\(\int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx\) [272]

Optimal result

Integrand size = 22, antiderivative size = 436 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {(b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^5}-\frac {(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^4}+\frac {(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{1920 b^4 d^3}+\frac {(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{320 b^4 d^2}+\frac {\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac {(9 b c+19 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac {(a+b x)^{9/2} (c+d x)^{5/2}}{7 b^2 d}-\frac {(b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{11/2}} \] Output:

1/1024*(-a*d+b*c)^4*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c) 
^(1/2)/b^4/d^5-1/1536*(-a*d+b*c)^3*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2)*(b*x+a 
)^(3/2)*(d*x+c)^(1/2)/b^4/d^4+1/1920*(-a*d+b*c)^2*(5*a^2*d^2+10*a*b*c*d+9* 
b^2*c^2)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^4/d^3+1/320*(-a*d+b*c)*(5*a^2*d^2+1 
0*a*b*c*d+9*b^2*c^2)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^4/d^2+1/120*(5*a^2*d^2+ 
10*a*b*c*d+9*b^2*c^2)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^3/d^2-1/84*(19*a*d+9*b 
*c)*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b^2/d^2+1/7*(b*x+a)^(9/2)*(d*x+c)^(5/2)/b^ 
2/d-1/1024*(-a*d+b*c)^5*(5*a^2*d^2+10*a*b*c*d+9*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^(11/2)
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.87 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-525 a^6 d^6+350 a^5 b d^5 (4 c+d x)-35 a^4 b^2 d^4 \left (15 c^2+26 c d x+8 d^2 x^2\right )+60 a^3 b^3 d^3 \left (-10 c^3+5 c^2 d x+12 c d^2 x^2+4 d^3 x^3\right )+a^2 b^4 d^2 \left (3689 c^4-2332 c^3 d x+1824 c^2 d^2 x^2+33520 c d^3 x^3+23680 d^4 x^4\right )+2 a b^5 d \left (-1680 c^5+1099 c^4 d x-872 c^3 d^2 x^2+744 c^2 d^3 x^3+24320 c d^4 x^4+18560 d^5 x^5\right )+3 b^6 \left (315 c^6-210 c^5 d x+168 c^4 d^2 x^2-144 c^3 d^3 x^3+128 c^2 d^4 x^4+6400 c d^5 x^5+5120 d^6 x^6\right )\right )}{107520 b^4 d^5}-\frac {(b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{11/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-525*a^6*d^6 + 350*a^5*b*d^5*(4*c + d*x) - 3 
5*a^4*b^2*d^4*(15*c^2 + 26*c*d*x + 8*d^2*x^2) + 60*a^3*b^3*d^3*(-10*c^3 + 
5*c^2*d*x + 12*c*d^2*x^2 + 4*d^3*x^3) + a^2*b^4*d^2*(3689*c^4 - 2332*c^3*d 
*x + 1824*c^2*d^2*x^2 + 33520*c*d^3*x^3 + 23680*d^4*x^4) + 2*a*b^5*d*(-168 
0*c^5 + 1099*c^4*d*x - 872*c^3*d^2*x^2 + 744*c^2*d^3*x^3 + 24320*c*d^4*x^4 
 + 18560*d^5*x^5) + 3*b^6*(315*c^6 - 210*c^5*d*x + 168*c^4*d^2*x^2 - 144*c 
^3*d^3*x^3 + 128*c^2*d^4*x^4 + 6400*c*d^5*x^5 + 5120*d^6*x^6)))/(107520*b^ 
4*d^5) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt 
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(11/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {101, 27, 90, 60, 60, 60, 60, 60, 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.

