\(\int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx\) [392]

Optimal result

Integrand size = 22, antiderivative size = 288 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {2 a^5}{b^5 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \left (b^5 c^5+3 a^5 d^5\right ) \sqrt {a+b x}}{3 b^5 d^4 (b c-a d)^2 (c+d x)^{3/2}}-\frac {2 \left (10 b^5 c^5-15 a b^4 c^4 d-3 a^5 d^5\right ) \sqrt {a+b x}}{3 b^4 d^4 (b c-a d)^3 \sqrt {c+d x}}-\frac {(13 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3 d^4}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b^2 d^4}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{9/2}} \] Output:

2*a^5/b^5/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^(3/2)+2/3*(3*a^5*d^5+b^5*c^5)*( 
b*x+a)^(1/2)/b^5/d^4/(-a*d+b*c)^2/(d*x+c)^(3/2)-2/3*(-3*a^5*d^5-15*a*b^4*c 
^4*d+10*b^5*c^5)*(b*x+a)^(1/2)/b^4/d^4/(-a*d+b*c)^3/(d*x+c)^(1/2)-1/4*(7*a 
*d+13*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^4+1/2*(b*x+a)^(1/2)*(d*x+c)^( 
3/2)/b^2/d^4+5/4*(3*a^2*d^2+6*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^(7/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\frac {45 a^5 d^4 (c+d x)^2+15 a^4 b d^3 (-2 c+d x) (c+d x)^2-6 a^3 b^2 d^2 (c+d x)^2 \left (6 c^2+2 c d x+d^2 x^2\right )-b^5 c^3 x \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )+a b^4 c^2 \left (-105 c^4+50 c^3 d x+237 c^2 d^2 x^2+48 c d^3 x^3-18 d^4 x^4\right )+2 a^2 b^3 c d \left (95 c^4+111 c^3 d x-6 c^2 d^2 x^2-9 c d^3 x^3+9 d^4 x^4\right )}{12 b^3 d^4 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {5 \left (7 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{7/2} d^{9/2}} \] Input:

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

Output:

(45*a^5*d^4*(c + d*x)^2 + 15*a^4*b*d^3*(-2*c + d*x)*(c + d*x)^2 - 6*a^3*b^ 
2*d^2*(c + d*x)^2*(6*c^2 + 2*c*d*x + d^2*x^2) - b^5*c^3*x*(105*c^3 + 140*c 
^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3) + a*b^4*c^2*(-105*c^4 + 50*c^3*d*x + 23 
7*c^2*d^2*x^2 + 48*c*d^3*x^3 - 18*d^4*x^4) + 2*a^2*b^3*c*d*(95*c^4 + 111*c 
^3*d*x - 6*c^2*d^2*x^2 - 9*c*d^3*x^3 + 9*d^4*x^4))/(12*b^3*d^4*(b*c - a*d) 
^3*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (5*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2) 
*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(7/2)*d^(9 
/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.36, 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, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {2 \int \frac {x^2 \left (6 a c (b c+3 a d)+\left (7 b^2 c^2-6 a b d c+15 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}}{b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\int \frac {x^2 \left (6 a c (b c+3 a d)+\left (7 b^2 c^2-6 a b d c+15 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}}{b (b c-a d)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {-\frac {2 \int -\frac {x \left (4 a c \left (7 b^2 c^2-12 a b d c-3 a^2 d^2\right )+\left (35 b^3 c^3-61 a b^2 d c^2+9 a^2 b d^2 c-15 a^3 d^3\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}}{b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\int \frac {x \left (4 a c \left (7 b^2 c^2-12 a b d c-3 a^2 d^2\right )+\left (35 b^3 c^3-61 a b^2 d c^2+9 a^2 b d^2 c-15 a^3 d^3\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}}{b (b c-a d)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {15 (b c-a d)^3 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{8 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}}{b (b c-a d)}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {15 (b c-a d)^3 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{4 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}}{b (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 a x^4}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}-\frac {\frac {2 c x^3 \sqrt {a+b x} (3 a d+b c)}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {\frac {\frac {15 (b c-a d)^3 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-2 b d x \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )-190 a b^3 c^3 d+105 b^4 c^4\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} \left (-3 a^2 d^2-12 a b c d+7 b^2 c^2\right )}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}}{b (b c-a d)}\)

Input:

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

Output:

