a^4*c*d^3)*x^3 + 2*(35*a^2*b^2*c^4 - 46* 
a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 - 8*(7*a^3*b*c^4 - 9*a^4*c^3*d)*x)*sqrt(b 
*x + a)*sqrt(d*x + c))/(a^5*c^3*x^4), 1/384*(3*(35*b^4*c^4 - 60*a*b^3*c^3* 
d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 - (105*a*b^3*c^4 
 - 145*a^2*b^2*c^3*d + 15*a^3*b*c^2*d^2 + 9*a^4*c*d^3)*x^3 + 2*(35*a^2*b^2 
*c^4 - 46*a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 - 8*(7*a^3*b*c^4 - 9*a^4*c^3*d) 
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^3*x^4)]
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate((d*x+c)**(3/2)/x**5/(b*x+a)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),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. 3834 vs. \(2 (228) = 456\).

Time = 1.46 (sec) , antiderivative size = 3834, normalized size of antiderivative = 14.41 \[ \int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x, algorithm="giac")
 

Output:

-1/192*(3*(35*sqrt(b*d)*b^5*c^4*abs(b) - 60*sqrt(b*d)*a*b^4*c^3*d*abs(b) + 
 18*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) + 4*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) + 
3*sqrt(b*d)*a^4*b*d^4*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))/(s 
qrt(-a*b*c*d)*a^4*b*c^2) - 2*(105*sqrt(b*d)*b^19*c^11*abs(b) - 985*sqrt(b* 
d)*a*b^18*c^10*d*abs(b) + 4115*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 10051*s 
qrt(b*d)*a^3*b^16*c^8*d^3*abs(b) + 15818*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) 
 - 16618*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) + 11606*sqrt(b*d)*a^6*b^13*c^5* 
d^6*abs(b) - 5110*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) + 1181*sqrt(b*d)*a^8*b 
^11*c^3*d^8*abs(b) - 13*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) - 57*sqrt(b*d)*a 
^10*b^9*c*d^10*abs(b) + 9*sqrt(b*d)*a^11*b^8*d^11*abs(b) - 735*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) + 4550*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^16*c^9*d*abs(b) - 10771*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) + 1 
0056*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) + 3602*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) - 17692*sq 
rt(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) + 17490*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq...
 

Mupad [F(-1)]

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

int((c + d*x)^(3/2)/(x^5*(a + b*x)^(1/2)),x)
 

Output:

int((c + d*x)^(3/2)/(x^5*(a + b*x)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 3.24 (sec) , antiderivative size = 1458, normalized size of antiderivative = 5.48 \[ \int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x)
 

Output:

( - 192*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**4*d - 288*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**5*c**3*d**2*x - 24*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**2*d**3*x 
**2 + 36*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c*d**4*x**3 - 192*sqrt(c + d*x)* 
sqrt(a + b*x)*a**4*b*c**5 - 64*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**4*d*x 
 + 344*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**3*d**2*x**2 + 96*sqrt(c + d*x 
)*sqrt(a + b*x)*a**4*b*c**2*d**3*x**3 + 224*sqrt(c + d*x)*sqrt(a + b*x)*a* 
*3*b**2*c**5*x + 88*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**4*d*x**2 - 52 
0*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**3*d**2*x**3 - 280*sqrt(c + d*x) 
*sqrt(a + b*x)*a**2*b**3*c**5*x**2 - 160*sqrt(c + d*x)*sqrt(a + b*x)*a**2* 
b**3*c**4*d*x**3 + 420*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**5*x**3 + 18*s 
qrt(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**5*x**4 + 42*sqrt( 
c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sq 
rt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**4*b*c*d**4*x**4 + 132*sqrt 
(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + s 
qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b**2*c**2*d**3*x**4 - 2 
52*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**2*b**3*c**3*d**2*x 
**4 - 150*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*b**4*c**4...