\(\int \frac {\sqrt {e x} (a+b x^2)^2}{(c+d x)^{5/2}} \, dx\) [846]

Optimal result

Integrand size = 26, antiderivative size = 245 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {\left (3 b^2 c^4+4 a b c^2 d^2+2 a^2 d^4\right ) (e x)^{3/2}}{3 c d^4 e (c+d x)^{3/2}}+\frac {b^2 (e x)^{9/2}}{3 d e^4 (c+d x)^{3/2}}+\frac {b c \left (9 b c^2+8 a d^2\right ) \sqrt {e x}}{d^5 \sqrt {c+d x}}+\frac {b \left (33 b c^2+16 a d^2\right ) \sqrt {e x} \sqrt {c+d x}}{8 d^5}-\frac {3 b^2 c (e x)^{3/2} \sqrt {c+d x}}{4 d^4 e}-\frac {5 b c \left (21 b c^2+16 a d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{8 d^{11/2}} \] Output:

1/3*(2*a^2*d^4+4*a*b*c^2*d^2+3*b^2*c^4)*(e*x)^(3/2)/c/d^4/e/(d*x+c)^(3/2)+ 
1/3*b^2*(e*x)^(9/2)/d/e^4/(d*x+c)^(3/2)+b*c*(8*a*d^2+9*b*c^2)*(e*x)^(1/2)/ 
d^5/(d*x+c)^(1/2)+1/8*b*(16*a*d^2+33*b*c^2)*(e*x)^(1/2)*(d*x+c)^(1/2)/d^5- 
3/4*b^2*c*(e*x)^(3/2)*(d*x+c)^(1/2)/d^4/e-5/8*b*c*(16*a*d^2+21*b*c^2)*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 = 0.66 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {e x} \left (\frac {\sqrt {d} \left (16 a^2 d^5 x+16 a b c d^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )+b^2 c \left (315 c^4+420 c^3 d x+63 c^2 d^2 x^2-18 c d^3 x^3+8 d^4 x^4\right )\right )}{(c+d x)^{3/2}}+\frac {30 b c^2 \left (21 b c^2+16 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )}{\sqrt {x}}\right )}{24 c d^{11/2}} \] Input:

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

Output:

(Sqrt[e*x]*((Sqrt[d]*(16*a^2*d^5*x + 16*a*b*c*d^2*(15*c^2 + 20*c*d*x + 3*d 
^2*x^2) + b^2*c*(315*c^4 + 420*c^3*d*x + 63*c^2*d^2*x^2 - 18*c*d^3*x^3 + 8 
*d^4*x^4)))/(c + d*x)^(3/2) + (30*b*c^2*(21*b*c^2 + 16*a*d^2)*ArcTanh[(Sqr 
t[d]*Sqrt[x])/(Sqrt[c] - Sqrt[c + d*x])])/Sqrt[x]))/(24*c*d^(11/2))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {519, 27, 2124, 27, 1194, 27, 90, 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.

