\(\int \sqrt {e x} (c+d x)^{5/2} (a+b x^2)^2 \, dx\) [812]

Optimal result

Integrand size = 26, antiderivative size = 429 \[ \int \sqrt {e x} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\frac {5 c^3 \left (9 b^2 c^4+64 a b c^2 d^2+256 a^2 d^4\right ) \sqrt {e x} \sqrt {c+d x}}{16384 d^5}+\frac {59 c^2 \left (9 b^2 c^4+64 a b c^2 d^2+256 a^2 d^4\right ) (e x)^{3/2} \sqrt {c+d x}}{24576 d^4 e}+\frac {17 c \left (9 b^2 c^4+64 a b c^2 d^2+256 a^2 d^4\right ) (e x)^{5/2} \sqrt {c+d x}}{6144 d^3 e^2}+\frac {\left (9 b^2 c^4+64 a b c^2 d^2+256 a^2 d^4\right ) (e x)^{7/2} \sqrt {c+d x}}{1024 d^2 e^3}-\frac {b c \left (9 b c^2+64 a d^2\right ) (e x)^{3/2} (c+d x)^{7/2}}{384 d^4 e}+\frac {b \left (9 b c^2+64 a d^2\right ) (e x)^{5/2} (c+d x)^{7/2}}{192 d^3 e^2}-\frac {9 b^2 c (e x)^{7/2} (c+d x)^{7/2}}{112 d^2 e^3}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}-\frac {5 c^4 \left (9 b^2 c^4+64 a b c^2 d^2+256 a^2 d^4\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{16384 d^{11/2}} \] Output:

5/16384*c^3*(256*a^2*d^4+64*a*b*c^2*d^2+9*b^2*c^4)*(e*x)^(1/2)*(d*x+c)^(1/ 
2)/d^5+59/24576*c^2*(256*a^2*d^4+64*a*b*c^2*d^2+9*b^2*c^4)*(e*x)^(3/2)*(d* 
x+c)^(1/2)/d^4/e+17/6144*c*(256*a^2*d^4+64*a*b*c^2*d^2+9*b^2*c^4)*(e*x)^(5 
/2)*(d*x+c)^(1/2)/d^3/e^2+1/1024*(256*a^2*d^4+64*a*b*c^2*d^2+9*b^2*c^4)*(e 
*x)^(7/2)*(d*x+c)^(1/2)/d^2/e^3-1/384*b*c*(64*a*d^2+9*b*c^2)*(e*x)^(3/2)*( 
d*x+c)^(7/2)/d^4/e+1/192*b*(64*a*d^2+9*b*c^2)*(e*x)^(5/2)*(d*x+c)^(7/2)/d^ 
3/e^2-9/112*b^2*c*(e*x)^(7/2)*(d*x+c)^(7/2)/d^2/e^3+1/8*b^2*(e*x)^(9/2)*(d 
*x+c)^(7/2)/d/e^4-5/16384*c^4*(256*a^2*d^4+64*a*b*c^2*d^2+9*b^2*c^4)*e^(1/ 
2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(11/2)
 

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.66 \[ \int \sqrt {e x} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\frac {\sqrt {e x} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (1792 a^2 d^4 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )+448 a b d^2 \left (15 c^5-10 c^4 d x+8 c^3 d^2 x^2+432 c^2 d^3 x^3+640 c d^4 x^4+256 d^5 x^5\right )+3 b^2 \left (315 c^7-210 c^6 d x+168 c^5 d^2 x^2-144 c^4 d^3 x^3+128 c^3 d^4 x^4+20736 c^2 d^5 x^5+33792 c d^6 x^6+14336 d^7 x^7\right )\right )+210 c^4 \left (9 b^2 c^4+64 a b c^2 d^2+256 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )\right )}{344064 d^{11/2} \sqrt {x}} \] Input:

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

Output:

(Sqrt[e*x]*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(1792*a^2*d^4*(15*c^3 + 118*c^2* 
d*x + 136*c*d^2*x^2 + 48*d^3*x^3) + 448*a*b*d^2*(15*c^5 - 10*c^4*d*x + 8*c 
^3*d^2*x^2 + 432*c^2*d^3*x^3 + 640*c*d^4*x^4 + 256*d^5*x^5) + 3*b^2*(315*c 
^7 - 210*c^6*d*x + 168*c^5*d^2*x^2 - 144*c^4*d^3*x^3 + 128*c^3*d^4*x^4 + 2 
0736*c^2*d^5*x^5 + 33792*c*d^6*x^6 + 14336*d^7*x^7)) + 210*c^4*(9*b^2*c^4 
+ 64*a*b*c^2*d^2 + 256*a^2*d^4)*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sqrt[ 
c + d*x])]))/(344064*d^(11/2)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.82, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {521, 27, 2125, 27, 521, 27, 90, 60, 60, 60, 60, 65, 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.

\(\displaystyle \int \sqrt {e x} \left (a+b x^2\right )^2 (c+d x)^{5/2} \, dx\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {1}{2} \sqrt {e x} (c+d x)^{5/2} \left (-9 b^2 c x^3 e^4+32 a b d x^2 e^4+16 a^2 d e^4\right )dx}{8 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {e x} (c+d x)^{5/2} \left (-9 b^2 c x^3 e^4+32 a b d x^2 e^4+16 a^2 d e^4\right )dx}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int \frac {7}{2} e^7 \sqrt {e x} (c+d x)^{5/2} \left (32 a^2 d^2+b \left (9 b c^2+64 a d^2\right ) x^2\right )dx}{7 d e^3}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \int \sqrt {e x} (c+d x)^{5/2} \left (32 a^2 d^2+b \left (9 b c^2+64 a d^2\right ) x^2\right )dx}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\frac {e^4 \left (\frac {\int \frac {1}{2} e^2 \sqrt {e x} (c+d x)^{5/2} \left (384 a^2 d^3-5 b c \left (9 b c^2+64 a d^2\right ) x\right )dx}{6 d e^2}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \left (\frac {\int \sqrt {e x} (c+d x)^{5/2} \left (384 a^2 d^3-5 b c \left (9 b c^2+64 a d^2\right ) x\right )dx}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \int \sqrt {e x} (c+d x)^{5/2}dx}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {5}{8} c \int \sqrt {e x} (c+d x)^{3/2}dx+\frac {(e x)^{3/2} (c+d x)^{5/2}}{4 e}\right )}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {5}{8} c \left (\frac {1}{2} c \int \sqrt {e x} \sqrt {c+d x}dx+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )+\frac {(e x)^{3/2} (c+d x)^{5/2}}{4 e}\right )}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {5}{8} c \left (\frac {1}{2} c \left (\frac {1}{4} c \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )+\frac {(e x)^{3/2} (c+d x)^{5/2}}{4 e}\right )}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {5}{8} c \left (\frac {1}{2} c \left (\frac {1}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )+\frac {(e x)^{3/2} (c+d x)^{5/2}}{4 e}\right )}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {5}{8} c \left (\frac {1}{2} c \left (\frac {1}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )+\frac {(e x)^{3/2} (c+d x)^{5/2}}{4 e}\right )}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {e^4 \left (\frac {\frac {3 \left (256 a^2 d^4+64 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {5}{8} c \left (\frac {1}{2} c \left (\frac {1}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )+\frac {(e x)^{3/2} (c+d x)^{5/2}}{4 e}\right )}{2 d}-\frac {b c (e x)^{3/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{d e}}{12 d}+\frac {b (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+9 b c^2\right )}{6 d e^2}\right )}{2 d}-\frac {9 b^2 c e (e x)^{7/2} (c+d x)^{7/2}}{7 d}}{16 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{7/2}}{8 d e^4}\)

Input:

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

Output:

(b^2*(e*x)^(9/2)*(c + d*x)^(7/2))/(8*d*e^4) + ((-9*b^2*c*e*(e*x)^(7/2)*(c 
+ d*x)^(7/2))/(7*d) + (e^4*((b*(9*b*c^2 + 64*a*d^2)*(e*x)^(5/2)*(c + d*x)^ 
(7/2))/(6*d*e^2) + (-((b*c*(9*b*c^2 + 64*a*d^2)*(e*x)^(3/2)*(c + d*x)^(7/2 
))/(d*e)) + (3*(9*b^2*c^4 + 64*a*b*c^2*d^2 + 256*a^2*d^4)*(((e*x)^(3/2)*(c 
 + d*x)^(5/2))/(4*e) + (5*c*(((e*x)^(3/2)*(c + d*x)^(3/2))/(3*e) + (c*(((e 
*x)^(3/2)*Sqrt[c + d*x])/(2*e) + (c*((Sqrt[e*x]*Sqrt[c + d*x])/d - (c*Sqrt 
[e]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/d^(3/2)))/4))/2) 
)/8))/(2*d))/(12*d)))/(2*d))/(16*d*e^4)
 

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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 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 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]
 

rule 521

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> Simp[b^p*(e*x)^(m + 2*p)*((c + d*x)^(n + 1)/(d*e^(2*p)*(m + n 
 + 2*p + 1))), x] + Simp[1/(d*e^(2*p)*(m + n + 2*p + 1))   Int[(e*x)^m*(c + 
 d*x)^n*ExpandToSum[d*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x^2)^p - b^p*(e*x)^ 
(2*p)) - b^p*(e*c)*(m + 2*p)*(e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] &&  !IntegerQ[m] &&  !I 
ntegerQ[n]
 

rule 2125

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x 
)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( 
m + n + q + 1))   Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q 
+ 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) 
^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, 
 x] && PolyQ[Px, x]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\left (43008 b^{2} x^{7} d^{7}+101376 b^{2} c \,x^{6} d^{6}+114688 a b \,d^{7} x^{5}+62208 b^{2} c^{2} d^{5} x^{5}+286720 a b c \,d^{6} x^{4}+384 b^{2} c^{3} d^{4} x^{4}+86016 a^{2} d^{7} x^{3}+193536 a b \,c^{2} d^{5} x^{3}-432 b^{2} c^{4} d^{3} x^{3}+243712 a^{2} c \,d^{6} x^{2}+3584 a b \,c^{3} d^{4} x^{2}+504 b^{2} c^{5} d^{2} x^{2}+211456 a^{2} c^{2} d^{5} x -4480 a b \,c^{4} d^{3} x -630 b^{2} c^{6} d x +26880 a^{2} c^{3} d^{4}+6720 a b \,c^{5} d^{2}+945 b^{2} c^{7}\right ) x \sqrt {d x +c}\, e}{344064 d^{5} \sqrt {e x}}-\frac {5 c^{4} \left (256 a^{2} d^{4}+64 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e \sqrt {\left (d x +c \right ) e x}}{32768 d^{5} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(326\)
default	\(-\frac {\sqrt {e x}\, \sqrt {d x +c}\, \left (-86016 b^{2} d^{7} x^{7} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-202752 b^{2} c \,d^{6} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-229376 a b \,d^{7} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-124416 b^{2} c^{2} d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-573440 a b c \,d^{6} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-768 b^{2} c^{3} d^{4} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-172032 a^{2} d^{7} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-387072 a b \,c^{2} d^{5} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+864 b^{2} c^{4} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-487424 a^{2} c \,d^{6} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-7168 a b \,c^{3} d^{4} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-1008 b^{2} c^{5} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+26880 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{4} d^{4} e +6720 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{6} d^{2} e +945 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{8} e -422912 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c^{2} d^{5} x +8960 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{4} d^{3} x +1260 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{6} d x -53760 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c^{3} d^{4}-13440 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{5} d^{2}-1890 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{7}\right )}{688128 d^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(649\)

