\(\int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx\) [291]

Optimal result

Integrand size = 22, antiderivative size = 369 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 a^4 c^4 x}-\frac {5 (b c-a d)^4 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{1536 a^3 c^4 x^2}+\frac {(b c-a d)^3 (b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{384 a^2 c^4 x^3}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{7/2}}{64 a c^4 x^4}+\frac {(b c-a d) (b c+a d) (a+b x)^{3/2} (c+d x)^{7/2}}{24 a c^3 x^5}+\frac {(b c+a d) (a+b x)^{5/2} (c+d x)^{7/2}}{12 a c^2 x^6}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}-\frac {5 (b c-a d)^6 (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{1024 a^{9/2} c^{9/2}} \] Output:

5/1024*(-a*d+b*c)^5*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^4/x-5/1536 
*(-a*d+b*c)^4*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(3/2)/a^3/c^4/x^2+1/384*(-a* 
d+b*c)^3*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(5/2)/a^2/c^4/x^3+1/64*(-a*d+b*c) 
^2*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(7/2)/a/c^4/x^4+1/24*(-a*d+b*c)*(a*d+b* 
c)*(b*x+a)^(3/2)*(d*x+c)^(7/2)/a/c^3/x^5+1/12*(a*d+b*c)*(b*x+a)^(5/2)*(d*x 
+c)^(7/2)/a/c^2/x^6-1/7*(b*x+a)^(7/2)*(d*x+c)^(7/2)/a/c/x^7-5/1024*(-a*d+b 
*c)^6*(a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/ 
2)/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\frac {(b c-a d)^6 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (-105 b c^7 (a+b x)^6-105 a c^6 d (a+b x)^6+700 a b c^6 (a+b x)^5 (c+d x)+700 a^2 c^5 d (a+b x)^5 (c+d x)-1981 a^2 b c^5 (a+b x)^4 (c+d x)^2-1981 a^3 c^4 d (a+b x)^4 (c+d x)^2+3072 a^3 b c^4 (a+b x)^3 (c+d x)^3-3072 a^4 c^3 d (a+b x)^3 (c+d x)^3+1981 a^4 b c^3 (a+b x)^2 (c+d x)^4+1981 a^5 c^2 d (a+b x)^2 (c+d x)^4-700 a^5 b c^2 (a+b x) (c+d x)^5-700 a^6 c d (a+b x) (c+d x)^5+105 a^6 b c (c+d x)^6+105 a^7 d (c+d x)^6\right )}{(-b c x+a d x)^7}-105 (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{21504 a^{9/2} c^{9/2}} \] Input:

Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^8,x]
 

Output:

((b*c - a*d)^6*((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b*c^7*( 
a + b*x)^6 - 105*a*c^6*d*(a + b*x)^6 + 700*a*b*c^6*(a + b*x)^5*(c + d*x) + 
 700*a^2*c^5*d*(a + b*x)^5*(c + d*x) - 1981*a^2*b*c^5*(a + b*x)^4*(c + d*x 
)^2 - 1981*a^3*c^4*d*(a + b*x)^4*(c + d*x)^2 + 3072*a^3*b*c^4*(a + b*x)^3* 
(c + d*x)^3 - 3072*a^4*c^3*d*(a + b*x)^3*(c + d*x)^3 + 1981*a^4*b*c^3*(a + 
 b*x)^2*(c + d*x)^4 + 1981*a^5*c^2*d*(a + b*x)^2*(c + d*x)^4 - 700*a^5*b*c 
^2*(a + b*x)*(c + d*x)^5 - 700*a^6*c*d*(a + b*x)*(c + d*x)^5 + 105*a^6*b*c 
*(c + d*x)^6 + 105*a^7*d*(c + d*x)^6))/(-(b*c*x) + a*d*x)^7 - 105*(b*c + a 
*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])]))/(21504*a^(9 
/2)*c^(9/2))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {107, 105, 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.

