\(\int (e x)^{3/2} \sqrt {c+d x} (a+b x^2)^2 \, dx\) [788]

Optimal result

Integrand size = 26, antiderivative size = 371 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )^2 \, dx=-\frac {c^2 \left (33 b^2 c^4+112 a b c^2 d^2+128 a^2 d^4\right ) e \sqrt {e x} \sqrt {c+d x}}{1024 d^6}+\frac {c \left (33 b^2 c^4+112 a b c^2 d^2+128 a^2 d^4\right ) (e x)^{3/2} \sqrt {c+d x}}{1536 d^5}+\frac {\left (33 b^2 c^4+112 a b c^2 d^2+128 a^2 d^4\right ) (e x)^{5/2} \sqrt {c+d x}}{384 d^4 e}-\frac {b c \left (33 b c^2+112 a d^2\right ) (e x)^{5/2} (c+d x)^{3/2}}{320 d^4 e}+\frac {b \left (33 b c^2+112 a d^2\right ) (e x)^{7/2} (c+d x)^{3/2}}{280 d^3 e^2}-\frac {11 b^2 c (e x)^{9/2} (c+d x)^{3/2}}{84 d^2 e^3}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}+\frac {c^3 \left (33 b^2 c^4+112 a b c^2 d^2+128 a^2 d^4\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{1024 d^{13/2}} \] Output:

-1/1024*c^2*(128*a^2*d^4+112*a*b*c^2*d^2+33*b^2*c^4)*e*(e*x)^(1/2)*(d*x+c) 
^(1/2)/d^6+1/1536*c*(128*a^2*d^4+112*a*b*c^2*d^2+33*b^2*c^4)*(e*x)^(3/2)*( 
d*x+c)^(1/2)/d^5+1/384*(128*a^2*d^4+112*a*b*c^2*d^2+33*b^2*c^4)*(e*x)^(5/2 
)*(d*x+c)^(1/2)/d^4/e-1/320*b*c*(112*a*d^2+33*b*c^2)*(e*x)^(5/2)*(d*x+c)^( 
3/2)/d^4/e+1/280*b*(112*a*d^2+33*b*c^2)*(e*x)^(7/2)*(d*x+c)^(3/2)/d^3/e^2- 
11/84*b^2*c*(e*x)^(9/2)*(d*x+c)^(3/2)/d^2/e^3+1/7*b^2*(e*x)^(11/2)*(d*x+c) 
^(3/2)/d/e^4+1/1024*c^3*(128*a^2*d^4+112*a*b*c^2*d^2+33*b^2*c^4)*e^(3/2)*a 
rctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(13/2)
 

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.67 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )^2 \, dx=\frac {(e x)^{3/2} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (4480 a^2 d^4 \left (-3 c^2+2 c d x+8 d^2 x^2\right )+112 a b d^2 \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )+b^2 \left (-3465 c^6+2310 c^5 d x-1848 c^4 d^2 x^2+1584 c^3 d^3 x^3-1408 c^2 d^4 x^4+1280 c d^5 x^5+15360 d^6 x^6\right )\right )+210 c^3 \left (33 b^2 c^4+112 a b c^2 d^2+128 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )\right )}{107520 d^{13/2} x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(4480*a^2*d^4*(-3*c^2 + 2*c*d* 
x + 8*d^2*x^2) + 112*a*b*d^2*(-105*c^4 + 70*c^3*d*x - 56*c^2*d^2*x^2 + 48* 
c*d^3*x^3 + 384*d^4*x^4) + b^2*(-3465*c^6 + 2310*c^5*d*x - 1848*c^4*d^2*x^ 
2 + 1584*c^3*d^3*x^3 - 1408*c^2*d^4*x^4 + 1280*c*d^5*x^5 + 15360*d^6*x^6)) 
 + 210*c^3*(33*b^2*c^4 + 112*a*b*c^2*d^2 + 128*a^2*d^4)*ArcTanh[(Sqrt[d]*S 
qrt[x])/(-Sqrt[c] + Sqrt[c + d*x])]))/(107520*d^(13/2)*x^(3/2))
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.88, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {521, 27, 2125, 27, 521, 27, 90, 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.

