\(\int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [411]

Optimal result

Integrand size = 22, antiderivative size = 343 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 a^6}{3 b^6 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 \left (b^6 c^6+a^6 d^6\right ) \sqrt {a+b x}}{3 b^6 d^4 (b c-a d)^3 (c+d x)^{3/2}}+\frac {4 a^5 (3 b c-2 a d)}{b^5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}-\frac {4 \left (5 b^6 c^6-9 a b^5 c^5 d-9 a^5 b c d^5+5 a^6 d^6\right ) \sqrt {a+b x}}{3 b^5 d^4 (b c-a d)^4 \sqrt {c+d x}}-\frac {11 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^4 d^4}+\frac {x \sqrt {a+b x} \sqrt {c+d x}}{2 b^3 d^3}+\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{9/2} d^{9/2}} \] Output:

-2/3*a^6/b^6/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)+2/3*(a^6*d^6+b^6*c^6)* 
(b*x+a)^(1/2)/b^6/d^4/(-a*d+b*c)^3/(d*x+c)^(3/2)+4*a^5*(-2*a*d+3*b*c)/b^5/ 
(-a*d+b*c)^3/(b*x+a)^(1/2)/(d*x+c)^(1/2)-4/3*(5*a^6*d^6-9*a^5*b*c*d^5-9*a* 
b^5*c^5*d+5*b^6*c^6)*(b*x+a)^(1/2)/b^5/d^4/(-a*d+b*c)^4/(d*x+c)^(1/2)-11/4 
*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^4+1/2*x*(b*x+a)^(1/2)*(d*x+c) 
^(1/2)/b^3/d^3+5/4*(7*a^2*d^2+10*a*b*c*d+7*b^2*c^2)*arctanh(d^(1/2)*(b*x+a 
)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.30 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {-105 a^7 d^5 (c+d x)^2+5 a^6 b d^4 (47 c-28 d x) (c+d x)^2-3 a^5 b^2 d^3 (c+d x)^2 \left (22 c^2-106 c d x+7 d^2 x^2\right )-b^7 c^4 x^2 \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )+3 a^4 b^3 d^2 (c+d x)^2 \left (-22 c^3-32 c^2 d x+17 c d^2 x^2+2 d^3 x^3\right )-3 a b^6 c^3 x \left (70 c^4+15 c^3 d x-92 c^2 d^2 x^2-21 c d^3 x^3+8 d^4 x^4\right )+a^3 b^4 c d \left (235 c^5+186 c^4 d x-207 c^3 d^2 x^2-168 c^2 d^3 x^3-42 c d^4 x^4-24 d^5 x^5\right )+3 a^2 b^5 c^2 \left (-35 c^5+110 c^4 d x+183 c^3 d^2 x^2+4 c^2 d^3 x^3-14 c d^4 x^4+12 d^5 x^5\right )}{12 b^4 d^4 (b c-a d)^4 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {5 \left (7 b^2 c^2+10 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{9/2} d^{9/2}} \] Input:

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

Output:

(-105*a^7*d^5*(c + d*x)^2 + 5*a^6*b*d^4*(47*c - 28*d*x)*(c + d*x)^2 - 3*a^ 
5*b^2*d^3*(c + d*x)^2*(22*c^2 - 106*c*d*x + 7*d^2*x^2) - b^7*c^4*x^2*(105* 
c^3 + 140*c^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3) + 3*a^4*b^3*d^2*(c + d*x)^2* 
(-22*c^3 - 32*c^2*d*x + 17*c*d^2*x^2 + 2*d^3*x^3) - 3*a*b^6*c^3*x*(70*c^4 
+ 15*c^3*d*x - 92*c^2*d^2*x^2 - 21*c*d^3*x^3 + 8*d^4*x^4) + a^3*b^4*c*d*(2 
35*c^5 + 186*c^4*d*x - 207*c^3*d^2*x^2 - 168*c^2*d^3*x^3 - 42*c*d^4*x^4 - 
24*d^5*x^5) + 3*a^2*b^5*c^2*(-35*c^5 + 110*c^4*d*x + 183*c^3*d^2*x^2 + 4*c 
^2*d^3*x^3 - 14*c*d^4*x^4 + 12*d^5*x^5))/(12*b^4*d^4*(b*c - a*d)^4*(a + b* 
x)^(3/2)*(c + d*x)^(3/2)) + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTan 
h[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(9/2)*d^(9/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.45, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 27, 167, 27, 167, 27, 167, 27, 164, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 \int \frac {x^3 \left (8 a c (13 b c-7 a d)-\left (3 b^2 c^2-62 a b d c+35 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\int \frac {x^3 \left (8 a c (13 b c-7 a d)-\left (3 b^2 c^2-62 a b d c+35 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}}dx}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {2 \int \frac {3 x^2 \left (6 a c \left (b^2 c^2+14 a b d c-7 a^2 d^2\right )+\left (7 b^3 c^3-9 a b^2 d c^2+69 a^2 b d^2 c-35 a^3 d^3\right ) x\right )}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {\int \frac {x^2 \left (6 a c \left (b^2 c^2+14 a b d c-7 a^2 d^2\right )+\left (7 b^3 c^3-9 a b^2 d c^2+69 a^2 b d^2 c-35 a^3 d^3\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {-\frac {2 \int -\frac {x \left (4 a c (b c+a d) \left (7 b^2 c^2-22 a b d c+7 a^2 d^2\right )+\left (35 b^4 c^4-76 a b^3 d c^3+18 a^2 b^2 d^2 c^2-76 a^3 b d^3 c+35 a^4 d^4\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\int \frac {x \left (4 a c (b c+a d) \left (7 b^2 c^2-22 a b d c+7 a^2 d^2\right )+\left (35 b^4 c^4-76 a b^3 d c^3+18 a^2 b^2 d^2 c^2-76 a^3 b d^3 c+35 a^4 d^4\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {15 (b c-a d)^4 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{8 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )-2 b d x \left (35 a^4 d^4-76 a^3 b c d^3+18 a^2 b^2 c^2 d^2-76 a b^3 c^3 d+35 b^4 c^4\right )\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {15 (b c-a d)^4 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{4 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )-2 b d x \left (35 a^4 d^4-76 a^3 b c d^3+18 a^2 b^2 c^2 d^2-76 a b^3 c^3 d+35 b^4 c^4\right )\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {2 c x^3 \sqrt {a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {15 (b c-a d)^4 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )-2 b d x \left (35 a^4 d^4-76 a^3 b c d^3+18 a^2 b^2 c^2 d^2-76 a b^3 c^3 d+35 b^4 c^4\right )\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{d (b c-a d)}}{b (b c-a d)}-\frac {2 a x^4 (13 b c-7 a d)}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{3 b (b c-a d)}\)