\(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx\)

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x^6}dx}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5}dx}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}}dx}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 c x^5}\right )}{12 c}-\frac {(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}\right )}{2 a c}-\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 a c x^7}\)

Input:

Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^8,x]
 

Output:

-1/7*((a + b*x)^(7/2)*(c + d*x)^(7/2))/(a*c*x^7) - ((b*c + a*d)*(-1/6*((a 
+ b*x)^(5/2)*(c + d*x)^(7/2))/(c*x^6) + (5*(b*c - a*d)*(-1/5*((a + b*x)^(3 
/2)*(c + d*x)^(7/2))/(c*x^5) + (3*(b*c - a*d)*(-1/4*(Sqrt[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)))/(12*c)))/(2*a*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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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. \(1357\) vs. \(2(313)=626\).

Time = 1.16 (sec) , antiderivative size = 1358, normalized size of antiderivative = 3.68

method	result	size
default	\(\text {Expression too large to display}\)	\(1358\)

Input:

int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-1/43008*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^4*(600*((b*x+a)*(d*x+c))^(1/2)* 
(a*c)^(1/2)*a^3*b^3*c^3*d^3*x^6-1582*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a 
^2*b^4*c^4*d^2*x^6+980*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^5*c^5*d*x^6 
-210*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^6*d^6*x^6-210*((b*x+a)*(d*x+c)) 
^(1/2)*(a*c)^(1/2)*b^6*c^6*x^6+6144*((b*x+a)*(d*x+c))^(1/2)*a^6*c^6*(a*c)^ 
(1/2)+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)* 
a^7*d^7*x^7+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^7*c^7*x^7-644*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*b*c^2*d^4*x^ 
5+1016*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^2*c^3*d^3*x^5+1016*(a*c)^ 
(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^3*c^4*d^2*x^5-644*(a*c)^(1/2)*((b*x+a) 
*(d*x+c))^(1/2)*a^2*b^4*c^5*d*x^5+512*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)* 
a^5*b*c^3*d^3*x^4+19680*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^2*c^4*d^ 
2*x^4+512*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^3*c^5*d*x^4+25504*(a*c 
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*b*c^4*d^2*x^3+25504*(a*c)^(1/2)*((b*x+ 
a)*(d*x+c))^(1/2)*a^4*b^2*c^5*d*x^3+37376*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 
/2)*a^5*b*c^5*d*x^2-112*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^4*c^6*x^ 
4+96*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^6*c^3*d^3*x^3+96*(a*c)^(1/2)*(( 
b*x+a)*(d*x+c))^(1/2)*a^3*b^3*c^6*x^3+9472*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 
1/2)*a^6*c^4*d^2*x^2+9472*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^2*c^6* 
x^2+14848*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^6*c^5*d*x+14848*(a*c)^(...
 

Fricas [A] (verification not implemented)

Time = 15.57 (sec) , antiderivative size = 1130, normalized size of antiderivative = 3.06 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^8,x, algorithm="fricas")
 

Output:

[1/86016*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4 
*d^3 - 5*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*sq 
rt(a*c)*x^7*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*(3072*a^7*c^7 - (105*a*b^6*c^7 - 490*a^2*b^5*c^6*d + 79 
1*a^3*b^4*c^5*d^2 - 300*a^4*b^3*c^4*d^3 + 791*a^5*b^2*c^3*d^4 - 490*a^6*b* 
c^2*d^5 + 105*a^7*c*d^6)*x^6 + 2*(35*a^2*b^5*c^7 - 161*a^3*b^4*c^6*d + 254 
*a^4*b^3*c^5*d^2 + 254*a^5*b^2*c^4*d^3 - 161*a^6*b*c^3*d^4 + 35*a^7*c^2*d^ 
5)*x^5 - 8*(7*a^3*b^4*c^7 - 32*a^4*b^3*c^6*d - 1230*a^5*b^2*c^5*d^2 - 32*a 
^6*b*c^4*d^3 + 7*a^7*c^3*d^4)*x^4 + 16*(3*a^4*b^3*c^7 + 797*a^5*b^2*c^6*d 
+ 797*a^6*b*c^5*d^2 + 3*a^7*c^4*d^3)*x^3 + 128*(37*a^5*b^2*c^7 + 146*a^6*b 
*c^6*d + 37*a^7*c^5*d^2)*x^2 + 7424*(a^6*b*c^7 + a^7*c^6*d)*x)*sqrt(b*x + 
a)*sqrt(d*x + c))/(a^5*c^5*x^7), 1/43008*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9 
*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d 
^5 - 5*a^6*b*c*d^6 + a^7*d^7)*sqrt(-a*c)*x^7*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*(3072*a^7*c^7 - (105*a*b^6*c^7 - 490*a^2*b^5*c^6*d 
 + 791*a^3*b^4*c^5*d^2 - 300*a^4*b^3*c^4*d^3 + 791*a^5*b^2*c^3*d^4 - 490*a 
^6*b*c^2*d^5 + 105*a^7*c*d^6)*x^6 + 2*(35*a^2*b^5*c^7 - 161*a^3*b^4*c^6*d 
+ 254*a^4*b^3*c^5*d^2 + 254*a^5*b^2*c^4*d^3 - 161*a^6*b*c^3*d^4 + 35*a^...
 