\(\displaystyle \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x} \, dx\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {1}{2} (e x)^{3/2} \sqrt {c+d x} \left (-11 b^2 c x^3 e^4+28 a b d x^2 e^4+14 a^2 d e^4\right )dx}{7 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{3/2} \sqrt {c+d x} \left (-11 b^2 c x^3 e^4+28 a b d x^2 e^4+14 a^2 d e^4\right )dx}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int \frac {3}{2} e^7 (e x)^{3/2} \sqrt {c+d x} \left (56 a^2 d^2+b \left (33 b c^2+112 a d^2\right ) x^2\right )dx}{6 d e^3}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \int (e x)^{3/2} \sqrt {c+d x} \left (56 a^2 d^2+b \left (33 b c^2+112 a d^2\right ) x^2\right )dx}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\frac {e^4 \left (\frac {\int \frac {7}{2} e^2 (e x)^{3/2} \sqrt {c+d x} \left (80 a^2 d^3-b c \left (33 b c^2+112 a d^2\right ) x\right )dx}{5 d e^2}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \int (e x)^{3/2} \sqrt {c+d x} \left (80 a^2 d^3-b c \left (33 b c^2+112 a d^2\right ) x\right )dx}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \left (\frac {5 \left (128 a^2 d^4+112 a b c^2 d^2+33 b^2 c^4\right ) \int (e x)^{3/2} \sqrt {c+d x}dx}{8 d}-\frac {b c (e x)^{5/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \left (\frac {5 \left (128 a^2 d^4+112 a b c^2 d^2+33 b^2 c^4\right ) \left (\frac {1}{6} c \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {b c (e x)^{5/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \left (\frac {5 \left (128 a^2 d^4+112 a b c^2 d^2+33 b^2 c^4\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {b c (e x)^{5/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \left (\frac {5 \left (128 a^2 d^4+112 a b c^2 d^2+33 b^2 c^4\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {b c (e x)^{5/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \left (\frac {5 \left (128 a^2 d^4+112 a b c^2 d^2+33 b^2 c^4\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {b c (e x)^{5/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {e^4 \left (\frac {7 \left (\frac {5 \left (128 a^2 d^4+112 a b c^2 d^2+33 b^2 c^4\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {b c (e x)^{5/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2} \left (112 a d^2+33 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{3/2}}{6 d}}{14 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{3/2}}{7 d e^4}\)

Input:

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

Output:

(b^2*(e*x)^(11/2)*(c + d*x)^(3/2))/(7*d*e^4) + ((-11*b^2*c*e*(e*x)^(9/2)*( 
c + d*x)^(3/2))/(6*d) + (e^4*((b*(33*b*c^2 + 112*a*d^2)*(e*x)^(7/2)*(c + d 
*x)^(3/2))/(5*d*e^2) + (7*(-1/4*(b*c*(33*b*c^2 + 112*a*d^2)*(e*x)^(5/2)*(c 
 + d*x)^(3/2))/(d*e) + (5*(33*b^2*c^4 + 112*a*b*c^2*d^2 + 128*a^2*d^4)*((( 
e*x)^(5/2)*Sqrt[c + d*x])/(3*e) + (c*(((e*x)^(3/2)*Sqrt[c + d*x])/(2*d) - 
(3*c*e*((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*d)))/6))/(8*d)))/(10*d)))/(4*d)) 
/(14*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 = 0.28 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\left (-15360 b^{2} d^{6} x^{6}-1280 b^{2} c \,d^{5} x^{5}-43008 x^{4} a b \,d^{6}+1408 b^{2} c^{2} d^{4} x^{4}-5376 x^{3} a b c \,d^{5}-1584 b^{2} c^{3} d^{3} x^{3}-35840 x^{2} a^{2} d^{6}+6272 x^{2} a b \,c^{2} d^{4}+1848 b^{2} c^{4} d^{2} x^{2}-8960 x \,a^{2} c \,d^{5}-7840 x a b \,c^{3} d^{3}-2310 b^{2} c^{5} d x +13440 a^{2} c^{2} d^{4}+11760 a b \,c^{4} d^{2}+3465 c^{6} b^{2}\right ) x \sqrt {d x +c}\, e^{2}}{107520 d^{6} \sqrt {e x}}+\frac {c^{3} \left (128 a^{2} d^{4}+112 b \,c^{2} d^{2} a +33 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e^{2} \sqrt {\left (d x +c \right ) e x}}{2048 d^{6} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(289\)
default	\(\frac {\sqrt {e x}\, \sqrt {d x +c}\, e \left (30720 b^{2} d^{6} x^{6} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+2560 b^{2} c \,d^{5} x^{5} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+86016 a b \,d^{6} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-2816 b^{2} c^{2} d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+10752 a b c \,d^{5} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+3168 b^{2} c^{3} d^{3} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+71680 a^{2} d^{6} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-12544 a b \,c^{2} d^{4} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-3696 b^{2} c^{4} d^{2} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+13440 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{3} d^{4} e +11760 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{5} d^{2} e +3465 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{7} e +17920 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{5} x +15680 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{3} x +4620 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5} d x -26880 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c^{2} d^{4}-23520 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{4} d^{2}-6930 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{6}\right )}{215040 \sqrt {\left (d x +c \right ) e x}\, d^{6} \sqrt {d e}}\)	\(564\)

