\(\int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx\) [281]

Optimal result

Integrand size = 22, antiderivative size = 333 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\frac {(b c-a d)^4 (5 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 a^3 c^4 x}+\frac {(b c-a d)^3 (5 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 a^3 c^3 x^2}+\frac {(b c-a d)^2 (5 b c+7 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 a^3 c^2 x^3}-\frac {(b c-a d) (5 b c+7 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 a^3 c x^4}+\frac {(5 b c+7 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 a^2 c x^5}-\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 a c x^6}+\frac {(b c-a d)^5 (5 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{512 a^{7/2} c^{9/2}} \] Output:

1/512*(-a*d+b*c)^4*(7*a*d+5*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4/x+1/7 
68*(-a*d+b*c)^3*(7*a*d+5*b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/a^3/c^3/x^2+1/96 
0*(-a*d+b*c)^2*(7*a*d+5*b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/a^3/c^2/x^3-1/160 
*(-a*d+b*c)*(7*a*d+5*b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/a^3/c/x^4+1/60*(7*a* 
d+5*b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/a^2/c/x^5-1/6*(b*x+a)^(7/2)*(d*x+c)^( 
5/2)/a/c/x^6+1/512*(-a*d+b*c)^5*(7*a*d+5*b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2 
)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\frac {(-b c+a d)^5 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (75 b^5 c^5 x^5-5 a b^4 c^4 x^4 (10 c+49 d x)+10 a^2 b^3 c^3 x^3 \left (4 c^2+16 c d x+15 d^2 x^2\right )+6 a^3 b^2 c^2 x^2 \left (360 c^3+564 c^2 d x+58 c d^2 x^2-91 d^3 x^3\right )+a^4 b c x \left (3200 c^4+4448 c^3 d x+216 c^2 d^2 x^2-272 c d^3 x^3+415 d^4 x^4\right )+a^5 \left (1280 c^5+1664 c^4 d x+48 c^3 d^2 x^2-56 c^2 d^3 x^3+70 c d^4 x^4-105 d^5 x^5\right )\right )}{(b c-a d)^5 x^6}-15 (5 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{7680 a^{7/2} c^{9/2}} \] Input:

Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^7,x]
 

Output:

((-(b*c) + a*d)^5*((Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(75*b^5*c^ 
5*x^5 - 5*a*b^4*c^4*x^4*(10*c + 49*d*x) + 10*a^2*b^3*c^3*x^3*(4*c^2 + 16*c 
*d*x + 15*d^2*x^2) + 6*a^3*b^2*c^2*x^2*(360*c^3 + 564*c^2*d*x + 58*c*d^2*x 
^2 - 91*d^3*x^3) + a^4*b*c*x*(3200*c^4 + 4448*c^3*d*x + 216*c^2*d^2*x^2 - 
272*c*d^3*x^3 + 415*d^4*x^4) + a^5*(1280*c^5 + 1664*c^4*d*x + 48*c^3*d^2*x 
^2 - 56*c^2*d^3*x^3 + 70*c*d^4*x^4 - 105*d^5*x^5)))/((b*c - a*d)^5*x^6) - 
15*(5*b*c + 7*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x]) 
]))/(7680*a^(7/2)*c^(9/2))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {107, 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.

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^7,x]
 

Output:

-1/6*((a + b*x)^(7/2)*(c + d*x)^(5/2))/(a*c*x^6) - ((5*b*c + 7*a*d)*(-1/5* 
((a + b*x)^(5/2)*(c + d*x)^(5/2))/(c*x^5) + ((b*c - a*d)*(-1/4*((a + b*x)^ 
(3/2)*(c + d*x)^(5/2))/(c*x^4) + (3*(b*c - a*d)*(-1/3*(Sqrt[a + b*x]*(c + 
d*x)^(5/2))/(c*x^3) + ((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)*Sq 
rt[c])))/(4*a)))/(6*c)))/(8*c)))/(2*c)))/(12*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. \(1067\) vs. \(2(283)=566\).

Time = 0.23 (sec) , antiderivative size = 1068, normalized size of antiderivative = 3.21

method	result	size
default	\(\text {Expression too large to display}\)	\(1068\)

Input:

int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(8896*((b*x+a)*(d*x+c))^(1/2) 
*(a*c)^(1/2)*a^4*b*c^4*d*x^2+300*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b 
^3*c^3*d^2*x^5-490*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^4*c^4*d*x^5+432 
0*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c^5*x^2+3328*((b*x+a)*(d*x+c 
))^(1/2)*(a*c)^(1/2)*a^5*c^4*d*x+6400*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)* 
a^4*b*c^5*x-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^5*b*c*d^5*x^6+675*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^2*c^2*d^4*x^6-300*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^3*c^3*d^3*x^6-225*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^2*b^4*c^4*d^2*x^6+270*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^5*c^5*d*x^6+ 
140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c*d^4*x^4-100*((b*x+a)*(d*x+c) 
)^(1/2)*(a*c)^(1/2)*a*b^4*c^5*x^4-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)* 
a^5*c^2*d^3*x^3+80*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^5*x^3+96* 
((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^5*c^3*d^2*x^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^6*d^6*x^6-75*ln((a*d*x+b*c 
*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^6*c^6*x^6+2560*((b*x+ 
a)*(d*x+c))^(1/2)*a^5*c^5*(a*c)^(1/2)-544*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1 
/2)*a^4*b*c^2*d^3*x^4+696*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b^2*c^3* 
d^2*x^4+320*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^4*d*x^4+432*(...
 