Sympy [F]

\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{8}}\, dx \] Input:

integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**8,x)
 

Output:

Integral((a + b*x)**(5/2)*(c + d*x)**(5/2)/x**8, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^8,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. 11549 vs. \(2 (313) = 626\).

Time = 60.51 (sec) , antiderivative size = 11549, normalized size of antiderivative = 31.30 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^8,x, algorithm="giac")
 

Output:

-1/21504*b^7*(105*(sqrt(b*d)*b^7*c^7*abs(b) - 5*sqrt(b*d)*a*b^6*c^6*d*abs( 
b) + 9*sqrt(b*d)*a^2*b^5*c^5*d^2*abs(b) - 5*sqrt(b*d)*a^3*b^4*c^4*d^3*abs( 
b) - 5*sqrt(b*d)*a^4*b^3*c^3*d^4*abs(b) + 9*sqrt(b*d)*a^5*b^2*c^2*d^5*abs( 
b) - 5*sqrt(b*d)*a^6*b*c*d^6*abs(b) + sqrt(b*d)*a^7*d^7*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^4*b^8*c^4) - 2*(105*sqrt 
(b*d)*b^33*c^20*abs(b) - 1960*sqrt(b*d)*a*b^32*c^19*d*abs(b) + 17206*sqrt( 
b*d)*a^2*b^31*c^18*d^2*abs(b) - 94184*sqrt(b*d)*a^3*b^30*c^17*d^3*abs(b) + 
 360437*sqrt(b*d)*a^4*b^29*c^16*d^4*abs(b) - 1027488*sqrt(b*d)*a^5*b^28*c^ 
15*d^5*abs(b) + 2276232*sqrt(b*d)*a^6*b^27*c^14*d^6*abs(b) - 4049696*sqrt( 
b*d)*a^7*b^26*c^13*d^7*abs(b) + 5952674*sqrt(b*d)*a^8*b^25*c^12*d^8*abs(b) 
 - 7409584*sqrt(b*d)*a^9*b^24*c^11*d^9*abs(b) + 7952516*sqrt(b*d)*a^10*b^2 
3*c^10*d^10*abs(b) - 7409584*sqrt(b*d)*a^11*b^22*c^9*d^11*abs(b) + 5952674 
*sqrt(b*d)*a^12*b^21*c^8*d^12*abs(b) - 4049696*sqrt(b*d)*a^13*b^20*c^7*d^1 
3*abs(b) + 2276232*sqrt(b*d)*a^14*b^19*c^6*d^14*abs(b) - 1027488*sqrt(b*d) 
*a^15*b^18*c^5*d^15*abs(b) + 360437*sqrt(b*d)*a^16*b^17*c^4*d^16*abs(b) - 
94184*sqrt(b*d)*a^17*b^16*c^3*d^17*abs(b) + 17206*sqrt(b*d)*a^18*b^15*c^2* 
d^18*abs(b) - 1960*sqrt(b*d)*a^19*b^14*c*d^19*abs(b) + 105*sqrt(b*d)*a^20* 
b^13*d^20*abs(b) - 1365*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d))^2*b^31*c^19*abs(b) + 21245*sqrt(b*d)*(sqrt(b*d)...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^8} \,d x \] Input:

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^8,x)
 

Output:

int(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^8} \, dx=\int \frac {\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {5}{2}}}{x^{8}}d x \] Input:

int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^8,x)
 

Output:

int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^8,x)