Input:

int((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/107520*(-15360*b^2*d^6*x^6-1280*b^2*c*d^5*x^5-43008*a*b*d^6*x^4+1408*b^ 
2*c^2*d^4*x^4-5376*a*b*c*d^5*x^3-1584*b^2*c^3*d^3*x^3-35840*a^2*d^6*x^2+62 
72*a*b*c^2*d^4*x^2+1848*b^2*c^4*d^2*x^2-8960*a^2*c*d^5*x-7840*a*b*c^3*d^3* 
x-2310*b^2*c^5*d*x+13440*a^2*c^2*d^4+11760*a*b*c^4*d^2+3465*b^2*c^6)*x*(d* 
x+c)^(1/2)/d^6*e^2/(e*x)^(1/2)+1/2048*c^3*(128*a^2*d^4+112*a*b*c^2*d^2+33* 
b^2*c^4)/d^6*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^2*((d*x+c)*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.47 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )^2 \, dx=\left [\frac {105 \, {\left (33 \, b^{2} c^{7} + 112 \, a b c^{5} d^{2} + 128 \, a^{2} c^{3} d^{4}\right )} e \sqrt {\frac {e}{d}} \log \left (2 \, d e x + 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (15360 \, b^{2} d^{6} e x^{6} + 1280 \, b^{2} c d^{5} e x^{5} - 128 \, {\left (11 \, b^{2} c^{2} d^{4} - 336 \, a b d^{6}\right )} e x^{4} + 48 \, {\left (33 \, b^{2} c^{3} d^{3} + 112 \, a b c d^{5}\right )} e x^{3} - 56 \, {\left (33 \, b^{2} c^{4} d^{2} + 112 \, a b c^{2} d^{4} - 640 \, a^{2} d^{6}\right )} e x^{2} + 70 \, {\left (33 \, b^{2} c^{5} d + 112 \, a b c^{3} d^{3} + 128 \, a^{2} c d^{5}\right )} e x - 105 \, {\left (33 \, b^{2} c^{6} + 112 \, a b c^{4} d^{2} + 128 \, a^{2} c^{2} d^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{215040 \, d^{6}}, -\frac {105 \, {\left (33 \, b^{2} c^{7} + 112 \, a b c^{5} d^{2} + 128 \, a^{2} c^{3} d^{4}\right )} e \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) - {\left (15360 \, b^{2} d^{6} e x^{6} + 1280 \, b^{2} c d^{5} e x^{5} - 128 \, {\left (11 \, b^{2} c^{2} d^{4} - 336 \, a b d^{6}\right )} e x^{4} + 48 \, {\left (33 \, b^{2} c^{3} d^{3} + 112 \, a b c d^{5}\right )} e x^{3} - 56 \, {\left (33 \, b^{2} c^{4} d^{2} + 112 \, a b c^{2} d^{4} - 640 \, a^{2} d^{6}\right )} e x^{2} + 70 \, {\left (33 \, b^{2} c^{5} d + 112 \, a b c^{3} d^{3} + 128 \, a^{2} c d^{5}\right )} e x - 105 \, {\left (33 \, b^{2} c^{6} + 112 \, a b c^{4} d^{2} + 128 \, a^{2} c^{2} d^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{107520 \, d^{6}}\right ] \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

[1/215040*(105*(33*b^2*c^7 + 112*a*b*c^5*d^2 + 128*a^2*c^3*d^4)*e*sqrt(e/d 
)*log(2*d*e*x + 2*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(15360*b^ 
2*d^6*e*x^6 + 1280*b^2*c*d^5*e*x^5 - 128*(11*b^2*c^2*d^4 - 336*a*b*d^6)*e* 
x^4 + 48*(33*b^2*c^3*d^3 + 112*a*b*c*d^5)*e*x^3 - 56*(33*b^2*c^4*d^2 + 112 
*a*b*c^2*d^4 - 640*a^2*d^6)*e*x^2 + 70*(33*b^2*c^5*d + 112*a*b*c^3*d^3 + 1 
28*a^2*c*d^5)*e*x - 105*(33*b^2*c^6 + 112*a*b*c^4*d^2 + 128*a^2*c^2*d^4)*e 
)*sqrt(d*x + c)*sqrt(e*x))/d^6, -1/107520*(105*(33*b^2*c^7 + 112*a*b*c^5*d 
^2 + 128*a^2*c^3*d^4)*e*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d*sqrt(- 
e/d)/(d*e*x + c*e)) - (15360*b^2*d^6*e*x^6 + 1280*b^2*c*d^5*e*x^5 - 128*(1 
1*b^2*c^2*d^4 - 336*a*b*d^6)*e*x^4 + 48*(33*b^2*c^3*d^3 + 112*a*b*c*d^5)*e 
*x^3 - 56*(33*b^2*c^4*d^2 + 112*a*b*c^2*d^4 - 640*a^2*d^6)*e*x^2 + 70*(33* 
b^2*c^5*d + 112*a*b*c^3*d^3 + 128*a^2*c*d^5)*e*x - 105*(33*b^2*c^6 + 112*a 
*b*c^4*d^2 + 128*a^2*c^2*d^4)*e)*sqrt(d*x + c)*sqrt(e*x))/d^6]
 

Sympy [F(-1)]

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

integrate((e*x)**(3/2)*(d*x+c)**(1/2)*(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)*(d*x+c)^(1/2)*(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. 774 vs. \(2 (315) = 630\).

