\(\int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx\) [240]

Optimal result

Integrand size = 22, antiderivative size = 242 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=-\frac {3 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^2 x}+\frac {(b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^2 x^2}-\frac {(b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/2}}{80 a c^2 x^3}-\frac {3 (b c-a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}+\frac {3 (b c-a d)^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{5/2}} \] Output:

-3/128*(-a*d+b*c)^4*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^2/x+1/64*(-a*d+b*c)^ 
3*(b*x+a)^(1/2)*(d*x+c)^(3/2)/a^2/c^2/x^2-1/80*(-a*d+b*c)^2*(b*x+a)^(1/2)* 
(d*x+c)^(5/2)/a/c^2/x^3-3/40*(-a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(7/2)/c^2/x^ 
4-1/5*(b*x+a)^(3/2)*(d*x+c)^(7/2)/c/x^5+3/128*(-a*d+b*c)^5*arctanh(c^(1/2) 
*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(5/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=\frac {(-b c+a d)^5 \left (\frac {\sqrt {a} \sqrt {c} (a+b x)^{9/2} \sqrt {c+d x} \left (-15 c^4+\frac {70 a c^3 (c+d x)}{a+b x}-\frac {128 a^2 c^2 (c+d x)^2}{(a+b x)^2}-\frac {70 a^3 c (c+d x)^3}{(a+b x)^3}+\frac {15 a^4 (c+d x)^4}{(a+b x)^4}\right )}{(-b c x+a d x)^5}-15 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{640 a^{7/2} c^{5/2}} \] Input:

Integrate[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^6,x]
 

Output:

((-(b*c) + a*d)^5*((Sqrt[a]*Sqrt[c]*(a + b*x)^(9/2)*Sqrt[c + d*x]*(-15*c^4 
 + (70*a*c^3*(c + d*x))/(a + b*x) - (128*a^2*c^2*(c + d*x)^2)/(a + b*x)^2 
- (70*a^3*c*(c + d*x)^3)/(a + b*x)^3 + (15*a^4*(c + d*x)^4)/(a + b*x)^4))/ 
(-(b*c*x) + a*d*x)^5 - 15*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a 
+ b*x])]))/(640*a^(7/2)*c^(5/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {105, 105, 105, 105, 105, 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.

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

\(\Big \downarrow \) 105

\(\displaystyle \frac {3 (b c-a d) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5}dx}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}}dx}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\)

Input:

Int[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^6,x]
 

Output:

-1/5*((a + b*x)^(3/2)*(c + d*x)^(7/2))/(c*x^5) + (3*(b*c - a*d)*(-1/4*(Sqr 
t[a + b*x]*(c + d*x)^(7/2))/(c*x^4) + ((b*c - a*d)*(-1/3*(Sqrt[a + b*x]*(c 
 + d*x)^(5/2))/(a*x^3) - (5*(b*c - a*d)*(-1/2*(Sqrt[a + b*x]*(c + d*x)^(3/ 
2))/(a*x^2) - (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + ((b 
*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/ 
2)*Sqrt[c])))/(4*a)))/(6*a)))/(8*c)))/(10*c)
 

Defintions of rubi rules used

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[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. \(812\) vs. \(2(198)=396\).

Time = 0.24 (sec) , antiderivative size = 813, normalized size of antiderivative = 3.36

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (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 ) a^{5} d^{5} x^{5}-75 \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} b c \,d^{4} x^{5}+150 \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^{2} c^{2} d^{3} x^{5}-150 \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^{3} c^{3} d^{2} x^{5}+75 \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^{4} c^{4} d \,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^{5} c^{5} x^{5}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{4} d^{4} x^{4}+140 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} b c \,d^{3} x^{4}+256 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}-140 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a \,b^{3} c^{3} d \,x^{4}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{4} c^{4} x^{4}+20 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}+932 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+92 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}-20 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}+496 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+1024 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+16 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+672 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x +352 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x +256 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{1280 a^{3} c^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{5} \sqrt {a c}}\)	\(813\)

