\(\int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [412]

Optimal result

Integrand size = 22, antiderivative size = 286 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 a^5}{3 b^5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {2 \left (b^5 c^5+a^5 d^5\right ) \sqrt {a+b x}}{3 b^5 d^3 (b c-a d)^3 (c+d x)^{3/2}}-\frac {2 a^4 (5 b c-3 a d)}{b^4 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 \left (7 b^5 c^5-15 a b^4 c^4 d-15 a^4 b c d^4+7 a^5 d^5\right ) \sqrt {a+b x}}{3 b^4 d^3 (b c-a d)^4 \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b^3 d^3}-\frac {5 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2} d^{7/2}} \] Output:

2/3*a^5/b^5/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-2/3*(a^5*d^5+b^5*c^5)*( 
b*x+a)^(1/2)/b^5/d^3/(-a*d+b*c)^3/(d*x+c)^(3/2)-2*a^4*(-3*a*d+5*b*c)/b^4/( 
-a*d+b*c)^3/(b*x+a)^(1/2)/(d*x+c)^(1/2)+2/3*(7*a^5*d^5-15*a^4*b*c*d^4-15*a 
*b^4*c^4*d+7*b^5*c^5)*(b*x+a)^(1/2)/b^4/d^3/(-a*d+b*c)^4/(d*x+c)^(1/2)+(b* 
x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^3-5*(a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2) 
/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.15 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {15 a^6 d^4 (c+d x)^2+20 a^5 b d^3 (-2 c+d x) (c+d x)^2+3 a^4 b^2 d^2 (c+d x)^2 \left (6 c^2-18 c d x+d^2 x^2\right )+b^6 c^4 x^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )+6 a b^5 c^3 x \left (5 c^3-8 c d^2 x^2-2 d^3 x^3\right )-2 a^3 b^3 c d \left (20 c^4+9 c^3 d x-24 c^2 d^2 x^2-6 c d^3 x^3+6 d^4 x^4\right )+3 a^2 b^4 c^2 \left (5 c^4-20 c^3 d x-29 c^2 d^2 x^2+4 c d^3 x^3+6 d^4 x^4\right )}{3 b^3 d^3 (b c-a d)^4 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {5 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{7/2} d^{7/2}} \] Input:

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

Output:

(15*a^6*d^4*(c + d*x)^2 + 20*a^5*b*d^3*(-2*c + d*x)*(c + d*x)^2 + 3*a^4*b^ 
2*d^2*(c + d*x)^2*(6*c^2 - 18*c*d*x + d^2*x^2) + b^6*c^4*x^2*(15*c^2 + 20* 
c*d*x + 3*d^2*x^2) + 6*a*b^5*c^3*x*(5*c^3 - 8*c*d^2*x^2 - 2*d^3*x^3) - 2*a 
^3*b^3*c*d*(20*c^4 + 9*c^3*d*x - 24*c^2*d^2*x^2 - 6*c*d^3*x^3 + 6*d^4*x^4) 
 + 3*a^2*b^4*c^2*(5*c^4 - 20*c^3*d*x - 29*c^2*d^2*x^2 + 4*c*d^3*x^3 + 6*d^ 
4*x^4))/(3*b^3*d^3*(b*c - a*d)^4*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (5*(b* 
c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(b^(7/2 
)*d^(7/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.38, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {109, 27, 167, 27, 167, 27, 160, 66, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 \int \frac {3 x^2 \left (2 a c (11 b c-5 a d)-\left (b^2 c^2-10 a b d c+5 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {3 \int \frac {x^2 \left (2 a c (11 b c-5 a d)-\left (b^2 c^2-10 a b d c+5 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}}dx}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {3 \left (\frac {2 c x^2 \sqrt {a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {2 \int \frac {x \left (4 a c \left (b^2 c^2+12 a b d c-5 a^2 d^2\right )+\left (5 b^3 c^3-9 a b^2 d c^2+35 a^2 b d^2 c-15 a^3 d^3\right ) x\right )}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}\right )}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {3 \left (\frac {2 c x^2 \sqrt {a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\int \frac {x \left (4 a c \left (b^2 c^2+12 a b d c-5 a^2 d^2\right )+\left (5 b^3 c^3-9 a b^2 d c^2+35 a^2 b d^2 c-15 a^3 d^3\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}\right )}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {3 \left (\frac {2 c x^2 \sqrt {a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\sqrt {a+b x} \left (d x (b c-a d) \left (-15 a^3 d^3+35 a^2 b c d^2-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{b d^2 \sqrt {c+d x} (b c-a d)}-\frac {15 (b c-a d)^3 (a d+b c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b d^2}}{3 d (b c-a d)}\right )}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {3 \left (\frac {2 c x^2 \sqrt {a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\sqrt {a+b x} \left (d x (b c-a d) \left (-15 a^3 d^3+35 a^2 b c d^2-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{b d^2 \sqrt {c+d x} (b c-a d)}-\frac {15 (b c-a d)^3 (a d+b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b d^2}}{3 d (b c-a d)}\right )}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {3 \left (\frac {2 c x^2 \sqrt {a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\sqrt {a+b x} \left (d x (b c-a d) \left (-15 a^3 d^3+35 a^2 b c d^2-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{b d^2 \sqrt {c+d x} (b c-a d)}-\frac {15 (b c-a d)^3 (a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}}}{3 d (b c-a d)}\right )}{b (b c-a d)}-\frac {2 a x^3 (11 b c-5 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