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

\(\Big \downarrow \) 519

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2124

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {2 \int -\frac {\sqrt {e x} \left (-\frac {2 b^2 e x c^3}{d^3}+\frac {b^2 e x^2 c^2}{d^2}+\frac {b \left (11 b c^2+10 a d^2\right ) e c^2}{d^4}\right )}{2 \sqrt {c+d x}}dx}{c e}+\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\int \frac {\sqrt {e x} \left (-\frac {2 b^2 e x c^3}{d^3}+\frac {b^2 e x^2 c^2}{d^2}+\frac {b \left (11 b c^2+10 a d^2\right ) e c^2}{d^4}\right )}{\sqrt {c+d x}}dx}{c e}}{c}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\frac {\int \frac {b c^2 e^3 \sqrt {e x} \left (6 \left (11 b c^2+10 a d^2\right )-17 b c d x\right )}{2 d^3 \sqrt {c+d x}}dx}{3 d e^2}+\frac {b^2 c^2 (e x)^{5/2} \sqrt {c+d x}}{3 d^3 e}}{c e}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\frac {b c^2 e \int \frac {\sqrt {e x} \left (6 \left (11 b c^2+10 a d^2\right )-17 b c d x\right )}{\sqrt {c+d x}}dx}{6 d^4}+\frac {b^2 c^2 (e x)^{5/2} \sqrt {c+d x}}{3 d^3 e}}{c e}}{c}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\frac {b c^2 e \left (\frac {15}{4} \left (16 a d^2+21 b c^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx-\frac {17 b c (e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{6 d^4}+\frac {b^2 c^2 (e x)^{5/2} \sqrt {c+d x}}{3 d^3 e}}{c e}}{c}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\frac {b c^2 e \left (\frac {15}{4} \left (16 a d^2+21 b c^2\right ) \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )-\frac {17 b c (e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{6 d^4}+\frac {b^2 c^2 (e x)^{5/2} \sqrt {c+d x}}{3 d^3 e}}{c e}}{c}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\frac {b c^2 e \left (\frac {15}{4} \left (16 a d^2+21 b c^2\right ) \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )-\frac {17 b c (e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{6 d^4}+\frac {b^2 c^2 (e x)^{5/2} \sqrt {c+d x}}{3 d^3 e}}{c e}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 (e x)^{3/2} \left (a d^2+b c^2\right )^2}{3 c d^4 e (c+d x)^{3/2}}-\frac {\frac {8 b c (e x)^{3/2} \left (a d^2+b c^2\right )}{d^4 e \sqrt {c+d x}}-\frac {\frac {b c^2 e \left (\frac {15}{4} \left (16 a d^2+21 b c^2\right ) \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )-\frac {17 b c (e x)^{3/2} \sqrt {c+d x}}{2 e}\right )}{6 d^4}+\frac {b^2 c^2 (e x)^{5/2} \sqrt {c+d x}}{3 d^3 e}}{c e}}{c}\)

Input:

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

Output:

(2*(b*c^2 + a*d^2)^2*(e*x)^(3/2))/(3*c*d^4*e*(c + d*x)^(3/2)) - ((8*b*c*(b 
*c^2 + a*d^2)*(e*x)^(3/2))/(d^4*e*Sqrt[c + d*x]) - ((b^2*c^2*(e*x)^(5/2)*S 
qrt[c + d*x])/(3*d^3*e) + (b*c^2*e*((-17*b*c*(e*x)^(3/2)*Sqrt[c + d*x])/(2 
*e) + (15*(21*b*c^2 + 16*a*d^2)*((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))/(6*d^4 
))/(c*e))/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 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.46

method	result	size
risch	\(\frac {b \left (8 b \,x^{2} d^{2}-34 b c d x +48 a \,d^{2}+123 b \,c^{2}\right ) x \sqrt {d x +c}\, e}{24 d^{5} \sqrt {e x}}-\frac {\left (\frac {2 \left (-16 a^{2} d^{4}-96 b \,c^{2} d^{2} a -80 b^{2} c^{4}\right ) \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{d c e \left (x +\frac {c}{d}\right )}+\frac {105 c^{3} b^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}+\frac {16 c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \left (\frac {2 \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{3 c e \left (x +\frac {c}{d}\right )^{2}}+\frac {4 d \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{3 e \,c^{2} \left (x +\frac {c}{d}\right )}\right )}{d^{2}}+\frac {80 a b c \,d^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}\right ) e \sqrt {\left (d x +c \right ) e x}}{16 d^{5} \sqrt {e x}\, \sqrt {d x +c}}\)	\(357\)
default	\(-\frac {\left (-16 b^{2} c \,d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+240 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{2} d^{4} e \,x^{2}+315 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} d^{2} e \,x^{2}+36 b^{2} c^{2} d^{3} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+480 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{3} d^{3} e x +630 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{5} d e x -96 a b c \,d^{4} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-126 b^{2} c^{3} d^{2} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+240 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{4} d^{2} e +315 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{6} e -32 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} d^{5} x -640 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{2} d^{3} x -840 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{4} d x -480 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{2}-630 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5}\right ) \sqrt {e x}}{48 \sqrt {d e}\, c \sqrt {\left (d x +c \right ) e x}\, d^{5} \left (d x +c \right )^{\frac {3}{2}}}\)	\(547\)

Input:

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

Output:

1/24*b*(8*b*d^2*x^2-34*b*c*d*x+48*a*d^2+123*b*c^2)*x*(d*x+c)^(1/2)/d^5*e/( 
e*x)^(1/2)-1/16/d^5*(2*(-16*a^2*d^4-96*a*b*c^2*d^2-80*b^2*c^4)/d/c/e/(x+c/ 
d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+105*c^3*b^2*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)+16*c*(a^2*d^4+2*a*b*c^2*d^2+b^2*c 
^4)/d^2*(2/3/c/e/(x+c/d)^2*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+4/3*d/e/c^2/( 
x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2))+80*a*b*c*d^2*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 = 0.12 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\left [\frac {15 \, {\left (21 \, b^{2} c^{6} + 16 \, a b c^{4} d^{2} + {\left (21 \, b^{2} c^{4} d^{2} + 16 \, a b c^{2} d^{4}\right )} x^{2} + 2 \, {\left (21 \, b^{2} c^{5} d + 16 \, a b c^{3} d^{3}\right )} x\right )} \sqrt {\frac {e}{d}} \log \left (2 \, d e x - 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (8 \, b^{2} c d^{4} x^{4} - 18 \, b^{2} c^{2} d^{3} x^{3} + 315 \, b^{2} c^{5} + 240 \, a b c^{3} d^{2} + 3 \, {\left (21 \, b^{2} c^{3} d^{2} + 16 \, a b c d^{4}\right )} x^{2} + 4 \, {\left (105 \, b^{2} c^{4} d + 80 \, a b c^{2} d^{3} + 4 \, a^{2} d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{48 \, {\left (c d^{7} x^{2} + 2 \, c^{2} d^{6} x + c^{3} d^{5}\right )}}, \frac {15 \, {\left (21 \, b^{2} c^{6} + 16 \, a b c^{4} d^{2} + {\left (21 \, b^{2} c^{4} d^{2} + 16 \, a b c^{2} d^{4}\right )} x^{2} + 2 \, {\left (21 \, b^{2} c^{5} d + 16 \, a b c^{3} d^{3}\right )} x\right )} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) + {\left (8 \, b^{2} c d^{4} x^{4} - 18 \, b^{2} c^{2} d^{3} x^{3} + 315 \, b^{2} c^{5} + 240 \, a b c^{3} d^{2} + 3 \, {\left (21 \, b^{2} c^{3} d^{2} + 16 \, a b c d^{4}\right )} x^{2} + 4 \, {\left (105 \, b^{2} c^{4} d + 80 \, a b c^{2} d^{3} + 4 \, a^{2} d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{24 \, {\left (c d^{7} x^{2} + 2 \, c^{2} d^{6} x + c^{3} d^{5}\right )}}\right ] \] Input:

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

Output:

[1/48*(15*(21*b^2*c^6 + 16*a*b*c^4*d^2 + (21*b^2*c^4*d^2 + 16*a*b*c^2*d^4) 
*x^2 + 2*(21*b^2*c^5*d + 16*a*b*c^3*d^3)*x)*sqrt(e/d)*log(2*d*e*x - 2*sqrt 
(d*x + c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(8*b^2*c*d^4*x^4 - 18*b^2*c^2*d 
^3*x^3 + 315*b^2*c^5 + 240*a*b*c^3*d^2 + 3*(21*b^2*c^3*d^2 + 16*a*b*c*d^4) 
*x^2 + 4*(105*b^2*c^4*d + 80*a*b*c^2*d^3 + 4*a^2*d^5)*x)*sqrt(d*x + c)*sqr 
t(e*x))/(c*d^7*x^2 + 2*c^2*d^6*x + c^3*d^5), 1/24*(15*(21*b^2*c^6 + 16*a*b 
*c^4*d^2 + (21*b^2*c^4*d^2 + 16*a*b*c^2*d^4)*x^2 + 2*(21*b^2*c^5*d + 16*a* 
b*c^3*d^3)*x)*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d*sqrt(-e/d)/(d*e* 
x + c*e)) + (8*b^2*c*d^4*x^4 - 18*b^2*c^2*d^3*x^3 + 315*b^2*c^5 + 240*a*b* 
c^3*d^2 + 3*(21*b^2*c^3*d^2 + 16*a*b*c*d^4)*x^2 + 4*(105*b^2*c^4*d + 80*a* 
b*c^2*d^3 + 4*a^2*d^5)*x)*sqrt(d*x + c)*sqrt(e*x))/(c*d^7*x^2 + 2*c^2*d^6* 
x + c^3*d^5)]
 

Sympy [F]

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

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

Output:

Integral(sqrt(e*x)*(a + b*x**2)**2/(c + d*x)**(5/2), x)
 

Maxima [F(-2)]

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

integrate((e*x)^(1/2)*(b*x^2+a)^2/(d*x+c)^(5/2),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. 527 vs. \(2 (203) = 406\).