Time = 0.31 (sec) , antiderivative size = 774, normalized size of antiderivative = 2.09 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a)^2,x, algorithm="giac")
 

Output:

-1/107520*(1120*(105*c^4*e*log(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*b*c*abs(d)/d^2 + 14*(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) - 11 
895*c^5/d^5)*sqrt(d*x + c))*b^2*c*abs(d)/d^2 + 26880*(3*c^2*d*e*log(abs(-s 
qrt(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)*(2*d*x - 3*c)*sqrt(d*x + c))*a^2*c*abs(d)/d^3 - 4480* 
(15*c^3*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)*(2*(4*d*x - 9*c)*(d*x + c) + 33 
*c^2)*sqrt(d*x + c))*a^2*abs(d)/d^3 - 336*(315*c^5*d*e*log(abs(-sqrt(d*e)* 
sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/sqrt(d*e) + (965*c^4 - 2*(74 
5*c^3 - 4*(2*(8*d*x - 33*c)*(d*x + c) + 171*c^2)*(d*x + c))*(d*x + c))*sqr 
t((d*x + c)*d*e - c*d*e)*sqrt(d*x + c))*a*b*abs(d)/d^5 - (45045*c^7*d*e*lo 
g(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/sqrt(d*e) + 
 (169995*c^6 - 2*(194215*c^5 - 4*(74053*c^4 - 2*(35463*c^3 - 8*(10*(12*d*x 
 - 73*c)*(d*x + c) + 2593*c^2)*(d*x + c))*(d*x + c))*(d*x + c))*(d*x + c)) 
*sqrt((d*x + c)*d*e - c*d*e)*sqrt(d*x + c))*b^2*abs(d)/d^7)*e/d
 

Mupad [F(-1)]

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

int((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)^(1/2),x)
 

Output:

int((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.05 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right )^2 \, dx=\frac {\sqrt {e}\, e \left (-13440 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{5}+8960 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{6} x +35840 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{7} x^{2}-11760 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{3}+7840 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{4} x -6272 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{5} x^{2}+5376 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{6} x^{3}+43008 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{7} x^{4}-3465 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d +2310 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{2} x -1848 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{3} x^{2}+1584 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{4} x^{3}-1408 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{5} x^{4}+1280 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{6} x^{5}+15360 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{7} x^{6}+13440 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{3} d^{4}+11760 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{5} d^{2}+3465 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{7}\right )}{107520 d^{7}} \] Input:

int((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a)^2,x)
 

Output:

(sqrt(e)*e*( - 13440*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**5 + 8960*sqrt(x)*s 
qrt(c + d*x)*a**2*c*d**6*x + 35840*sqrt(x)*sqrt(c + d*x)*a**2*d**7*x**2 - 
11760*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d**3 + 7840*sqrt(x)*sqrt(c + d*x)*a*b 
*c**3*d**4*x - 6272*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**5*x**2 + 5376*sqrt(x 
)*sqrt(c + d*x)*a*b*c*d**6*x**3 + 43008*sqrt(x)*sqrt(c + d*x)*a*b*d**7*x** 
4 - 3465*sqrt(x)*sqrt(c + d*x)*b**2*c**6*d + 2310*sqrt(x)*sqrt(c + d*x)*b* 
*2*c**5*d**2*x - 1848*sqrt(x)*sqrt(c + d*x)*b**2*c**4*d**3*x**2 + 1584*sqr 
t(x)*sqrt(c + d*x)*b**2*c**3*d**4*x**3 - 1408*sqrt(x)*sqrt(c + d*x)*b**2*c 
**2*d**5*x**4 + 1280*sqrt(x)*sqrt(c + d*x)*b**2*c*d**6*x**5 + 15360*sqrt(x 
)*sqrt(c + d*x)*b**2*d**7*x**6 + 13440*sqrt(d)*log((sqrt(c + d*x) + sqrt(x 
)*sqrt(d))/sqrt(c))*a**2*c**3*d**4 + 11760*sqrt(d)*log((sqrt(c + d*x) + sq 
rt(x)*sqrt(d))/sqrt(c))*a*b*c**5*d**2 + 3465*sqrt(d)*log((sqrt(c + d*x) + 
sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**7))/(107520*d**7)