\(\int \frac {(e x)^{3/2} (a+b x^2)^2}{\sqrt {c+d x}} \, dx\) [824]

Optimal result

Integrand size = 26, antiderivative size = 315 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=-\frac {c \left (231 b^2 c^4+560 a b c^2 d^2+384 a^2 d^4\right ) e \sqrt {e x} \sqrt {c+d x}}{512 d^6}+\frac {\left (231 b^2 c^4+560 a b c^2 d^2+384 a^2 d^4\right ) (e x)^{3/2} \sqrt {c+d x}}{768 d^5}-\frac {7 b c \left (33 b c^2+80 a d^2\right ) (e x)^{5/2} \sqrt {c+d x}}{960 d^4 e}+\frac {b \left (33 b c^2+80 a d^2\right ) (e x)^{7/2} \sqrt {c+d x}}{160 d^3 e^2}-\frac {11 b^2 c (e x)^{9/2} \sqrt {c+d x}}{60 d^2 e^3}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}+\frac {c^2 \left (231 b^2 c^4+560 a b c^2 d^2+384 a^2 d^4\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{512 d^{13/2}} \] Output:

-1/512*c*(384*a^2*d^4+560*a*b*c^2*d^2+231*b^2*c^4)*e*(e*x)^(1/2)*(d*x+c)^( 
1/2)/d^6+1/768*(384*a^2*d^4+560*a*b*c^2*d^2+231*b^2*c^4)*(e*x)^(3/2)*(d*x+ 
c)^(1/2)/d^5-7/960*b*c*(80*a*d^2+33*b*c^2)*(e*x)^(5/2)*(d*x+c)^(1/2)/d^4/e 
+1/160*b*(80*a*d^2+33*b*c^2)*(e*x)^(7/2)*(d*x+c)^(1/2)/d^3/e^2-11/60*b^2*c 
*(e*x)^(9/2)*(d*x+c)^(1/2)/d^2/e^3+1/6*b^2*(e*x)^(11/2)*(d*x+c)^(1/2)/d/e^ 
4+1/512*c^2*(384*a^2*d^4+560*a*b*c^2*d^2+231*b^2*c^4)*e^(3/2)*arctanh(d^(1 
/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(13/2)
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {(e x)^{3/2} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (1920 a^2 d^4 (-3 c+2 d x)+80 a b d^2 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )+b^2 \left (-3465 c^5+2310 c^4 d x-1848 c^3 d^2 x^2+1584 c^2 d^3 x^3-1408 c d^4 x^4+1280 d^5 x^5\right )\right )+30 c^2 \left (231 b^2 c^4+560 a b c^2 d^2+384 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )\right )}{7680 d^{13/2} x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(1920*a^2*d^4*(-3*c + 2*d*x) + 
 80*a*b*d^2*(-105*c^3 + 70*c^2*d*x - 56*c*d^2*x^2 + 48*d^3*x^3) + b^2*(-34 
65*c^5 + 2310*c^4*d*x - 1848*c^3*d^2*x^2 + 1584*c^2*d^3*x^3 - 1408*c*d^4*x 
^4 + 1280*d^5*x^5)) + 30*c^2*(231*b^2*c^4 + 560*a*b*c^2*d^2 + 384*a^2*d^4) 
*ArcTanh[(Sqrt[d]*Sqrt[x])/(-Sqrt[c] + Sqrt[c + d*x])]))/(7680*d^(13/2)*x^ 
(3/2))
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {521, 27, 2125, 27, 521, 27, 90, 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.