Input:

int((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^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^5*d^5*x^5-75*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^4*x^5+150*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^3*x^5-1 
50*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*d^2*x^5+75*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^4*d*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^5*c^5*x^5-30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*d^4* 
x^4+140*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c*d^3*x^4+256*(a*c)^(1/2 
)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^2*c^2*d^2*x^4-140*(a*c)^(1/2)*((b*x+a)*(d* 
x+c))^(1/2)*a*b^3*c^3*d*x^4+30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^4*c^4 
*x^4+20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3+932*((b*x+a)*(d* 
x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3+92*((b*x+a)*(d*x+c))^(1/2)*(a*c) 
^(1/2)*a^2*b^2*c^3*d*x^3-20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^3*c^4* 
x^3+496*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2+1024*((b*x+a)* 
(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2+16*((b*x+a)*(d*x+c))^(1/2)*(a*c 
)^(1/2)*a^2*b^2*c^4*x^2+672*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^3*d* 
x+352*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^4*x+256*((b*x+a)*(d*x+c) 
)^(1/2)*a^4*c^4*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^5/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 2.66 (sec) , antiderivative size = 728, normalized size of antiderivative = 3.01 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=\left [-\frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d + 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 23 \, a^{3} b^{2} c^{4} d - 233 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + 31 \, a^{5} c^{3} d^{2}\right )} x^{2} + 16 \, {\left (11 \, a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, a^{4} c^{3} x^{5}}, -\frac {15 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d + 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 23 \, a^{3} b^{2} c^{4} d - 233 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + 31 \, a^{5} c^{3} d^{2}\right )} x^{2} + 16 \, {\left (11 \, a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, a^{4} c^{3} x^{5}}\right ] \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^6,x, algorithm="fricas")
 

Output:

[-1/2560*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^ 
2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(a*c)*x^5*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*(128*a^5*c^5 + (15*a 
*b^4*c^5 - 70*a^2*b^3*c^4*d + 128*a^3*b^2*c^3*d^2 + 70*a^4*b*c^2*d^3 - 15* 
a^5*c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 23*a^3*b^2*c^4*d - 233*a^4*b*c^3*d^2 - 
 5*a^5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^5 + 64*a^4*b*c^4*d + 31*a^5*c^3*d^2)*x^ 
2 + 16*(11*a^4*b*c^5 + 21*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4* 
c^3*x^5), -1/1280*(15*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a 
^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(-a*c)*x^5*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*(128*a^5*c^5 + (15*a*b^4*c^5 - 70*a^2* 
b^3*c^4*d + 128*a^3*b^2*c^3*d^2 + 70*a^4*b*c^2*d^3 - 15*a^5*c*d^4)*x^4 - 2 
*(5*a^2*b^3*c^5 - 23*a^3*b^2*c^4*d - 233*a^4*b*c^3*d^2 - 5*a^5*c^2*d^3)*x^ 
3 + 8*(a^3*b^2*c^5 + 64*a^4*b*c^4*d + 31*a^5*c^3*d^2)*x^2 + 16*(11*a^4*b*c 
^5 + 21*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^5)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{6}}\, dx \] Input:

integrate((b*x+a)**(3/2)*(d*x+c)**(5/2)/x**6,x)
 

Output:

Integral((a + b*x)**(3/2)*(c + d*x)**(5/2)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^6,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. 5922 vs. \(2 (198) = 396\).