Input:

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

Output:

(2*a*x^4)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - ((-2*a*(11*b 
*c - 5*a*d)*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (3*((2*c* 
(b^2*c^2 + 12*a*b*c*d - 5*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c 
+ d*x)^(3/2)) - ((Sqrt[a + b*x]*(c*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b 
^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*a^4*d^4) + d*(b*c - a*d)*(5*b^3*c^3 - 9*a 
*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*x))/(b*d^2*(b*c - a*d)*Sqrt[c + 
d*x]) - (15*(b*c - a*d)^3*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqr 
t[b]*Sqrt[c + d*x])])/(b^(3/2)*d^(5/2)))/(3*d*(b*c - a*d))))/(b*(b*c - a*d 
)))/(3*b*(b*c - a*d))
 

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g 
+ e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* 
(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)   Int[(a + b*x)^(m + 1)*(c + d*x)^n, 
x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && 
NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

rule 167

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] && LtQ[m, -1] && GtQ[n, 0] && 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. \(2747\) vs. \(2(248)=496\).

Time = 0.30 (sec) , antiderivative size = 2748, normalized size of antiderivative = 9.61

method	result	size
default	\(\text {Expression too large to display}\)	\(2748\)

Input:

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

Output:

-1/6*(36*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b^3*c^4*d^2*x+96*((b*x+a) 
*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4*b^2*c*d^5*x^3+174*((b*x+a)*(d*x+c))^(1/2)* 
(d*b)^(1/2)*a^4*b^2*c^2*d^4*x^2-96*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3 
*b^3*c^3*d^3*x^2+174*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^4*c^4*d^2*x 
^2+24*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a*b^5*c^3*d^3*x^4+15*ln(1/2*(2*b 
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^7*d^7*x 
^2+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b) 
^(1/2))*b^7*c^7*x^2+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/ 
2)+a*d+b*c)/(d*b)^(1/2))*a^7*c^2*d^5+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c) 
)^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*b^5*c^7-40*((b*x+a)*(d*x+c)) 
^(1/2)*(d*b)^(1/2)*b^6*c^5*d*x^3-6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4 
*b^2*d^6*x^4-6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^6*c^4*d^2*x^4+80*((b* 
x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^5*b*c^3*d^3-36*((b*x+a)*(d*x+c))^(1/2)*( 
d*b)^(1/2)*a^4*b^2*c^4*d^2+80*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b^3* 
c^5*d-135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/( 
d*b)^(1/2))*a^2*b^5*c^5*d^2*x^2+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/ 
2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^6*c^6*d*x^2-60*ln(1/2*(2*b*d*x+2* 
((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^6*b*c^2*d^5*x- 
30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1 
/2))*a^5*b^2*c^3*d^4*x+120*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (248) = 496\).