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

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {(e x)^{3/2} \left (-11 b^2 c x^3 e^4+24 a b d x^2 e^4+12 a^2 d e^4\right )}{2 \sqrt {c+d x}}dx}{6 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(e x)^{3/2} \left (-11 b^2 c x^3 e^4+24 a b d x^2 e^4+12 a^2 d e^4\right )}{\sqrt {c+d x}}dx}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int \frac {3 e^7 (e x)^{3/2} \left (40 a^2 d^2+b \left (33 b c^2+80 a d^2\right ) x^2\right )}{2 \sqrt {c+d x}}dx}{5 d e^3}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 e^4 \int \frac {(e x)^{3/2} \left (40 a^2 d^2+b \left (33 b c^2+80 a d^2\right ) x^2\right )}{\sqrt {c+d x}}dx}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\int \frac {e^2 (e x)^{3/2} \left (320 a^2 d^3-7 b c \left (33 b c^2+80 a d^2\right ) x\right )}{2 \sqrt {c+d x}}dx}{4 d e^2}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\int \frac {(e x)^{3/2} \left (320 a^2 d^3-7 b c \left (33 b c^2+80 a d^2\right ) x\right )}{\sqrt {c+d x}}dx}{8 d}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\frac {5 \left (384 a^2 d^4+560 a b c^2 d^2+231 b^2 c^4\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}-\frac {7 b c (e x)^{5/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{3 d e}}{8 d}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\frac {5 \left (384 a^2 d^4+560 a b c^2 d^2+231 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}-\frac {7 b c (e x)^{5/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{3 d e}}{8 d}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\frac {5 \left (384 a^2 d^4+560 a b c^2 d^2+231 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}-\frac {7 b c (e x)^{5/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{3 d e}}{8 d}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\frac {5 \left (384 a^2 d^4+560 a b c^2 d^2+231 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}-\frac {7 b c (e x)^{5/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{3 d e}}{8 d}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3 e^4 \left (\frac {\frac {5 \left (384 a^2 d^4+560 a b c^2 d^2+231 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )}{6 d}-\frac {7 b c (e x)^{5/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{3 d e}}{8 d}+\frac {b (e x)^{7/2} \sqrt {c+d x} \left (80 a d^2+33 b c^2\right )}{4 d e^2}\right )}{10 d}-\frac {11 b^2 c e (e x)^{9/2} \sqrt {c+d x}}{5 d}}{12 d e^4}+\frac {b^2 (e x)^{11/2} \sqrt {c+d x}}{6 d e^4}\)

Input:

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

Output:

(b^2*(e*x)^(11/2)*Sqrt[c + d*x])/(6*d*e^4) + ((-11*b^2*c*e*(e*x)^(9/2)*Sqr 
t[c + d*x])/(5*d) + (3*e^4*((b*(33*b*c^2 + 80*a*d^2)*(e*x)^(7/2)*Sqrt[c + 
d*x])/(4*d*e^2) + ((-7*b*c*(33*b*c^2 + 80*a*d^2)*(e*x)^(5/2)*Sqrt[c + d*x] 
)/(3*d*e) + (5*(231*b^2*c^4 + 560*a*b*c^2*d^2 + 384*a^2*d^4)*(((e*x)^(3/2) 
*Sqrt[c + d*x])/(2*d) - (3*c*e*((Sqrt[e*x]*Sqrt[c + d*x])/d - (c*Sqrt[e]*A 
rcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/d^(3/2)))/(4*d)))/(6* 
d))/(8*d)))/(10*d))/(12*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 = 0.30 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {\left (-1280 b^{2} x^{5} d^{5}+1408 b^{2} c \,x^{4} d^{4}-3840 a b \,d^{5} x^{3}-1584 c^{2} d^{3} x^{3} b^{2}+4480 a b c \,d^{4} x^{2}+1848 b^{2} c^{3} d^{2} x^{2}-3840 a^{2} x \,d^{5}-5600 a b \,c^{2} d^{3} x -2310 b^{2} c^{4} d x +5760 a^{2} c \,d^{4}+8400 a \,c^{3} d^{2} b +3465 c^{5} b^{2}\right ) x \sqrt {d x +c}\, e^{2}}{7680 d^{6} \sqrt {e x}}+\frac {c^{2} \left (384 a^{2} d^{4}+560 b \,c^{2} d^{2} a +231 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e^{2} \sqrt {\left (d x +c \right ) e x}}{1024 d^{6} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(248\)
default	\(\frac {\sqrt {e x}\, \sqrt {d x +c}\, e \left (2560 b^{2} d^{5} x^{5} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-2816 b^{2} c \,d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+7680 a b \,d^{5} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+3168 b^{2} c^{2} d^{3} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-8960 a b c \,d^{4} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-3696 b^{2} c^{3} d^{2} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+5760 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{2} d^{4} e +8400 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{4} d^{2} e +3465 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{6} e +7680 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} d^{5} x +11200 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{2} d^{3} x +4620 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{4} d x -11520 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{4}-16800 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{2}-6930 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5}\right )}{15360 d^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(478\)