Fricas [A] (verification not implemented)

Time = 8.09 (sec) , antiderivative size = 920, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^7,x, algorithm="fricas")
 

Output:

[-1/30720*(15*(5*b^6*c^6 - 18*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^ 
3*c^3*d^3 - 45*a^4*b^2*c^2*d^4 + 30*a^5*b*c*d^5 - 7*a^6*d^6)*sqrt(a*c)*x^6 
*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*(1280*a^6*c^6 + (75*a*b^5*c^6 - 245*a^2*b^4*c^5*d + 150*a^3*b^3*c^ 
4*d^2 - 546*a^4*b^2*c^3*d^3 + 415*a^5*b*c^2*d^4 - 105*a^6*c*d^5)*x^5 - 2*( 
25*a^2*b^4*c^6 - 80*a^3*b^3*c^5*d - 174*a^4*b^2*c^4*d^2 + 136*a^5*b*c^3*d^ 
3 - 35*a^6*c^2*d^4)*x^4 + 8*(5*a^3*b^3*c^6 + 423*a^4*b^2*c^5*d + 27*a^5*b* 
c^4*d^2 - 7*a^6*c^3*d^3)*x^3 + 16*(135*a^4*b^2*c^6 + 278*a^5*b*c^5*d + 3*a 
^6*c^4*d^2)*x^2 + 128*(25*a^5*b*c^6 + 13*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt( 
d*x + c))/(a^4*c^5*x^6), -1/15360*(15*(5*b^6*c^6 - 18*a*b^5*c^5*d + 15*a^2 
*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 - 45*a^4*b^2*c^2*d^4 + 30*a^5*b*c*d^5 - 
7*a^6*d^6)*sqrt(-a*c)*x^6*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sq 
rt(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*(1280*a^6*c^6 + (75*a*b^5*c^6 - 245*a^2*b^4*c^5*d + 150*a^3*b^3*c^4*d 
^2 - 546*a^4*b^2*c^3*d^3 + 415*a^5*b*c^2*d^4 - 105*a^6*c*d^5)*x^5 - 2*(25* 
a^2*b^4*c^6 - 80*a^3*b^3*c^5*d - 174*a^4*b^2*c^4*d^2 + 136*a^5*b*c^3*d^3 - 
 35*a^6*c^2*d^4)*x^4 + 8*(5*a^3*b^3*c^6 + 423*a^4*b^2*c^5*d + 27*a^5*b*c^4 
*d^2 - 7*a^6*c^3*d^3)*x^3 + 16*(135*a^4*b^2*c^6 + 278*a^5*b*c^5*d + 3*a^6* 
c^4*d^2)*x^2 + 128*(25*a^5*b*c^6 + 13*a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(...
 

Sympy [F]

\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{7}}\, dx \] Input:

integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**7,x)
 

Output:

Integral((a + b*x)**(5/2)*(c + d*x)**(3/2)/x**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^7,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. 8502 vs. \(2 (283) = 566\).

Time = 3.12 (sec) , antiderivative size = 8502, normalized size of antiderivative = 25.53 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^7,x, algorithm="giac")
 

Output:

1/7680*(15*(5*sqrt(b*d)*b^7*c^6*abs(b) - 18*sqrt(b*d)*a*b^6*c^5*d*abs(b) + 
 15*sqrt(b*d)*a^2*b^5*c^4*d^2*abs(b) + 20*sqrt(b*d)*a^3*b^4*c^3*d^3*abs(b) 
 - 45*sqrt(b*d)*a^4*b^3*c^2*d^4*abs(b) + 30*sqrt(b*d)*a^5*b^2*c*d^5*abs(b) 
 - 7*sqrt(b*d)*a^6*b*d^6*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*s 
qrt(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*c^4) - 2*(75*sqrt(b*d)*b^29*c^17*abs(b) - 1145*sqrt 
(b*d)*a*b^28*c^16*d*abs(b) + 8040*sqrt(b*d)*a^2*b^27*c^15*d^2*abs(b) - 350 
16*sqrt(b*d)*a^3*b^26*c^14*d^3*abs(b) + 107892*sqrt(b*d)*a^4*b^25*c^13*d^4 
*abs(b) - 254796*sqrt(b*d)*a^5*b^24*c^12*d^5*abs(b) + 486360*sqrt(b*d)*a^6 
*b^23*c^11*d^6*abs(b) - 773080*sqrt(b*d)*a^7*b^22*c^10*d^7*abs(b) + 103072 
2*sqrt(b*d)*a^8*b^21*c^9*d^8*abs(b) - 1141734*sqrt(b*d)*a^9*b^20*c^8*d^9*a 
bs(b) + 1032152*sqrt(b*d)*a^10*b^19*c^7*d^10*abs(b) - 746040*sqrt(b*d)*a^1 
1*b^18*c^6*d^11*abs(b) + 421620*sqrt(b*d)*a^12*b^17*c^5*d^12*abs(b) - 1813 
56*sqrt(b*d)*a^13*b^16*c^4*d^13*abs(b) + 57192*sqrt(b*d)*a^14*b^15*c^3*d^1 
4*abs(b) - 12456*sqrt(b*d)*a^15*b^14*c^2*d^15*abs(b) + 1675*sqrt(b*d)*a^16 
*b^13*c*d^16*abs(b) - 105*sqrt(b*d)*a^17*b^12*d^17*abs(b) - 825*sqrt(b*d)* 
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^27*c^1 
6*abs(b) + 10020*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^26*c^15*d*abs(b) - 53460*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^25*c^14*d^2*abs...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^7} \,d x \] Input:

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^7,x)
 

Output:

int(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^7, x)
 

Reduce [F]

\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^7} \, dx=\int \frac {\left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}}}{x^{7}}d x \] Input:

int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^7,x)
 

Output:

int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^7,x)