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

\(\Big \downarrow \) 101

\(\displaystyle \frac {\int -\frac {1}{2} (a+b x)^{5/2} (c+d x)^{3/2} (2 a c+(9 b c+7 a d) x)dx}{7 b d}+\frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\int (a+b x)^{5/2} (c+d x)^{3/2} (2 a c+(9 b c+7 a d) x)dx}{14 b d}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \int (a+b x)^{5/2} (c+d x)^{3/2}dx}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\)

Input:

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

Output:

(x*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(7*b*d) - (((9*b*c + 7*a*d)*(a + b*x)^ 
(7/2)*(c + d*x)^(5/2))/(6*b*d) - (7*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*( 
((a + b*x)^(7/2)*(c + d*x)^(3/2))/(5*b) + (3*(b*c - a*d)*(((a + b*x)^(7/2) 
*Sqrt[c + d*x])/(4*b) + ((b*c - a*d)*(((a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d 
) - (5*(b*c - a*d)*(((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*d) - (3*(b*c - a*d) 
*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + 
 b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(4*d)))/(6*d)))/(8*b) 
))/(10*b)))/(12*b*d))/(14*b*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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 90

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

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. \(1320\) vs. \(2(380)=760\).

Time = 0.23 (sec) , antiderivative size = 1321, normalized size of antiderivative = 3.03

method	result	size
default	\(\text {Expression too large to display}\)	\(1321\)

Input:

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

Output:

1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-864*b^6*c^3*d^3*x^3*((b*x+a)*(d*x+c 
))^(1/2)*(d*b)^(1/2)+700*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^5*b*d^6*x-1 
260*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^6*c^5*d*x+2800*((b*x+a)*(d*x+c)) 
^(1/2)*(d*b)^(1/2)*a^5*b*c*d^5-1050*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^ 
4*b^2*c^2*d^4-1200*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b^3*c^3*d^3+737 
8*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*b^4*c^4*d^2-1050*((b*x+a)*(d*x+c 
))^(1/2)*(d*b)^(1/2)*a^6*d^6+1890*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^6* 
c^6+30720*b^6*d^6*x^6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+525*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^7-94 
5*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^7-4725*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+3675*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-1575*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*d^6+9 
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^5*b^2*c^2*d^5+525*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^3*c^3*d^4+1575*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^4*c^4*d^3-1820*((b 
*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4*b^2*c*d^5*x+600*((b*x+a)*(d*x+c))^(1/ 
2)*(d*b)^(1/2)*a^3*b^3*c^2*d^4*x-4664*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/...
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1110, normalized size of antiderivative = 2.55 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/430080*(105*(9*b^7*c^7 - 35*a*b^6*c^6*d + 45*a^2*b^5*c^5*d^2 - 15*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 + 15*a^6*b*c*d^6 - 5*a 
^7*d^7)*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*(15360*b^7*d^7*x^6 + 945*b^7*c^6*d - 3360*a*b^6*c^5*d^2 + 3 
689*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 - 525*a^4*b^3*c^2*d^5 + 1400*a^5 
*b^2*c*d^6 - 525*a^6*b*d^7 + 1280*(15*b^7*c*d^6 + 29*a*b^6*d^7)*x^5 + 128* 
(3*b^7*c^2*d^5 + 380*a*b^6*c*d^6 + 185*a^2*b^5*d^7)*x^4 - 16*(27*b^7*c^3*d 
^4 - 93*a*b^6*c^2*d^5 - 2095*a^2*b^5*c*d^6 - 15*a^3*b^4*d^7)*x^3 + 8*(63*b 
^7*c^4*d^3 - 218*a*b^6*c^3*d^4 + 228*a^2*b^5*c^2*d^5 + 90*a^3*b^4*c*d^6 - 
35*a^4*b^3*d^7)*x^2 - 2*(315*b^7*c^5*d^2 - 1099*a*b^6*c^4*d^3 + 1166*a^2*b 
^5*c^3*d^4 - 150*a^3*b^4*c^2*d^5 + 455*a^4*b^3*c*d^6 - 175*a^5*b^2*d^7)*x) 
*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^6), 1/215040*(105*(9*b^7*c^7 - 35*a*b 
^6*c^6*d + 45*a^2*b^5*c^5*d^2 - 15*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 + 15*a^6*b*c*d^6 - 5*a^7*d^7)*sqrt(-b*d)*arctan(1/2*(2*b* 
d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b 
*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(15360*b^7*d^7*x^6 + 945*b^7*c^6*d - 33 
60*a*b^6*c^5*d^2 + 3689*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 - 525*a^4*b^ 
3*c^2*d^5 + 1400*a^5*b^2*c*d^6 - 525*a^6*b*d^7 + 1280*(15*b^7*c*d^6 + 29*a 
*b^6*d^7)*x^5 + 128*(3*b^7*c^2*d^5 + 380*a*b^6*c*d^6 + 185*a^2*b^5*d^7)...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),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. 2985 vs. \(2 (380) = 760\).

