\(\int \frac {(e x)^{3/2} (a+b x^2)^2}{(c+d x)^{3/2}} \, dx\) [835]

Optimal result

Integrand size = 26, antiderivative size = 309 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=-\frac {\left (11 b^2 c^4+20 a b c^2 d^2+10 a^2 d^4\right ) (e x)^{3/2}}{5 d^5 \sqrt {c+d x}}+\frac {b^2 (e x)^{11/2}}{5 d e^4 \sqrt {c+d x}}+\frac {\left (693 b^2 c^4+1120 a b c^2 d^2+384 a^2 d^4\right ) e \sqrt {e x} \sqrt {c+d x}}{128 d^6}-\frac {11 b c \left (123 b c^2+160 a d^2\right ) (e x)^{3/2} \sqrt {c+d x}}{960 d^5}+\frac {b \left (231 b c^2+160 a d^2\right ) (e x)^{5/2} \sqrt {c+d x}}{240 d^4 e}-\frac {11 b^2 c (e x)^{5/2} (c+d x)^{3/2}}{40 d^4 e}-\frac {c \left (693 b^2 c^4+1120 a b c^2 d^2+384 a^2 d^4\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{128 d^{13/2}} \] Output:

-1/5*(10*a^2*d^4+20*a*b*c^2*d^2+11*b^2*c^4)*(e*x)^(3/2)/d^5/(d*x+c)^(1/2)+ 
1/5*b^2*(e*x)^(11/2)/d/e^4/(d*x+c)^(1/2)+1/128*(384*a^2*d^4+1120*a*b*c^2*d 
^2+693*b^2*c^4)*e*(e*x)^(1/2)*(d*x+c)^(1/2)/d^6-11/960*b*c*(160*a*d^2+123* 
b*c^2)*(e*x)^(3/2)*(d*x+c)^(1/2)/d^5+1/240*b*(160*a*d^2+231*b*c^2)*(e*x)^( 
5/2)*(d*x+c)^(1/2)/d^4/e-11/40*b^2*c*(e*x)^(5/2)*(d*x+c)^(3/2)/d^4/e-1/128 
*c*(384*a^2*d^4+1120*a*b*c^2*d^2+693*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 = 0.99 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {e \sqrt {e x} \left (\frac {\sqrt {d} \left (1920 a^2 d^4 (3 c+d x)+160 a b d^2 \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )+3 b^2 \left (3465 c^5+1155 c^4 d x-462 c^3 d^2 x^2+264 c^2 d^3 x^3-176 c d^4 x^4+128 d^5 x^5\right )\right )}{\sqrt {c+d x}}+\frac {30 c \left (693 b^2 c^4+1120 a b c^2 d^2+384 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )}{\sqrt {x}}\right )}{1920 d^{13/2}} \] Input:

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

Output:

(e*Sqrt[e*x]*((Sqrt[d]*(1920*a^2*d^4*(3*c + d*x) + 160*a*b*d^2*(105*c^3 + 
35*c^2*d*x - 14*c*d^2*x^2 + 8*d^3*x^3) + 3*b^2*(3465*c^5 + 1155*c^4*d*x - 
462*c^3*d^2*x^2 + 264*c^2*d^3*x^3 - 176*c*d^4*x^4 + 128*d^5*x^5)))/Sqrt[c 
+ d*x] + (30*c*(693*b^2*c^4 + 1120*a*b*c^2*d^2 + 384*a^2*d^4)*ArcTanh[(Sqr 
t[d]*Sqrt[x])/(Sqrt[c] - Sqrt[c + d*x])])/Sqrt[x]))/(1920*d^(13/2))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 293, 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 = {519, 27, 2125, 27, 1194, 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.