Input:

int((e*x)^(1/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/344064/d^5*(43008*b^2*d^7*x^7+101376*b^2*c*d^6*x^6+114688*a*b*d^7*x^5+62 
208*b^2*c^2*d^5*x^5+286720*a*b*c*d^6*x^4+384*b^2*c^3*d^4*x^4+86016*a^2*d^7 
*x^3+193536*a*b*c^2*d^5*x^3-432*b^2*c^4*d^3*x^3+243712*a^2*c*d^6*x^2+3584* 
a*b*c^3*d^4*x^2+504*b^2*c^5*d^2*x^2+211456*a^2*c^2*d^5*x-4480*a*b*c^4*d^3* 
x-630*b^2*c^6*d*x+26880*a^2*c^3*d^4+6720*a*b*c^5*d^2+945*b^2*c^7)*x*(d*x+c 
)^(1/2)*e/(e*x)^(1/2)-5/32768*c^4/d^5*(256*a^2*d^4+64*a*b*c^2*d^2+9*b^2*c^ 
4)*ln((1/2*c*e+d*e*x)/(d*e)^(1/2)+(d*e*x^2+c*e*x)^(1/2))/(d*e)^(1/2)*e*((d 
*x+c)*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.39 \[ \int \sqrt {e x} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\left [\frac {105 \, {\left (9 \, b^{2} c^{8} + 64 \, a b c^{6} d^{2} + 256 \, a^{2} c^{4} d^{4}\right )} \sqrt {\frac {e}{d}} \log \left (2 \, d e x - 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (43008 \, b^{2} d^{7} x^{7} + 101376 \, b^{2} c d^{6} x^{6} + 945 \, b^{2} c^{7} + 6720 \, a b c^{5} d^{2} + 26880 \, a^{2} c^{3} d^{4} + 256 \, {\left (243 \, b^{2} c^{2} d^{5} + 448 \, a b d^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{2} c^{3} d^{4} + 2240 \, a b c d^{6}\right )} x^{4} - 48 \, {\left (9 \, b^{2} c^{4} d^{3} - 4032 \, a b c^{2} d^{5} - 1792 \, a^{2} d^{7}\right )} x^{3} + 56 \, {\left (9 \, b^{2} c^{5} d^{2} + 64 \, a b c^{3} d^{4} + 4352 \, a^{2} c d^{6}\right )} x^{2} - 14 \, {\left (45 \, b^{2} c^{6} d + 320 \, a b c^{4} d^{3} - 15104 \, a^{2} c^{2} d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{688128 \, d^{5}}, \frac {105 \, {\left (9 \, b^{2} c^{8} + 64 \, a b c^{6} d^{2} + 256 \, a^{2} c^{4} d^{4}\right )} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) + {\left (43008 \, b^{2} d^{7} x^{7} + 101376 \, b^{2} c d^{6} x^{6} + 945 \, b^{2} c^{7} + 6720 \, a b c^{5} d^{2} + 26880 \, a^{2} c^{3} d^{4} + 256 \, {\left (243 \, b^{2} c^{2} d^{5} + 448 \, a b d^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{2} c^{3} d^{4} + 2240 \, a b c d^{6}\right )} x^{4} - 48 \, {\left (9 \, b^{2} c^{4} d^{3} - 4032 \, a b c^{2} d^{5} - 1792 \, a^{2} d^{7}\right )} x^{3} + 56 \, {\left (9 \, b^{2} c^{5} d^{2} + 64 \, a b c^{3} d^{4} + 4352 \, a^{2} c d^{6}\right )} x^{2} - 14 \, {\left (45 \, b^{2} c^{6} d + 320 \, a b c^{4} d^{3} - 15104 \, a^{2} c^{2} d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{344064 \, d^{5}}\right ] \] Input:

integrate((e*x)^(1/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

[1/688128*(105*(9*b^2*c^8 + 64*a*b*c^6*d^2 + 256*a^2*c^4*d^4)*sqrt(e/d)*lo 
g(2*d*e*x - 2*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(43008*b^2*d^ 
7*x^7 + 101376*b^2*c*d^6*x^6 + 945*b^2*c^7 + 6720*a*b*c^5*d^2 + 26880*a^2* 
c^3*d^4 + 256*(243*b^2*c^2*d^5 + 448*a*b*d^7)*x^5 + 128*(3*b^2*c^3*d^4 + 2 
240*a*b*c*d^6)*x^4 - 48*(9*b^2*c^4*d^3 - 4032*a*b*c^2*d^5 - 1792*a^2*d^7)* 
x^3 + 56*(9*b^2*c^5*d^2 + 64*a*b*c^3*d^4 + 4352*a^2*c*d^6)*x^2 - 14*(45*b^ 
2*c^6*d + 320*a*b*c^4*d^3 - 15104*a^2*c^2*d^5)*x)*sqrt(d*x + c)*sqrt(e*x)) 
/d^5, 1/344064*(105*(9*b^2*c^8 + 64*a*b*c^6*d^2 + 256*a^2*c^4*d^4)*sqrt(-e 
/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d*sqrt(-e/d)/(d*e*x + c*e)) + (43008*b^ 
2*d^7*x^7 + 101376*b^2*c*d^6*x^6 + 945*b^2*c^7 + 6720*a*b*c^5*d^2 + 26880* 
a^2*c^3*d^4 + 256*(243*b^2*c^2*d^5 + 448*a*b*d^7)*x^5 + 128*(3*b^2*c^3*d^4 
 + 2240*a*b*c*d^6)*x^4 - 48*(9*b^2*c^4*d^3 - 4032*a*b*c^2*d^5 - 1792*a^2*d 
^7)*x^3 + 56*(9*b^2*c^5*d^2 + 64*a*b*c^3*d^4 + 4352*a^2*c*d^6)*x^2 - 14*(4 
5*b^2*c^6*d + 320*a*b*c^4*d^3 - 15104*a^2*c^2*d^5)*x)*sqrt(d*x + c)*sqrt(e 
*x))/d^5]
 

Sympy [F(-1)]

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

integrate((e*x)**(1/2)*(d*x+c)**(5/2)*(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(1/2)*(d*x+c)^(5/2)*(b*x^2+a)^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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1580 vs. \(2 (367) = 734\).

Time = 0.54 (sec) , antiderivative size = 1580, normalized size of antiderivative = 3.68 \[ \int \sqrt {e x} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(1/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x, algorithm="giac")
 

Output:

1/1720320*(1720320*(c*d*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c 
)*d*e - c*d*e)))/sqrt(d*e) + sqrt((d*x + c)*d*e - c*d*e)*sqrt(d*x + c))*a^ 
2*c^3*abs(d)/d^2 - 53760*(105*c^4*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqr 
t((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^2) - sqrt((d*x + c)*d*e - c*d*e)*( 
2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 - 25*c/d^3) + 163*c^2/d^3) - 279 
*c^3/d^3)*sqrt(d*x + c))*a*b*c^2*abs(d)/d - 672*(3465*c^6*e*log(abs(-sqrt( 
d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^4) - sqrt( 
(d*x + c)*d*e - c*d*e)*(2*(4*(2*(d*x + c)*(8*(d*x + c)*(10*(d*x + c)/d^5 - 
 61*c/d^5) + 1251*c^2/d^5) - 3481*c^3/d^5)*(d*x + c) + 11395*c^4/d^5)*(d*x 
 + c) - 11895*c^5/d^5)*sqrt(d*x + c))*b^2*c^2*abs(d)/d - 8960*(105*c^4*e*l 
og(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e) 
*d^2) - sqrt((d*x + c)*d*e - c*d*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c) 
/d^3 - 25*c/d^3) + 163*c^2/d^3) - 279*c^3/d^3)*sqrt(d*x + c))*a^2*d*abs(d) 
 - 448*(3465*c^6*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - 
 c*d*e)))/(sqrt(d*e)*d^4) - sqrt((d*x + c)*d*e - c*d*e)*(2*(4*(2*(d*x + c) 
*(8*(d*x + c)*(10*(d*x + c)/d^5 - 61*c/d^5) + 1251*c^2/d^5) - 3481*c^3/d^5 
)*(d*x + c) + 11395*c^4/d^5)*(d*x + c) - 11895*c^5/d^5)*sqrt(d*x + c))*a*b 
*d*abs(d) - 15*(45045*c^8*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + 
 c)*d*e - c*d*e)))/(sqrt(d*e)*d^6) - sqrt((d*x + c)*d*e - c*d*e)*(2*(4*(2* 
(8*(2*(d*x + c)*(4*(d*x + c)*(14*(d*x + c)/d^7 - 113*c/d^7) + 1601*c^2/...
 