Time = 1.11 (sec) , antiderivative size = 2474, normalized size of antiderivative = 8.65 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(15*(a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c 
^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5 + (b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 
2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 
 + 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - 
a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x^3 + (b^7*c^7 + a*b^6*c^6* 
d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^4 - 9*a^5*b^2* 
c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^6*d - a^ 
3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3 - a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^5 + a^ 
7*c*d^6)*x)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 
4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d 
 + a*b*d^2)*x) + 4*(15*a^2*b^5*c^6*d - 40*a^3*b^4*c^5*d^2 + 18*a^4*b^3*c^4 
*d^3 - 40*a^5*b^2*c^3*d^4 + 15*a^6*b*c^2*d^5 + 3*(b^7*c^4*d^3 - 4*a*b^6*c^ 
3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^4 + 4*(5*b^7* 
c^5*d^2 - 12*a*b^6*c^4*d^3 + 3*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 - 12*a^ 
4*b^3*c*d^6 + 5*a^5*b^2*d^7)*x^3 + 3*(5*b^7*c^6*d - 29*a^2*b^5*c^4*d^3 + 1 
6*a^3*b^4*c^3*d^4 - 29*a^4*b^3*c^2*d^5 + 5*a^6*b*d^7)*x^2 + 6*(5*a*b^6*c^6 
*d - 10*a^2*b^5*c^5*d^2 - 3*a^3*b^4*c^4*d^3 - 3*a^4*b^3*c^3*d^4 - 10*a^5*b 
^2*c^2*d^5 + 5*a^6*b*c*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^8*c^6*d 
^4 - 4*a^3*b^7*c^5*d^5 + 6*a^4*b^6*c^4*d^6 - 4*a^5*b^5*c^3*d^7 + a^6*b^4*c 
^2*d^8 + (b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^5/(b*x+a)^(5/2)/(d*x+c)^(5/2),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. 1249 vs. \(2 (248) = 496\).

Time = 0.70 (sec) , antiderivative size = 1249, normalized size of antiderivative = 4.37 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*((b*x + a)*(3*(b^13*c^7*d^4*abs(b) - 7*a*b^12*c^6*d^5*abs(b) + 21*a^2* 
b^11*c^5*d^6*abs(b) - 35*a^3*b^10*c^4*d^7*abs(b) + 35*a^4*b^9*c^3*d^8*abs( 
b) - 21*a^5*b^8*c^2*d^9*abs(b) + 7*a^6*b^7*c*d^10*abs(b) - a^7*b^6*d^11*ab 
s(b))*(b*x + a)/(b^16*c^7*d^5 - 7*a*b^15*c^6*d^6 + 21*a^2*b^14*c^5*d^7 - 3 
5*a^3*b^13*c^4*d^8 + 35*a^4*b^12*c^3*d^9 - 21*a^5*b^11*c^2*d^10 + 7*a^6*b^ 
10*c*d^11 - a^7*b^9*d^12) + 2*(10*b^14*c^8*d^3*abs(b) - 60*a*b^13*c^7*d^4* 
abs(b) + 150*a^2*b^12*c^6*d^5*abs(b) - 220*a^3*b^11*c^5*d^6*abs(b) + 225*a 
^4*b^10*c^4*d^7*abs(b) - 168*a^5*b^9*c^3*d^8*abs(b) + 84*a^6*b^8*c^2*d^9*a 
bs(b) - 24*a^7*b^7*c*d^10*abs(b) + 3*a^8*b^6*d^11*abs(b))/(b^16*c^7*d^5 - 
7*a*b^15*c^6*d^6 + 21*a^2*b^14*c^5*d^7 - 35*a^3*b^13*c^4*d^8 + 35*a^4*b^12 
*c^3*d^9 - 21*a^5*b^11*c^2*d^10 + 7*a^6*b^10*c*d^11 - a^7*b^9*d^12)) + 3*( 
5*b^15*c^9*d^2*abs(b) - 35*a*b^14*c^8*d^3*abs(b) + 100*a^2*b^13*c^7*d^4*ab 
s(b) - 160*a^3*b^12*c^6*d^5*abs(b) + 170*a^4*b^11*c^5*d^6*abs(b) - 136*a^5 
*b^10*c^4*d^7*abs(b) + 84*a^6*b^9*c^3*d^8*abs(b) - 36*a^7*b^8*c^2*d^9*abs( 
b) + 9*a^8*b^7*c*d^10*abs(b) - a^9*b^6*d^11*abs(b))/(b^16*c^7*d^5 - 7*a*b^ 
15*c^6*d^6 + 21*a^2*b^14*c^5*d^7 - 35*a^3*b^13*c^4*d^8 + 35*a^4*b^12*c^3*d 
^9 - 21*a^5*b^11*c^2*d^10 + 7*a^6*b^10*c*d^11 - a^7*b^9*d^12))*sqrt(b*x + 
a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 4/3*(15*a^4*b^5*c^3*d - 37*a^5* 
b^4*c^2*d^2 + 29*a^6*b^3*c*d^3 - 7*a^7*b^2*d^4 - 30*(sqrt(b*d)*sqrt(b*x + 
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^3*c^2*d + 42*(sqrt(b*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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