(2*a*x^4)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - ((2*c*(b*c + 3*a 
*d)*x^3*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) - ((-2*c*(7*b^2*c 
^2 - 12*a*b*c*d - 3*a^2*d^2)*x^2*Sqrt[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d* 
x]) + (-1/4*(Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^4*c^4 - 190*a*b^3*c^3*d + 
36*a^2*b^2*c^2*d^2 + 30*a^3*b*c*d^3 - 45*a^4*d^4 - 2*b*d*(35*b^3*c^3 - 61* 
a*b^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*x))/(b^2*d^2) + (15*(b*c - a*d)^ 
3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqr 
t[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(5/2)))/(d*(b*c - a*d)))/(3*d*(b*c - a* 
d)))/(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. \(2227\) vs. \(2(246)=492\).

Time = 0.30 (sec) , antiderivative size = 2228, normalized size of antiderivative = 7.74

method	result	size
default	\(\text {Expression too large to display}\)	\(2228\)

Input:

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

Output:

1/24*(-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^4*b^2*c^4*d^3-90*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^3*b^3*c^5*d^2+225*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^4*c^6*d+45*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*d^7*x^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))*b^6*c^7*x+45*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*c^2*d^5-105*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*b^5*c^7-36*a^2*b^3*c* 
d^5*x^4*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+36*a*b^4*c^2*d^4*x^4*(d*b)^(1/ 
2)*((b*x+a)*(d*x+c))^(1/2)+48*a^3*b^2*c*d^5*x^3*(d*b)^(1/2)*((b*x+a)*(d*x+ 
c))^(1/2)+36*a^2*b^3*c^2*d^4*x^3*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-96*a* 
b^4*c^3*d^3*x^3*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+132*a^3*b^2*c^2*d^4*x^ 
2*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+24*a^2*b^3*c^3*d^3*x^2*(d*b)^(1/2)*( 
(b*x+a)*(d*x+c))^(1/2)-474*a*b^4*c^4*d^2*x^2*(d*b)^(1/2)*((b*x+a)*(d*x+c)) 
^(1/2)+90*a^4*b*c^2*d^4*x*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+168*a^3*b^2* 
c^3*d^3*x*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-444*a^2*b^3*c^4*d^2*x*(d*b)^ 
(1/2)*((b*x+a)*(d*x+c))^(1/2)-100*a*b^4*c^5*d*x*(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^4*b^2*c^3*d^4*x-210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (246) = 492\).

Time = 0.97 (sec) , antiderivative size = 1960, normalized size of antiderivative = 6.81 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(15*(7*a*b^5*c^7 - 15*a^2*b^4*c^6*d + 6*a^3*b^3*c^5*d^2 + 2*a^4*b^2* 
c^4*d^3 + 3*a^5*b*c^3*d^4 - 3*a^6*c^2*d^5 + (7*b^6*c^5*d^2 - 15*a*b^5*c^4* 
d^3 + 6*a^2*b^4*c^3*d^4 + 2*a^3*b^3*c^2*d^5 + 3*a^4*b^2*c*d^6 - 3*a^5*b*d^ 
7)*x^3 + (14*b^6*c^6*d - 23*a*b^5*c^5*d^2 - 3*a^2*b^4*c^4*d^3 + 10*a^3*b^3 
*c^3*d^4 + 8*a^4*b^2*c^2*d^5 - 3*a^5*b*c*d^6 - 3*a^6*d^7)*x^2 + (7*b^6*c^7 
 - a*b^5*c^6*d - 24*a^2*b^4*c^5*d^2 + 14*a^3*b^3*c^4*d^3 + 7*a^4*b^2*c^3*d 
^4 + 3*a^5*b*c^2*d^5 - 6*a^6*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*(105*a*b^5*c^6*d - 190*a^2*b 
^4*c^5*d^2 + 36*a^3*b^3*c^4*d^3 + 30*a^4*b^2*c^3*d^4 - 45*a^5*b*c^2*d^5 - 
6*(b^6*c^3*d^4 - 3*a*b^5*c^2*d^5 + 3*a^2*b^4*c*d^6 - a^3*b^3*d^7)*x^4 + 3* 
(7*b^6*c^4*d^3 - 16*a*b^5*c^3*d^4 + 6*a^2*b^4*c^2*d^5 + 8*a^3*b^3*c*d^6 - 
5*a^4*b^2*d^7)*x^3 + (140*b^6*c^5*d^2 - 237*a*b^5*c^4*d^3 + 12*a^2*b^4*c^3 
*d^4 + 66*a^3*b^3*c^2*d^5 - 45*a^5*b*d^7)*x^2 + (105*b^6*c^6*d - 50*a*b^5* 
c^5*d^2 - 222*a^2*b^4*c^4*d^3 + 84*a^3*b^3*c^3*d^4 + 45*a^4*b^2*c^2*d^5 - 
90*a^5*b*c*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^7*c^5*d^5 - 3*a^2*b^6 
*c^4*d^6 + 3*a^3*b^5*c^3*d^7 - a^4*b^4*c^2*d^8 + (b^8*c^3*d^7 - 3*a*b^7*c^ 
2*d^8 + 3*a^2*b^6*c*d^9 - a^3*b^5*d^10)*x^3 + (2*b^8*c^4*d^6 - 5*a*b^7*c^3 
*d^7 + 3*a^2*b^6*c^2*d^8 + a^3*b^5*c*d^9 - a^4*b^4*d^10)*x^2 + (b^8*c^5*d^ 
5 - a*b^7*c^4*d^6 - 3*a^2*b^6*c^3*d^7 + 5*a^3*b^5*c^2*d^8 - 2*a^4*b^4*c...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^5/(b*x+a)^(3/2)/(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. 982 vs. \(2 (246) = 492\).