Mupad [F(-1)]

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

int((e*x)^(1/2)*(a + b*x^2)^2*(c + d*x)^(5/2),x)
 

Output:

int((e*x)^(1/2)*(a + b*x^2)^2*(c + d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.06 \[ \int \sqrt {e x} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\frac {\sqrt {e}\, \left (26880 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{5}+211456 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{6} x +243712 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{7} x^{2}+86016 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{8} x^{3}+6720 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d^{3}-4480 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{4} x +3584 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{5} x^{2}+193536 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{6} x^{3}+286720 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{7} x^{4}+114688 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{8} x^{5}+945 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{7} d -630 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d^{2} x +504 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{3} x^{2}-432 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{4} x^{3}+384 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{5} x^{4}+62208 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{6} x^{5}+101376 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{7} x^{6}+43008 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{8} x^{7}-26880 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{4} d^{4}-6720 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{6} d^{2}-945 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{8}\right )}{344064 d^{6}} \] Input:

int((e*x)^(1/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x)
 

Output:

(sqrt(e)*(26880*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**5 + 211456*sqrt(x)*sqrt 
(c + d*x)*a**2*c**2*d**6*x + 243712*sqrt(x)*sqrt(c + d*x)*a**2*c*d**7*x**2 
 + 86016*sqrt(x)*sqrt(c + d*x)*a**2*d**8*x**3 + 6720*sqrt(x)*sqrt(c + d*x) 
*a*b*c**5*d**3 - 4480*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d**4*x + 3584*sqrt(x) 
*sqrt(c + d*x)*a*b*c**3*d**5*x**2 + 193536*sqrt(x)*sqrt(c + d*x)*a*b*c**2* 
d**6*x**3 + 286720*sqrt(x)*sqrt(c + d*x)*a*b*c*d**7*x**4 + 114688*sqrt(x)* 
sqrt(c + d*x)*a*b*d**8*x**5 + 945*sqrt(x)*sqrt(c + d*x)*b**2*c**7*d - 630* 
sqrt(x)*sqrt(c + d*x)*b**2*c**6*d**2*x + 504*sqrt(x)*sqrt(c + d*x)*b**2*c* 
*5*d**3*x**2 - 432*sqrt(x)*sqrt(c + d*x)*b**2*c**4*d**4*x**3 + 384*sqrt(x) 
*sqrt(c + d*x)*b**2*c**3*d**5*x**4 + 62208*sqrt(x)*sqrt(c + d*x)*b**2*c**2 
*d**6*x**5 + 101376*sqrt(x)*sqrt(c + d*x)*b**2*c*d**7*x**6 + 43008*sqrt(x) 
*sqrt(c + d*x)*b**2*d**8*x**7 - 26880*sqrt(d)*log((sqrt(c + d*x) + sqrt(x) 
*sqrt(d))/sqrt(c))*a**2*c**4*d**4 - 6720*sqrt(d)*log((sqrt(c + d*x) + sqrt 
(x)*sqrt(d))/sqrt(c))*a*b*c**6*d**2 - 945*sqrt(d)*log((sqrt(c + d*x) + sqr 
t(x)*sqrt(d))/sqrt(c))*b**2*c**8))/(344064*d**6)