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

\(\Big \downarrow \) 519

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {2 \int \frac {(e x)^{3/2} \left (\frac {5 b^2 c^4}{d^4}+\frac {b^2 x^2 c^2}{d^2}+\frac {10 a b c^2}{d^2}-\frac {b^2 x^3 c}{d}-\frac {b \left (b c^2+2 a d^2\right ) x c}{d^3}+4 a^2\right )}{2 \sqrt {c+d x}}dx}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\int \frac {(e x)^{3/2} \left (\frac {5 b^2 c^4}{d^4}+\frac {b^2 x^2 c^2}{d^2}+\frac {10 a b c^2}{d^2}-\frac {b^2 x^3 c}{d}-\frac {b \left (b c^2+2 a d^2\right ) x c}{d^3}+4 a^2\right )}{\sqrt {c+d x}}dx}{c}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\int \frac {(e x)^{3/2} \left (\frac {19 b^2 c^2 x^2 e^3}{d}+\frac {10 \left (5 b^2 c^4+10 a b d^2 c^2+4 a^2 d^4\right ) e^3}{d^3}-10 b c \left (\frac {b c^2}{d^2}+2 a\right ) x e^3\right )}{2 \sqrt {c+d x}}dx}{5 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\int \frac {(e x)^{3/2} \left (\frac {19 b^2 c^2 x^2 e^3}{d}+\frac {10 \left (5 b^2 c^4+10 a b d^2 c^2+4 a^2 d^4\right ) e^3}{d^3}-10 b c \left (\frac {b c^2}{d^2}+2 a\right ) x e^3\right )}{\sqrt {c+d x}}dx}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {\int \frac {e^5 (e x)^{3/2} \left (80 \left (5 b^2 c^4+10 a b d^2 c^2+4 a^2 d^4\right )-b c d^2 \left (\frac {213 b c^2}{d}+160 a d\right ) x\right )}{2 d^2 \sqrt {c+d x}}dx}{4 d e^2}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {e^3 \int \frac {(e x)^{3/2} \left (80 \left (5 b^2 c^4+10 a b d^2 c^2+4 a^2 d^4\right )-b c d \left (213 b c^2+160 a d^2\right ) x\right )}{\sqrt {c+d x}}dx}{8 d^3}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {e^3 \left (\frac {5}{6} \left (384 a^2 d^4+1120 a b c^2 d^2+693 b^2 c^4\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx-\frac {b c (e x)^{5/2} \sqrt {c+d x} \left (160 a d^2+213 b c^2\right )}{3 e}\right )}{8 d^3}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {e^3 \left (\frac {5}{6} \left (384 a^2 d^4+1120 a b c^2 d^2+693 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )-\frac {b c (e x)^{5/2} \sqrt {c+d x} \left (160 a d^2+213 b c^2\right )}{3 e}\right )}{8 d^3}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {e^3 \left (\frac {5}{6} \left (384 a^2 d^4+1120 a b c^2 d^2+693 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )-\frac {b c (e x)^{5/2} \sqrt {c+d x} \left (160 a d^2+213 b c^2\right )}{3 e}\right )}{8 d^3}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {e^3 \left (\frac {5}{6} \left (384 a^2 d^4+1120 a b c^2 d^2+693 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )-\frac {b c (e x)^{5/2} \sqrt {c+d x} \left (160 a d^2+213 b c^2\right )}{3 e}\right )}{8 d^3}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 (e x)^{5/2} \left (a d^2+b c^2\right )^2}{c d^4 e \sqrt {c+d x}}-\frac {\frac {\frac {e^3 \left (\frac {5}{6} \left (384 a^2 d^4+1120 a b c^2 d^2+693 b^2 c^4\right ) \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )-\frac {b c (e x)^{5/2} \sqrt {c+d x} \left (160 a d^2+213 b c^2\right )}{3 e}\right )}{8 d^3}+\frac {19 b^2 c^2 e (e x)^{7/2} \sqrt {c+d x}}{4 d^2}}{10 d e^3}-\frac {b^2 c (e x)^{9/2} \sqrt {c+d x}}{5 d^2 e^3}}{c}\)

Input:

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

Output:

(2*(b*c^2 + a*d^2)^2*(e*x)^(5/2))/(c*d^4*e*Sqrt[c + d*x]) - (-1/5*(b^2*c*( 
e*x)^(9/2)*Sqrt[c + d*x])/(d^2*e^3) + ((19*b^2*c^2*e*(e*x)^(7/2)*Sqrt[c + 
d*x])/(4*d^2) + (e^3*(-1/3*(b*c*(213*b*c^2 + 160*a*d^2)*(e*x)^(5/2)*Sqrt[c 
 + d*x])/e + (5*(693*b^2*c^4 + 1120*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] 
*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/d^(3/2)))/(4*d)))/6 
))/(8*d^3))/(10*d*e^3))/c
 