Time = 9.91 (sec) , antiderivative size = 5922, normalized size of antiderivative = 24.47 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^6,x, algorithm="giac")
 

Output:

1/640*b^5*(15*(sqrt(b*d)*b^5*c^5*abs(b) - 5*sqrt(b*d)*a*b^4*c^4*d*abs(b) + 
 10*sqrt(b*d)*a^2*b^3*c^3*d^2*abs(b) - 10*sqrt(b*d)*a^3*b^2*c^2*d^3*abs(b) 
 + 5*sqrt(b*d)*a^4*b*c*d^4*abs(b) - sqrt(b*d)*a^5*d^5*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^3*b^6*c^2) - 2*(15*sqrt(b* 
d)*b^23*c^14*abs(b) - 220*sqrt(b*d)*a*b^22*c^13*d*abs(b) + 1503*sqrt(b*d)* 
a^2*b^21*c^12*d^2*abs(b) - 6160*sqrt(b*d)*a^3*b^20*c^11*d^3*abs(b) + 16595 
*sqrt(b*d)*a^4*b^19*c^10*d^4*abs(b) - 30540*sqrt(b*d)*a^5*b^18*c^9*d^5*abs 
(b) + 38595*sqrt(b*d)*a^6*b^17*c^8*d^6*abs(b) - 32256*sqrt(b*d)*a^7*b^16*c 
^7*d^7*abs(b) + 15165*sqrt(b*d)*a^8*b^15*c^6*d^8*abs(b) - 180*sqrt(b*d)*a^ 
9*b^14*c^5*d^9*abs(b) - 5075*sqrt(b*d)*a^10*b^13*c^4*d^10*abs(b) + 3600*sq 
rt(b*d)*a^11*b^12*c^3*d^11*abs(b) - 1247*sqrt(b*d)*a^12*b^11*c^2*d^12*abs( 
b) + 220*sqrt(b*d)*a^13*b^10*c*d^13*abs(b) - 15*sqrt(b*d)*a^14*b^9*d^14*ab 
s(b) - 135*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d 
 - a*b*d))^2*b^21*c^13*abs(b) + 1555*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^20*c^12*d*abs(b) - 8010*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 
^19*c^11*d^2*abs(b) + 23450*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^18*c^10*d^3*abs(b) - 41765*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^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x^6} \,d x \] Input:

int(((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^6,x)
 

Output:

int(((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 13.24 (sec) , antiderivative size = 1643, normalized size of antiderivative = 6.79 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^6,x)
 

Output:

( - 256*sqrt(c + d*x)*sqrt(a + b*x)*a**6*c**5*d - 672*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**6*c**4*d**2*x - 496*sqrt(c + d*x)*sqrt(a + b*x)*a**6*c**3*d**3* 
x**2 - 20*sqrt(c + d*x)*sqrt(a + b*x)*a**6*c**2*d**4*x**3 + 30*sqrt(c + d* 
x)*sqrt(a + b*x)*a**6*c*d**5*x**4 - 256*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b 
*c**6 - 1024*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b*c**5*d*x - 1520*sqrt(c + d 
*x)*sqrt(a + b*x)*a**5*b*c**4*d**2*x**2 - 952*sqrt(c + d*x)*sqrt(a + b*x)* 
a**5*b*c**3*d**3*x**3 - 110*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b*c**2*d**4*x 
**4 - 352*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**2*c**6*x - 1040*sqrt(c + d*x 
)*sqrt(a + b*x)*a**4*b**2*c**5*d*x**2 - 1024*sqrt(c + d*x)*sqrt(a + b*x)*a 
**4*b**2*c**4*d**2*x**3 - 396*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**2*c**3*d 
**3*x**4 - 16*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**3*c**6*x**2 - 72*sqrt(c 
+ d*x)*sqrt(a + b*x)*a**3*b**3*c**5*d*x**3 - 116*sqrt(c + d*x)*sqrt(a + b* 
x)*a**3*b**3*c**4*d**2*x**4 + 20*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c** 
6*x**3 + 110*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c**5*d*x**4 - 30*sqrt(c 
 + d*x)*sqrt(a + b*x)*a*b**5*c**6*x**4 + 15*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) + sqr 
t(b)*sqrt(c + d*x))*a**6*d**6*x**5 - 60*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*b*c*d**5*x**5 + 75*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...