Time = 0.34 (sec) , antiderivative size = 982, normalized size of antiderivative = 3.41 \[ \int \frac {x^5}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

4*a^5*d/((sqrt(b*d)*b^3*c^2*abs(b) - 2*sqrt(b*d)*a*b^2*c*d*abs(b) + sqrt(b 
*d)*a^2*b*d^2*abs(b))*(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)) + 1/12*((3*(b*x + a)*(2*(b^14*c^5*d^6 - 5 
*a*b^13*c^4*d^7 + 10*a^2*b^12*c^3*d^8 - 10*a^3*b^11*c^2*d^9 + 5*a^4*b^10*c 
*d^10 - a^5*b^9*d^11)*(b*x + a)/(b^15*c^5*d^7*abs(b) - 5*a*b^14*c^4*d^8*ab 
s(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12*c^2*d^10*abs(b) + 5*a^4*b^ 
11*c*d^11*abs(b) - a^5*b^10*d^12*abs(b)) - (7*b^15*c^6*d^5 - 22*a*b^14*c^5 
*d^6 + 5*a^2*b^13*c^4*d^7 + 60*a^3*b^12*c^3*d^8 - 95*a^4*b^11*c^2*d^9 + 58 
*a^5*b^10*c*d^10 - 13*a^6*b^9*d^11)/(b^15*c^5*d^7*abs(b) - 5*a*b^14*c^4*d^ 
8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12*c^2*d^10*abs(b) + 5*a^ 
4*b^11*c*d^11*abs(b) - a^5*b^10*d^12*abs(b))) - 20*(7*b^16*c^7*d^4 - 29*a* 
b^15*c^6*d^5 + 43*a^2*b^14*c^5*d^6 - 21*a^3*b^13*c^4*d^7 - 15*a^4*b^12*c^3 
*d^8 + 27*a^5*b^11*c^2*d^9 - 15*a^6*b^10*c*d^10 + 3*a^7*b^9*d^11)/(b^15*c^ 
5*d^7*abs(b) - 5*a*b^14*c^4*d^8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a 
^3*b^12*c^2*d^10*abs(b) + 5*a^4*b^11*c*d^11*abs(b) - a^5*b^10*d^12*abs(b)) 
)*(b*x + a) - 3*(35*b^17*c^8*d^3 - 180*a*b^16*c^7*d^4 + 360*a^2*b^15*c^6*d 
^5 - 340*a^3*b^14*c^5*d^6 + 110*a^4*b^13*c^4*d^7 + 84*a^5*b^12*c^3*d^8 - 1 
12*a^6*b^11*c^2*d^9 + 52*a^7*b^10*c*d^10 - 9*a^8*b^9*d^11)/(b^15*c^5*d^7*a 
bs(b) - 5*a*b^14*c^4*d^8*abs(b) + 10*a^2*b^13*c^3*d^9*abs(b) - 10*a^3*b^12 
*c^2*d^10*abs(b) + 5*a^4*b^11*c*d^11*abs(b) - a^5*b^10*d^12*abs(b)))*sq...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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