\(\int \sqrt {e x} (c+d x)^{3/2} (a+b x^2)^2 \, dx\) [800]

Optimal result

Integrand size = 26, antiderivative size = 373 \[ \int \sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2 \, dx=\frac {c^2 \left (9 b^2 c^4+48 a b c^2 d^2+128 a^2 d^4\right ) \sqrt {e x} \sqrt {c+d x}}{1024 d^5}+\frac {7 c \left (9 b^2 c^4+48 a b c^2 d^2+128 a^2 d^4\right ) (e x)^{3/2} \sqrt {c+d x}}{1536 d^4 e}+\frac {\left (9 b^2 c^4+48 a b c^2 d^2+128 a^2 d^4\right ) (e x)^{5/2} \sqrt {c+d x}}{384 d^3 e^2}-\frac {b c \left (3 b c^2+16 a d^2\right ) (e x)^{3/2} (c+d x)^{5/2}}{64 d^4 e}+\frac {b \left (3 b c^2+16 a d^2\right ) (e x)^{5/2} (c+d x)^{5/2}}{40 d^3 e^2}-\frac {3 b^2 c (e x)^{7/2} (c+d x)^{5/2}}{28 d^2 e^3}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}-\frac {c^3 \left (9 b^2 c^4+48 a b c^2 d^2+128 a^2 d^4\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{1024 d^{11/2}} \] Output:

1/1024*c^2*(128*a^2*d^4+48*a*b*c^2*d^2+9*b^2*c^4)*(e*x)^(1/2)*(d*x+c)^(1/2 
)/d^5+7/1536*c*(128*a^2*d^4+48*a*b*c^2*d^2+9*b^2*c^4)*(e*x)^(3/2)*(d*x+c)^ 
(1/2)/d^4/e+1/384*(128*a^2*d^4+48*a*b*c^2*d^2+9*b^2*c^4)*(e*x)^(5/2)*(d*x+ 
c)^(1/2)/d^3/e^2-1/64*b*c*(16*a*d^2+3*b*c^2)*(e*x)^(3/2)*(d*x+c)^(5/2)/d^4 
/e+1/40*b*(16*a*d^2+3*b*c^2)*(e*x)^(5/2)*(d*x+c)^(5/2)/d^3/e^2-3/28*b^2*c* 
(e*x)^(7/2)*(d*x+c)^(5/2)/d^2/e^3+1/7*b^2*(e*x)^(9/2)*(d*x+c)^(5/2)/d/e^4- 
1/1024*c^3*(128*a^2*d^4+48*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 = 1.33 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.67 \[ \int \sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2 \, dx=\frac {\sqrt {e x} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (4480 a^2 d^4 \left (3 c^2+14 c d x+8 d^2 x^2\right )+336 a b d^2 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )+3 b^2 \left (315 c^6-210 c^5 d x+168 c^4 d^2 x^2-144 c^3 d^3 x^3+128 c^2 d^4 x^4+6400 c d^5 x^5+5120 d^6 x^6\right )\right )+210 c^3 \left (9 b^2 c^4+48 a b c^2 d^2+128 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )\right )}{107520 d^{11/2} \sqrt {x}} \] Input:

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

Output:

(Sqrt[e*x]*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(4480*a^2*d^4*(3*c^2 + 14*c*d*x 
+ 8*d^2*x^2) + 336*a*b*d^2*(15*c^4 - 10*c^3*d*x + 8*c^2*d^2*x^2 + 176*c*d^ 
3*x^3 + 128*d^4*x^4) + 3*b^2*(315*c^6 - 210*c^5*d*x + 168*c^4*d^2*x^2 - 14 
4*c^3*d^3*x^3 + 128*c^2*d^4*x^4 + 6400*c*d^5*x^5 + 5120*d^6*x^6)) + 210*c^ 
3*(9*b^2*c^4 + 48*a*b*c^2*d^2 + 128*a^2*d^4)*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sq 
rt[c] - Sqrt[c + d*x])]))/(107520*d^(11/2)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.87, 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.