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 519

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, c + d*x, x], R = 
 PolynomialRemainder[(a + b*x^2)^p, c + d*x, x]}, Simp[(-R)*(e*x)^(m + 1)*( 
(c + d*x)^(n + 1)/(c*e*(n + 1))), x] + Simp[1/(c*(n + 1))   Int[(e*x)^m*(c 
+ d*x)^(n + 1)*ExpandToSum[c*(n + 1)*Qx + R*(m + n + 2), x], x], x]] /; Fre 
eQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && LtQ[n, -1] &&  !IntegerQ[m]
 

rule 1194

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

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 = 0.32 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {\left (384 b^{2} d^{4} x^{4}-912 b^{2} c \,d^{3} x^{3}+1280 a b \,d^{4} x^{2}+1704 d^{2} c^{2} x^{2} b^{2}-3520 a b c \,d^{3} x -3090 b^{2} c^{3} d x +1920 a^{2} d^{4}+9120 b \,c^{2} d^{2} a +6555 b^{2} c^{4}\right ) x \sqrt {d x +c}\, e^{2}}{1920 d^{6} \sqrt {e x}}-\frac {c \left (\frac {384 a^{2} d^{4} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}+\frac {693 b^{2} c^{4} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}-\frac {512 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{d e \left (x +\frac {c}{d}\right )}+\frac {1120 b \,c^{2} d^{2} a \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}\right ) e^{2} \sqrt {\left (d x +c \right ) e x}}{256 d^{6} \sqrt {e x}\, \sqrt {d x +c}}\)	\(347\)
default	\(-\frac {\left (-768 b^{2} d^{5} x^{5} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+1056 b^{2} c \,d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-2560 a b \,d^{5} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-1584 b^{2} c^{2} d^{3} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+5760 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c \,d^{5} e x +16800 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{3} d^{3} e x +10395 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{5} d e x +4480 a b c \,d^{4} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+2772 b^{2} c^{3} d^{2} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+5760 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{2} d^{4} e +16800 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{4} d^{2} e +10395 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{6} e -3840 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} d^{5} x -11200 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{2} d^{3} x -6930 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{4} d x -11520 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{4}-33600 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{2}-20790 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5}\right ) e \sqrt {e x}}{3840 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, d^{6} \sqrt {d x +c}}\)	\(614\)

Input:

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

Output:

1/1920*(384*b^2*d^4*x^4-912*b^2*c*d^3*x^3+1280*a*b*d^4*x^2+1704*b^2*c^2*d^ 
2*x^2-3520*a*b*c*d^3*x-3090*b^2*c^3*d*x+1920*a^2*d^4+9120*a*b*c^2*d^2+6555 
*b^2*c^4)*x*(d*x+c)^(1/2)/d^6*e^2/(e*x)^(1/2)-1/256*c/d^6*(384*a^2*d^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)+693*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)-512*(a^ 
2*d^4+2*a*b*c^2*d^2+b^2*c^4)/d/e/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2) 
+1120*b*c^2*d^2*a*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 = 0.12 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.79 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\left [\frac {15 \, {\left ({\left (693 \, b^{2} c^{5} d + 1120 \, a b c^{3} d^{3} + 384 \, a^{2} c d^{5}\right )} e x + {\left (693 \, b^{2} c^{6} + 1120 \, a b c^{4} d^{2} + 384 \, a^{2} c^{2} d^{4}\right )} e\right )} \sqrt {\frac {e}{d}} \log \left (2 \, d e x - 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (384 \, b^{2} d^{5} e x^{5} - 528 \, b^{2} c d^{4} e x^{4} + 8 \, {\left (99 \, b^{2} c^{2} d^{3} + 160 \, a b d^{5}\right )} e x^{3} - 14 \, {\left (99 \, b^{2} c^{3} d^{2} + 160 \, a b c d^{4}\right )} e x^{2} + 5 \, {\left (693 \, b^{2} c^{4} d + 1120 \, a b c^{2} d^{3} + 384 \, a^{2} d^{5}\right )} e x + 15 \, {\left (693 \, b^{2} c^{5} + 1120 \, a b c^{3} d^{2} + 384 \, a^{2} c d^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{3840 \, {\left (d^{7} x + c d^{6}\right )}}, \frac {15 \, {\left ({\left (693 \, b^{2} c^{5} d + 1120 \, a b c^{3} d^{3} + 384 \, a^{2} c d^{5}\right )} e x + {\left (693 \, b^{2} c^{6} + 1120 \, a b c^{4} d^{2} + 384 \, a^{2} c^{2} d^{4}\right )} e\right )} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) + {\left (384 \, b^{2} d^{5} e x^{5} - 528 \, b^{2} c d^{4} e x^{4} + 8 \, {\left (99 \, b^{2} c^{2} d^{3} + 160 \, a b d^{5}\right )} e x^{3} - 14 \, {\left (99 \, b^{2} c^{3} d^{2} + 160 \, a b c d^{4}\right )} e x^{2} + 5 \, {\left (693 \, b^{2} c^{4} d + 1120 \, a b c^{2} d^{3} + 384 \, a^{2} d^{5}\right )} e x + 15 \, {\left (693 \, b^{2} c^{5} + 1120 \, a b c^{3} d^{2} + 384 \, a^{2} c d^{4}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{1920 \, {\left (d^{7} x + c d^{6}\right )}}\right ] \] Input:

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

Output:

[1/3840*(15*((693*b^2*c^5*d + 1120*a*b*c^3*d^3 + 384*a^2*c*d^5)*e*x + (693 
*b^2*c^6 + 1120*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*(384*b^2*d^5*e*x^5 - 528* 
b^2*c*d^4*e*x^4 + 8*(99*b^2*c^2*d^3 + 160*a*b*d^5)*e*x^3 - 14*(99*b^2*c^3* 
d^2 + 160*a*b*c*d^4)*e*x^2 + 5*(693*b^2*c^4*d + 1120*a*b*c^2*d^3 + 384*a^2 
*d^5)*e*x + 15*(693*b^2*c^5 + 1120*a*b*c^3*d^2 + 384*a^2*c*d^4)*e)*sqrt(d* 
x + c)*sqrt(e*x))/(d^7*x + c*d^6), 1/1920*(15*((693*b^2*c^5*d + 1120*a*b*c 
^3*d^3 + 384*a^2*c*d^5)*e*x + (693*b^2*c^6 + 1120*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*sqrt(-e/d)/(d*e*x + 
c*e)) + (384*b^2*d^5*e*x^5 - 528*b^2*c*d^4*e*x^4 + 8*(99*b^2*c^2*d^3 + 160 
*a*b*d^5)*e*x^3 - 14*(99*b^2*c^3*d^2 + 160*a*b*c*d^4)*e*x^2 + 5*(693*b^2*c 
^4*d + 1120*a*b*c^2*d^3 + 384*a^2*d^5)*e*x + 15*(693*b^2*c^5 + 1120*a*b*c^ 
3*d^2 + 384*a^2*c*d^4)*e)*sqrt(d*x + c)*sqrt(e*x))/(d^7*x + c*d^6)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.18 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {1}{3840} \, {\left (2 \, \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, {\left (d x + c\right )} {\left (6 \, {\left (d x + c\right )} {\left (\frac {8 \, {\left (d x + c\right )} b^{2} {\left | d \right |}}{d^{8}} - \frac {51 \, b^{2} c {\left | d \right |}}{d^{8}}\right )} + \frac {843 \, b^{2} c^{2} d^{39} {\left | d \right |} + 160 \, a b d^{41} {\left | d \right |}}{d^{47}}\right )} - \frac {5 \, {\left (1077 \, b^{2} c^{3} d^{39} {\left | d \right |} + 608 \, a b c d^{41} {\left | d \right |}\right )}}{d^{47}}\right )} {\left (d x + c\right )} + \frac {15 \, {\left (843 \, b^{2} c^{4} d^{39} {\left | d \right |} + 928 \, a b c^{2} d^{41} {\left | d \right |} + 128 \, a^{2} d^{43} {\left | d \right |}\right )}}{d^{47}}\right )} \sqrt {d x + c} + \frac {15 \, {\left (693 \, \sqrt {d e} b^{2} c^{5} {\left | d \right |} + 1120 \, \sqrt {d e} a b c^{3} d^{2} {\left | d \right |} + 384 \, \sqrt {d e} a^{2} c d^{4} {\left | d \right |}\right )} \log \left ({\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}{d^{8}} + \frac {15360 \, {\left (\sqrt {d e} b^{2} c^{6} e {\left | d \right |} + 2 \, \sqrt {d e} a b c^{4} d^{2} e {\left | d \right |} + \sqrt {d e} a^{2} c^{2} d^{4} e {\left | d \right |}\right )}}{{\left (c d e + {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )} d^{7}}\right )} e \] Input:

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

Output:

1/3840*(2*sqrt((d*x + c)*d*e - c*d*e)*(2*(4*(d*x + c)*(6*(d*x + c)*(8*(d*x 
 + c)*b^2*abs(d)/d^8 - 51*b^2*c*abs(d)/d^8) + (843*b^2*c^2*d^39*abs(d) + 1 
60*a*b*d^41*abs(d))/d^47) - 5*(1077*b^2*c^3*d^39*abs(d) + 608*a*b*c*d^41*a 
bs(d))/d^47)*(d*x + c) + 15*(843*b^2*c^4*d^39*abs(d) + 928*a*b*c^2*d^41*ab 
s(d) + 128*a^2*d^43*abs(d))/d^47)*sqrt(d*x + c) + 15*(693*sqrt(d*e)*b^2*c^ 
5*abs(d) + 1120*sqrt(d*e)*a*b*c^3*d^2*abs(d) + 384*sqrt(d*e)*a^2*c*d^4*abs 
(d))*log((sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2)/d^8 + 
15360*(sqrt(d*e)*b^2*c^6*e*abs(d) + 2*sqrt(d*e)*a*b*c^4*d^2*e*abs(d) + sqr 
t(d*e)*a^2*c^2*d^4*e*abs(d))/((c*d*e + (sqrt(d*e)*sqrt(d*x + c) - sqrt((d* 
x + c)*d*e - c*d*e))^2)*d^7))*e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.06 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {e}\, e \left (-23040 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c \,d^{4}-67200 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{3} d^{2}-41580 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{5}+17280 \sqrt {d}\, \sqrt {d x +c}\, a^{2} c \,d^{4}+42000 \sqrt {d}\, \sqrt {d x +c}\, a b \,c^{3} d^{2}+24255 \sqrt {d}\, \sqrt {d x +c}\, b^{2} c^{5}+23040 \sqrt {x}\, a^{2} c \,d^{5}+7680 \sqrt {x}\, a^{2} d^{6} x +67200 \sqrt {x}\, a b \,c^{3} d^{3}+22400 \sqrt {x}\, a b \,c^{2} d^{4} x -8960 \sqrt {x}\, a b c \,d^{5} x^{2}+5120 \sqrt {x}\, a b \,d^{6} x^{3}+41580 \sqrt {x}\, b^{2} c^{5} d +13860 \sqrt {x}\, b^{2} c^{4} d^{2} x -5544 \sqrt {x}\, b^{2} c^{3} d^{3} x^{2}+3168 \sqrt {x}\, b^{2} c^{2} d^{4} x^{3}-2112 \sqrt {x}\, b^{2} c \,d^{5} x^{4}+1536 \sqrt {x}\, b^{2} d^{6} x^{5}\right )}{7680 \sqrt {d x +c}\, d^{7}} \] Input:

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

Output:

(sqrt(e)*e*( - 23040*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) + sqrt(x)*sq 
rt(d))/sqrt(c))*a**2*c*d**4 - 67200*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d* 
x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c**3*d**2 - 41580*sqrt(d)*sqrt(c + d*x) 
*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**5 + 17280*sqrt(d)* 
sqrt(c + d*x)*a**2*c*d**4 + 42000*sqrt(d)*sqrt(c + d*x)*a*b*c**3*d**2 + 24 
255*sqrt(d)*sqrt(c + d*x)*b**2*c**5 + 23040*sqrt(x)*a**2*c*d**5 + 7680*sqr 
t(x)*a**2*d**6*x + 67200*sqrt(x)*a*b*c**3*d**3 + 22400*sqrt(x)*a*b*c**2*d* 
*4*x - 8960*sqrt(x)*a*b*c*d**5*x**2 + 5120*sqrt(x)*a*b*d**6*x**3 + 41580*s 
qrt(x)*b**2*c**5*d + 13860*sqrt(x)*b**2*c**4*d**2*x - 5544*sqrt(x)*b**2*c* 
*3*d**3*x**2 + 3168*sqrt(x)*b**2*c**2*d**4*x**3 - 2112*sqrt(x)*b**2*c*d**5 
*x**4 + 1536*sqrt(x)*b**2*d**6*x**5))/(7680*sqrt(c + d*x)*d**7)