Input:

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

Output:

(2*a*x^5)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - ((-2*a*(13*b 
*c - 7*a*d)*x^4)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + ((2*c*(b^ 
2*c^2 + 14*a*b*c*d - 7*a^2*d^2)*x^3*Sqrt[a + b*x])/(d*(b*c - a*d)*(c + d*x 
)^(3/2)) - ((-2*c*(b*c + a*d)*(7*b^2*c^2 - 22*a*b*c*d + 7*a^2*d^2)*x^2*Sqr 
t[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d*x]) + (-1/4*(Sqrt[a + b*x]*Sqrt[c + 
d*x]*((b*c + a*d)*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^2*b^2*c^2*d^2 - 3 
40*a^3*b*c*d^3 + 105*a^4*d^4) - 2*b*d*(35*b^4*c^4 - 76*a*b^3*c^3*d + 18*a^ 
2*b^2*c^2*d^2 - 76*a^3*b*c*d^3 + 35*a^4*d^4)*x))/(b^2*d^2) + (15*(b*c - a* 
d)^4*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/ 
(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(5/2)))/(d*(b*c - a*d)))/(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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

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. \(3424\) vs. \(2(295)=590\).

Time = 0.32 (sec) , antiderivative size = 3425, normalized size of antiderivative = 9.99

method	result	size
default	\(\text {Expression too large to display}\)	\(3425\)

Input:

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

Output:

1/24*(-48*a^3*b^4*c*d^6*x^5*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+72*a^2*b^5 
*c^2*d^5*x^5*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-48*a*b^6*c^3*d^4*x^5*(d*b 
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1098*a^5*b^2*c^2*d^5*x^2*(d*b)^(1/2)*((b*x 
+a)*(d*x+c))^(1/2)+1098*a^2*b^5*c^5*d^2*x^2*(d*b)^(1/2)*((b*x+a)*(d*x+c))^ 
(1/2)-456*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^3*c^4*d^3*x+372*(d*b)^ 
(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^5*d^2*x+660*(d*b)^(1/2)*((b*x+a)*( 
d*x+c))^(1/2)*a^2*b^5*c^6*d*x-414*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3* 
b^4*c^4*d^3*x^2-90*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^6*c^6*d*x^2-90* 
(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^6*b*c*d^6*x^2-414*(d*b)^(1/2)*((b*x+ 
a)*(d*x+c))^(1/2)*a^4*b^3*c^3*d^4*x^2+552*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1 
/2)*a^5*b^2*c*d^6*x^3+24*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^3*c^2*d 
^5*x^3-336*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*c^3*d^4*x^3-210*a^7 
*c^2*d^5*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*a^2*b^5*c^7*(d*b)^(1/2)*( 
(b*x+a)*(d*x+c))^(1/2)+105*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^2*d^8*x^4+105*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^8*c^6*d^2*x^4+210*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*b*d^8*x^3+210*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^8*c^7*d*x^3+210*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^8*c*d^7*x+210*ln(1/2*(2*b*d*x+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1470 vs. \(2 (295) = 590\).