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

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {1}{2} \sqrt {e x} (c+d x)^{3/2} \left (-9 b^2 c x^3 e^4+28 a b d x^2 e^4+14 a^2 d e^4\right )dx}{7 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {e x} (c+d x)^{3/2} \left (-9 b^2 c x^3 e^4+28 a b d x^2 e^4+14 a^2 d e^4\right )dx}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int \frac {21}{2} e^7 \sqrt {e x} (c+d x)^{3/2} \left (8 a^2 d^2+b \left (3 b c^2+16 a d^2\right ) x^2\right )dx}{6 d e^3}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {7 e^4 \int \sqrt {e x} (c+d x)^{3/2} \left (8 a^2 d^2+b \left (3 b c^2+16 a d^2\right ) x^2\right )dx}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\int \frac {5}{2} e^2 \sqrt {e x} (c+d x)^{3/2} \left (16 a^2 d^3-b c \left (3 b c^2+16 a d^2\right ) x\right )dx}{5 d e^2}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\int \sqrt {e x} (c+d x)^{3/2} \left (16 a^2 d^3-b c \left (3 b c^2+16 a d^2\right ) x\right )dx}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\frac {\left (128 a^2 d^4+48 a b c^2 d^2+9 b^2 c^4\right ) \int \sqrt {e x} (c+d x)^{3/2}dx}{8 d}-\frac {b c (e x)^{3/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{4 d e}}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\frac {\left (128 a^2 d^4+48 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {1}{2} c \int \sqrt {e x} \sqrt {c+d x}dx+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )}{8 d}-\frac {b c (e x)^{3/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{4 d e}}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\frac {\left (128 a^2 d^4+48 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {1}{4} c \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )}{8 d}-\frac {b c (e x)^{3/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{4 d e}}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\frac {\left (128 a^2 d^4+48 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {1}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )}{8 d}-\frac {b c (e x)^{3/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{4 d e}}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\frac {\left (128 a^2 d^4+48 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {1}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )}{8 d}-\frac {b c (e x)^{3/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{4 d e}}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {7 e^4 \left (\frac {\frac {\left (128 a^2 d^4+48 a b c^2 d^2+9 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {1}{4} c \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )+\frac {(e x)^{3/2} \sqrt {c+d x}}{2 e}\right )+\frac {(e x)^{3/2} (c+d x)^{3/2}}{3 e}\right )}{8 d}-\frac {b c (e x)^{3/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{4 d e}}{2 d}+\frac {b (e x)^{5/2} (c+d x)^{5/2} \left (16 a d^2+3 b c^2\right )}{5 d e^2}\right )}{4 d}-\frac {3 b^2 c e (e x)^{7/2} (c+d x)^{5/2}}{2 d}}{14 d e^4}+\frac {b^2 (e x)^{9/2} (c+d x)^{5/2}}{7 d e^4}\)

Input:

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

Output:

(b^2*(e*x)^(9/2)*(c + d*x)^(5/2))/(7*d*e^4) + ((-3*b^2*c*e*(e*x)^(7/2)*(c 
+ d*x)^(5/2))/(2*d) + (7*e^4*((b*(3*b*c^2 + 16*a*d^2)*(e*x)^(5/2)*(c + d*x 
)^(5/2))/(5*d*e^2) + (-1/4*(b*c*(3*b*c^2 + 16*a*d^2)*(e*x)^(3/2)*(c + d*x) 
^(5/2))/(d*e) + ((9*b^2*c^4 + 48*a*b*c^2*d^2 + 128*a^2*d^4)*(((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]*S 
qrt[c + d*x])])/d^(3/2)))/4))/2))/(8*d))/(2*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 = 0.26 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\left (15360 b^{2} d^{6} x^{6}+19200 b^{2} c \,d^{5} x^{5}+43008 x^{4} a b \,d^{6}+384 b^{2} c^{2} d^{4} x^{4}+59136 x^{3} a b c \,d^{5}-432 b^{2} c^{3} d^{3} x^{3}+35840 x^{2} a^{2} d^{6}+2688 x^{2} a b \,c^{2} d^{4}+504 b^{2} c^{4} d^{2} x^{2}+62720 x \,a^{2} c \,d^{5}-3360 x a b \,c^{3} d^{3}-630 b^{2} c^{5} d x +13440 a^{2} c^{2} d^{4}+5040 a b \,c^{4} d^{2}+945 c^{6} b^{2}\right ) x \sqrt {d x +c}\, e}{107520 d^{5} \sqrt {e x}}-\frac {c^{3} \left (128 a^{2} d^{4}+48 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e \sqrt {\left (d x +c \right ) e x}}{2048 d^{5} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(285\)
default	\(-\frac {\sqrt {e x}\, \sqrt {d x +c}\, \left (-30720 b^{2} d^{6} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-38400 b^{2} c \,d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-86016 a b \,d^{6} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-768 b^{2} c^{2} d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-118272 a b c \,d^{5} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+864 b^{2} c^{3} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-71680 a^{2} d^{6} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-5376 a b \,c^{2} d^{4} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-1008 b^{2} c^{4} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+13440 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{3} d^{4} e +5040 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{5} d^{2} e +945 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{7} e -125440 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{5} x +6720 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{3} x +1260 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5} d x -26880 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c^{2} d^{4}-10080 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{4} d^{2}-1890 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{6}\right )}{215040 d^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(563\)