Time = 0.46 (sec) , antiderivative size = 2985, normalized size of antiderivative = 6.85 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/107520*(1680*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + 
 a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13* 
c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3* 
d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))* 
sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3* 
b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x 
 + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*a^2*c*abs(b)/b + 14*(sqrt(b^2*c 
+ (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 
 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b 
^30*c*d^9 - 417*a^2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31 
*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) 
 - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3 
*b^30*c*d^9 - 2279*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5 
*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8 + 105 
*a^4*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b 
^6*c^6 + 14*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4 
*b^2*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*log(abs(-sqrt(b*d)*sqrt(b*x 
+ a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^5))*b*c*abs( 
b) + 42*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + 
a)*(10*(b*x + a)/b^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 1098, normalized size of antiderivative = 2.52 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 525*sqrt(c + d*x)*sqrt(a + b*x)*a**6*b*d**7 + 1400*sqrt(c + d*x)*sqrt( 
a + b*x)*a**5*b**2*c*d**6 + 350*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b**2*d**7 
*x - 525*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**3*c**2*d**5 - 910*sqrt(c + d* 
x)*sqrt(a + b*x)*a**4*b**3*c*d**6*x - 280*sqrt(c + d*x)*sqrt(a + b*x)*a**4 
*b**3*d**7*x**2 - 600*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**4*c**3*d**4 + 30 
0*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**4*c**2*d**5*x + 720*sqrt(c + d*x)*sq 
rt(a + b*x)*a**3*b**4*c*d**6*x**2 + 240*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b 
**4*d**7*x**3 + 3689*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**5*c**4*d**3 - 233 
2*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**5*c**3*d**4*x + 1824*sqrt(c + d*x)*s 
qrt(a + b*x)*a**2*b**5*c**2*d**5*x**2 + 33520*sqrt(c + d*x)*sqrt(a + b*x)* 
a**2*b**5*c*d**6*x**3 + 23680*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**5*d**7*x 
**4 - 3360*sqrt(c + d*x)*sqrt(a + b*x)*a*b**6*c**5*d**2 + 2198*sqrt(c + d* 
x)*sqrt(a + b*x)*a*b**6*c**4*d**3*x - 1744*sqrt(c + d*x)*sqrt(a + b*x)*a*b 
**6*c**3*d**4*x**2 + 1488*sqrt(c + d*x)*sqrt(a + b*x)*a*b**6*c**2*d**5*x** 
3 + 48640*sqrt(c + d*x)*sqrt(a + b*x)*a*b**6*c*d**6*x**4 + 37120*sqrt(c + 
d*x)*sqrt(a + b*x)*a*b**6*d**7*x**5 + 945*sqrt(c + d*x)*sqrt(a + b*x)*b**7 
*c**6*d - 630*sqrt(c + d*x)*sqrt(a + b*x)*b**7*c**5*d**2*x + 504*sqrt(c + 
d*x)*sqrt(a + b*x)*b**7*c**4*d**3*x**2 - 432*sqrt(c + d*x)*sqrt(a + b*x)*b 
**7*c**3*d**4*x**3 + 384*sqrt(c + d*x)*sqrt(a + b*x)*b**7*c**2*d**5*x**4 + 
 19200*sqrt(c + d*x)*sqrt(a + b*x)*b**7*c*d**6*x**5 + 15360*sqrt(c + d*...