Input:

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

Output:

-1/7680*(-1280*b^2*d^5*x^5+1408*b^2*c*d^4*x^4-3840*a*b*d^5*x^3-1584*b^2*c^ 
2*d^3*x^3+4480*a*b*c*d^4*x^2+1848*b^2*c^3*d^2*x^2-3840*a^2*d^5*x-5600*a*b* 
c^2*d^3*x-2310*b^2*c^4*d*x+5760*a^2*c*d^4+8400*a*b*c^3*d^2+3465*b^2*c^5)*x 
*(d*x+c)^(1/2)/d^6*e^2/(e*x)^(1/2)+1/1024*c^2*(384*a^2*d^4+560*a*b*c^2*d^2 
+231*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 = 0.14 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.48 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\left [\frac {15 \, {\left (231 \, b^{2} c^{6} + 560 \, a b c^{4} d^{2} + 384 \, a^{2} c^{2} d^{4}\right )} e \sqrt {\frac {e}{d}} \log \left (2 \, d e x + 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (1280 \, b^{2} d^{5} e x^{5} - 1408 \, b^{2} c d^{4} e x^{4} + 48 \, {\left (33 \, b^{2} c^{2} d^{3} + 80 \, a b d^{5}\right )} e x^{3} - 56 \, {\left (33 \, b^{2} c^{3} d^{2} + 80 \, a b c d^{4}\right )} e x^{2} + 10 \, {\left (231 \, b^{2} c^{4} d + 560 \, a b c^{2} d^{3} + 384 \, a^{2} d^{5}\right )} e x - 15 \, {\left (231 \, b^{2} c^{5} + 560 \, a b c^{3} d^{2} + 384 \, a^{2} c d^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{15360 \, d^{6}}, -\frac {15 \, {\left (231 \, b^{2} c^{6} + 560 \, a b c^{4} d^{2} + 384 \, a^{2} c^{2} d^{4}\right )} e \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) - {\left (1280 \, b^{2} d^{5} e x^{5} - 1408 \, b^{2} c d^{4} e x^{4} + 48 \, {\left (33 \, b^{2} c^{2} d^{3} + 80 \, a b d^{5}\right )} e x^{3} - 56 \, {\left (33 \, b^{2} c^{3} d^{2} + 80 \, a b c d^{4}\right )} e x^{2} + 10 \, {\left (231 \, b^{2} c^{4} d + 560 \, a b c^{2} d^{3} + 384 \, a^{2} d^{5}\right )} e x - 15 \, {\left (231 \, b^{2} c^{5} + 560 \, a b c^{3} d^{2} + 384 \, a^{2} c d^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{7680 \, d^{6}}\right ] \] Input:

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

Output:

[1/15360*(15*(231*b^2*c^6 + 560*a*b*c^4*d^2 + 384*a^2*c^2*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*(1280*b^2* 
d^5*e*x^5 - 1408*b^2*c*d^4*e*x^4 + 48*(33*b^2*c^2*d^3 + 80*a*b*d^5)*e*x^3 
- 56*(33*b^2*c^3*d^2 + 80*a*b*c*d^4)*e*x^2 + 10*(231*b^2*c^4*d + 560*a*b*c 
^2*d^3 + 384*a^2*d^5)*e*x - 15*(231*b^2*c^5 + 560*a*b*c^3*d^2 + 384*a^2*c* 
d^4)*e)*sqrt(d*x + c)*sqrt(e*x))/d^6, -1/7680*(15*(231*b^2*c^6 + 560*a*b*c 
^4*d^2 + 384*a^2*c^2*d^4)*e*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d*sq 
rt(-e/d)/(d*e*x + c*e)) - (1280*b^2*d^5*e*x^5 - 1408*b^2*c*d^4*e*x^4 + 48* 
(33*b^2*c^2*d^3 + 80*a*b*d^5)*e*x^3 - 56*(33*b^2*c^3*d^2 + 80*a*b*c*d^4)*e 
*x^2 + 10*(231*b^2*c^4*d + 560*a*b*c^2*d^3 + 384*a^2*d^5)*e*x - 15*(231*b^ 
2*c^5 + 560*a*b*c^3*d^2 + 384*a^2*c*d^4)*e)*sqrt(d*x + c)*sqrt(e*x))/d^6]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)*(b*x^2+a)^2/(d*x+c)^(1/2),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 [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.24 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=-\frac {{\left (\frac {80 \, {\left (\frac {105 \, c^{4} e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{2}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{3}} - \frac {25 \, c}{d^{3}}\right )} + \frac {163 \, c^{2}}{d^{3}}\right )} - \frac {279 \, c^{3}}{d^{3}}\right )} \sqrt {d x + c}\right )} a b {\left | d \right |}}{d^{2}} + \frac {{\left (\frac {3465 \, c^{6} e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (d x + c\right )} {\left (8 \, {\left (d x + c\right )} {\left (\frac {10 \, {\left (d x + c\right )}}{d^{5}} - \frac {61 \, c}{d^{5}}\right )} + \frac {1251 \, c^{2}}{d^{5}}\right )} - \frac {3481 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {11395 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {11895 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c}\right )} b^{2} {\left | d \right |}}{d^{2}} + \frac {1920 \, {\left (\frac {3 \, c^{2} d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, d x - 3 \, c\right )} \sqrt {d x + c}\right )} a^{2} {\left | d \right |}}{d^{3}}\right )} e}{7680 \, d} \] Input:

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

Output:

-1/7680*(80*(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)*sqr 
t(d*x + c))*a*b*abs(d)/d^2 + (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*abs(d)/d^2 + 1920*(3*c^2*d*e*log(abs(-sqrt(d*e)*sqr 
t(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*abs(d)/d^3)*e/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2}{\sqrt {c+d\,x}} \,d x \] 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 = 0.27 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.03 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, e \left (-5760 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{5}+3840 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{6} x -8400 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{3}+5600 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{4} x -4480 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{5} x^{2}+3840 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{6} x^{3}-3465 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d +2310 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{2} x -1848 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{3} x^{2}+1584 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{4} x^{3}-1408 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{5} x^{4}+1280 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{6} x^{5}+5760 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{2} d^{4}+8400 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{4} d^{2}+3465 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{6}\right )}{7680 d^{7}} \] Input:

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

Output:

(sqrt(e)*e*( - 5760*sqrt(x)*sqrt(c + d*x)*a**2*c*d**5 + 3840*sqrt(x)*sqrt( 
c + d*x)*a**2*d**6*x - 8400*sqrt(x)*sqrt(c + d*x)*a*b*c**3*d**3 + 5600*sqr 
t(x)*sqrt(c + d*x)*a*b*c**2*d**4*x - 4480*sqrt(x)*sqrt(c + d*x)*a*b*c*d**5 
*x**2 + 3840*sqrt(x)*sqrt(c + d*x)*a*b*d**6*x**3 - 3465*sqrt(x)*sqrt(c + d 
*x)*b**2*c**5*d + 2310*sqrt(x)*sqrt(c + d*x)*b**2*c**4*d**2*x - 1848*sqrt( 
x)*sqrt(c + d*x)*b**2*c**3*d**3*x**2 + 1584*sqrt(x)*sqrt(c + d*x)*b**2*c** 
2*d**4*x**3 - 1408*sqrt(x)*sqrt(c + d*x)*b**2*c*d**5*x**4 + 1280*sqrt(x)*s 
qrt(c + d*x)*b**2*d**6*x**5 + 5760*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sq 
rt(d))/sqrt(c))*a**2*c**2*d**4 + 8400*sqrt(d)*log((sqrt(c + d*x) + sqrt(x) 
*sqrt(d))/sqrt(c))*a*b*c**4*d**2 + 3465*sqrt(d)*log((sqrt(c + d*x) + sqrt( 
x)*sqrt(d))/sqrt(c))*b**2*c**6))/(7680*d**7)