Input:

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

Output:

1/107520*(15360*b^2*d^6*x^6+19200*b^2*c*d^5*x^5+43008*a*b*d^6*x^4+384*b^2* 
c^2*d^4*x^4+59136*a*b*c*d^5*x^3-432*b^2*c^3*d^3*x^3+35840*a^2*d^6*x^2+2688 
*a*b*c^2*d^4*x^2+504*b^2*c^4*d^2*x^2+62720*a^2*c*d^5*x-3360*a*b*c^3*d^3*x- 
630*b^2*c^5*d*x+13440*a^2*c^2*d^4+5040*a*b*c^4*d^2+945*b^2*c^6)*x*(d*x+c)^ 
(1/2)/d^5*e/(e*x)^(1/2)-1/2048*c^3*(128*a^2*d^4+48*a*b*c^2*d^2+9*b^2*c^4)/ 
d^5*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 = 520, normalized size of antiderivative = 1.39 \[ \int \sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2 \, dx=\left [\frac {105 \, {\left (9 \, b^{2} c^{7} + 48 \, a b c^{5} d^{2} + 128 \, a^{2} c^{3} d^{4}\right )} \sqrt {\frac {e}{d}} \log \left (2 \, d e x - 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (15360 \, b^{2} d^{6} x^{6} + 19200 \, b^{2} c d^{5} x^{5} + 945 \, b^{2} c^{6} + 5040 \, a b c^{4} d^{2} + 13440 \, a^{2} c^{2} d^{4} + 384 \, {\left (b^{2} c^{2} d^{4} + 112 \, a b d^{6}\right )} x^{4} - 48 \, {\left (9 \, b^{2} c^{3} d^{3} - 1232 \, a b c d^{5}\right )} x^{3} + 56 \, {\left (9 \, b^{2} c^{4} d^{2} + 48 \, a b c^{2} d^{4} + 640 \, a^{2} d^{6}\right )} x^{2} - 70 \, {\left (9 \, b^{2} c^{5} d + 48 \, a b c^{3} d^{3} - 896 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{215040 \, d^{5}}, \frac {105 \, {\left (9 \, b^{2} c^{7} + 48 \, a b c^{5} d^{2} + 128 \, a^{2} c^{3} d^{4}\right )} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) + {\left (15360 \, b^{2} d^{6} x^{6} + 19200 \, b^{2} c d^{5} x^{5} + 945 \, b^{2} c^{6} + 5040 \, a b c^{4} d^{2} + 13440 \, a^{2} c^{2} d^{4} + 384 \, {\left (b^{2} c^{2} d^{4} + 112 \, a b d^{6}\right )} x^{4} - 48 \, {\left (9 \, b^{2} c^{3} d^{3} - 1232 \, a b c d^{5}\right )} x^{3} + 56 \, {\left (9 \, b^{2} c^{4} d^{2} + 48 \, a b c^{2} d^{4} + 640 \, a^{2} d^{6}\right )} x^{2} - 70 \, {\left (9 \, b^{2} c^{5} d + 48 \, a b c^{3} d^{3} - 896 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{107520 \, d^{5}}\right ] \] Input:

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

Output:

[1/215040*(105*(9*b^2*c^7 + 48*a*b*c^5*d^2 + 128*a^2*c^3*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*(15360*b^2*d^ 
6*x^6 + 19200*b^2*c*d^5*x^5 + 945*b^2*c^6 + 5040*a*b*c^4*d^2 + 13440*a^2*c 
^2*d^4 + 384*(b^2*c^2*d^4 + 112*a*b*d^6)*x^4 - 48*(9*b^2*c^3*d^3 - 1232*a* 
b*c*d^5)*x^3 + 56*(9*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + 640*a^2*d^6)*x^2 - 70* 
(9*b^2*c^5*d + 48*a*b*c^3*d^3 - 896*a^2*c*d^5)*x)*sqrt(d*x + c)*sqrt(e*x)) 
/d^5, 1/107520*(105*(9*b^2*c^7 + 48*a*b*c^5*d^2 + 128*a^2*c^3*d^4)*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*x^6 + 19200*b^2*c*d^5*x^5 + 945*b^2*c^6 + 5040*a*b*c^4*d^2 + 13440*a 
^2*c^2*d^4 + 384*(b^2*c^2*d^4 + 112*a*b*d^6)*x^4 - 48*(9*b^2*c^3*d^3 - 123 
2*a*b*c*d^5)*x^3 + 56*(9*b^2*c^4*d^2 + 48*a*b*c^2*d^4 + 640*a^2*d^6)*x^2 - 
 70*(9*b^2*c^5*d + 48*a*b*c^3*d^3 - 896*a^2*c*d^5)*x)*sqrt(d*x + c)*sqrt(e 
*x))/d^5]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(1/2)*(d*x+c)^(3/2)*(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. 1085 vs. \(2 (317) = 634\).

Time = 0.40 (sec) , antiderivative size = 1085, normalized size of antiderivative = 2.91 \[ \int \sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/107520*(107520*(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^2*abs(d)/d^2 - 2240*(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 - 28*(3465*c^6*e*log(abs(-sqrt(d*e)*s 
qrt(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*abs(d)/d - 53760*(3*c^2*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*d*x - 3*c)*sqrt(d*x + c))*a^2*c*abs(d)/d^2 + 896 
0*(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*b*c^2*abs(d)/d^4 + 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^2 + 168*(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*(745*c^3 - 4*(2*(8*d*x - 33* 
c)*(d*x + c) + 171*c^2)*(d*x + c))*(d*x + c))*sqrt((d*x + c)*d*e - c*d*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.05 \[ \int \sqrt {e x} (c+d x)^{3/2} \left (a+b x^2\right )^2 \, dx=\frac {\sqrt {e}\, \left (13440 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{5}+62720 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{6} x +35840 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{7} x^{2}+5040 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{3}-3360 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{4} x +2688 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{5} x^{2}+59136 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{6} x^{3}+43008 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{7} x^{4}+945 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d -630 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{2} x +504 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{3} x^{2}-432 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{4} x^{3}+384 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{5} x^{4}+19200 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{6} x^{5}+15360 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{7} x^{6}-13440 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{3} d^{4}-5040 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{5} d^{2}-945 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{7}\right )}{107520 d^{6}} \] Input:

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

Output:

(sqrt(e)*(13440*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**5 + 62720*sqrt(x)*sqrt( 
c + d*x)*a**2*c*d**6*x + 35840*sqrt(x)*sqrt(c + d*x)*a**2*d**7*x**2 + 5040 
*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d**3 - 3360*sqrt(x)*sqrt(c + d*x)*a*b*c**3 
*d**4*x + 2688*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**5*x**2 + 59136*sqrt(x)*sq 
rt(c + d*x)*a*b*c*d**6*x**3 + 43008*sqrt(x)*sqrt(c + d*x)*a*b*d**7*x**4 + 
945*sqrt(x)*sqrt(c + d*x)*b**2*c**6*d - 630*sqrt(x)*sqrt(c + d*x)*b**2*c** 
5*d**2*x + 504*sqrt(x)*sqrt(c + d*x)*b**2*c**4*d**3*x**2 - 432*sqrt(x)*sqr 
t(c + d*x)*b**2*c**3*d**4*x**3 + 384*sqrt(x)*sqrt(c + d*x)*b**2*c**2*d**5* 
x**4 + 19200*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 - 5040*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqr 
t(d))/sqrt(c))*a*b*c**5*d**2 - 945*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sq 
rt(d))/sqrt(c))*b**2*c**7))/(107520*d**6)