Time = 2.31 (sec) , antiderivative size = 2954, normalized size of antiderivative = 8.61 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(15*(7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^ 
3*c^5*d^3 + 9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6 + (7*b^8* 
c^6*d^2 - 18*a*b^7*c^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4 
*b^4*c^2*d^6 - 18*a^5*b^3*c*d^7 + 7*a^6*b^2*d^8)*x^4 + 2*(7*b^8*c^7*d - 11 
*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3 + 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d 
^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7 + 7*a^7*b*d^8)*x^3 + (7*b^8*c^8 
+ 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*b^5*c^5*d^3 + 34*a^4*b^4*c^ 
4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*a^7*b*c*d^7 + 7*a^8*d 
^8)*x^2 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6*d^2 + 13*a^4*b 
^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b*c^2*d^6 + 7 
*a^8*c*d^7)*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*(105*a^2*b^6*c^7*d - 235*a^3*b^5*c^6*d^2 + 66*a^4*b^ 
4*c^5*d^3 + 66*a^5*b^3*c^4*d^4 - 235*a^6*b^2*c^3*d^5 + 105*a^7*b*c^2*d^6 - 
 6*(b^8*c^4*d^4 - 4*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 - 4*a^3*b^5*c*d^7 + 
a^4*b^4*d^8)*x^5 + 21*(b^8*c^5*d^3 - 3*a*b^7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 
 2*a^3*b^5*c^2*d^6 - 3*a^4*b^4*c*d^7 + a^5*b^3*d^8)*x^4 + 4*(35*b^8*c^6*d^ 
2 - 69*a*b^7*c^5*d^3 - 3*a^2*b^6*c^4*d^4 + 42*a^3*b^5*c^3*d^5 - 3*a^4*b^4* 
c^2*d^6 - 69*a^5*b^3*c*d^7 + 35*a^6*b^2*d^8)*x^3 + 3*(35*b^8*c^7*d + 15*a* 
b^7*c^6*d^2 - 183*a^2*b^6*c^5*d^3 + 69*a^3*b^5*c^4*d^4 + 69*a^4*b^4*c^3...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^6/(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. 1922 vs. \(2 (295) = 590\).

Time = 1.24 (sec) , antiderivative size = 1922, normalized size of antiderivative = 5.60 \[ \int \frac {x^6}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(2*((3*(b*x + a)*(2*(b^16*c^7*d^6 - 7*a*b^15*c^6*d^7 + 21*a^2*b^14*c^ 
5*d^8 - 35*a^3*b^13*c^4*d^9 + 35*a^4*b^12*c^3*d^10 - 21*a^5*b^11*c^2*d^11 
+ 7*a^6*b^10*c*d^12 - a^7*b^9*d^13)*(b*x + a)/(b^12*c^7*d^7*abs(b) - 7*a*b 
^11*c^6*d^8*abs(b) + 21*a^2*b^10*c^5*d^9*abs(b) - 35*a^3*b^9*c^4*d^10*abs( 
b) + 35*a^4*b^8*c^3*d^11*abs(b) - 21*a^5*b^7*c^2*d^12*abs(b) + 7*a^6*b^6*c 
*d^13*abs(b) - a^7*b^5*d^14*abs(b)) - (24*b^18*c^14*d^37 - 168*a*b^17*c^13 
*d^38 + 504*a^2*b^16*c^12*d^39 - 840*a^3*b^15*c^11*d^40 + 840*a^4*b^14*c^1 
0*d^41 - 504*a^5*b^13*c^9*d^42 + 168*a^6*b^12*c^8*d^43 - 24*a^7*b^11*c^7*d 
^44 + 7*b^17*c^8*d^5 - 56*a*b^16*c^7*d^6 + 196*a^2*b^15*c^6*d^7 - 392*a^3* 
b^14*c^5*d^8 + 490*a^4*b^13*c^4*d^9 - 392*a^5*b^12*c^3*d^10 + 196*a^6*b^11 
*c^2*d^11 - 56*a^7*b^10*c*d^12 + 7*a^8*b^9*d^13)/(b^12*c^7*d^7*abs(b) - 7* 
a*b^11*c^6*d^8*abs(b) + 21*a^2*b^10*c^5*d^9*abs(b) - 35*a^3*b^9*c^4*d^10*a 
bs(b) + 35*a^4*b^8*c^3*d^11*abs(b) - 21*a^5*b^7*c^2*d^12*abs(b) + 7*a^6*b^ 
6*c*d^13*abs(b) - a^7*b^5*d^14*abs(b))) - 4*(120*b^19*c^15*d^36 - 840*a*b^ 
18*c^14*d^37 + 2520*a^2*b^17*c^13*d^38 - 4200*a^3*b^16*c^12*d^39 + 4200*a^ 
4*b^15*c^11*d^40 - 2520*a^5*b^14*c^10*d^41 + 840*a^6*b^13*c^9*d^42 - 120*a 
^7*b^12*c^8*d^43 + 35*b^18*c^9*d^4 - 315*a*b^17*c^8*d^5 + 1260*a^2*b^16*c^ 
7*d^6 - 2900*a^3*b^15*c^6*d^7 + 4110*a^4*b^14*c^5*d^8 - 3570*a^5*b^13*c^4* 
d^9 + 1764*a^6*b^12*c^3*d^10 - 372*a^7*b^11*c^2*d^11 - 33*a^8*b^10*c*d^12 
+ 21*a^9*b^9*d^13)/(b^12*c^7*d^7*abs(b) - 7*a*b^11*c^6*d^8*abs(b) + 21*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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