Time = 0.28 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {1}{24} \, \sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {d x + c} {\left (2 \, {\left (d x + c\right )} {\left (\frac {4 \, {\left (d x + c\right )} b^{2} {\left | d \right |}}{d^{7}} - \frac {25 \, b^{2} c {\left | d \right |}}{d^{7}}\right )} + \frac {3 \, {\left (55 \, b^{2} c^{2} d^{20} {\left | d \right |} + 16 \, a b d^{22} {\left | d \right |}\right )}}{d^{27}}\right )} + \frac {5 \, {\left (21 \, \sqrt {d e} b^{2} c^{3} {\left | d \right |} + 16 \, \sqrt {d e} a b c d^{2} {\left | d \right |}\right )} \log \left ({\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}{16 \, d^{7}} + \frac {4 \, {\left (13 \, \sqrt {d e} b^{2} c^{6} d^{2} e^{3} {\left | d \right |} + 14 \, \sqrt {d e} a b c^{4} d^{4} e^{3} {\left | d \right |} + \sqrt {d e} a^{2} c^{2} d^{6} e^{3} {\left | d \right |} + 24 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} b^{2} c^{5} d e^{2} {\left | d \right |} + 24 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} a b c^{3} d^{3} e^{2} {\left | d \right |} + 15 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} b^{2} c^{4} e {\left | d \right |} + 18 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} a b c^{2} d^{2} e {\left | d \right |} + 3 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} a^{2} d^{4} e {\left | d \right |}\right )}}{3 \, {\left (c d e + {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}^{3} d^{6}} \] Input:

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

Output:

1/24*sqrt((d*x + c)*d*e - c*d*e)*sqrt(d*x + c)*(2*(d*x + c)*(4*(d*x + c)*b 
^2*abs(d)/d^7 - 25*b^2*c*abs(d)/d^7) + 3*(55*b^2*c^2*d^20*abs(d) + 16*a*b* 
d^22*abs(d))/d^27) + 5/16*(21*sqrt(d*e)*b^2*c^3*abs(d) + 16*sqrt(d*e)*a*b* 
c*d^2*abs(d))*log((sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^ 
2)/d^7 + 4/3*(13*sqrt(d*e)*b^2*c^6*d^2*e^3*abs(d) + 14*sqrt(d*e)*a*b*c^4*d 
^4*e^3*abs(d) + sqrt(d*e)*a^2*c^2*d^6*e^3*abs(d) + 24*sqrt(d*e)*(sqrt(d*e) 
*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*b^2*c^5*d*e^2*abs(d) + 24* 
sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*a*b*c^ 
3*d^3*e^2*abs(d) + 15*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)* 
d*e - c*d*e))^4*b^2*c^4*e*abs(d) + 18*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - 
 sqrt((d*x + c)*d*e - c*d*e))^4*a*b*c^2*d^2*e*abs(d) + 3*sqrt(d*e)*(sqrt(d 
*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*a^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)^3*d^6)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\int \frac {\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2}{{\left (c+d\,x\right )}^{5/2}} \,d x \] 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 = 0.24 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {e}\, \left (-1920 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{3} d^{2}-1920 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{2} d^{3} x -2520 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{5}-2520 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} d x +128 \sqrt {d}\, \sqrt {d x +c}\, a^{2} c \,d^{4}+128 \sqrt {d}\, \sqrt {d x +c}\, a^{2} d^{5} x -320 \sqrt {d}\, \sqrt {d x +c}\, a b \,c^{3} d^{2}-320 \sqrt {d}\, \sqrt {d x +c}\, a b \,c^{2} d^{3} x -567 \sqrt {d}\, \sqrt {d x +c}\, b^{2} c^{5}-567 \sqrt {d}\, \sqrt {d x +c}\, b^{2} c^{4} d x +128 \sqrt {x}\, a^{2} d^{6} x +1920 \sqrt {x}\, a b \,c^{3} d^{3}+2560 \sqrt {x}\, a b \,c^{2} d^{4} x +384 \sqrt {x}\, a b c \,d^{5} x^{2}+2520 \sqrt {x}\, b^{2} c^{5} d +3360 \sqrt {x}\, b^{2} c^{4} d^{2} x +504 \sqrt {x}\, b^{2} c^{3} d^{3} x^{2}-144 \sqrt {x}\, b^{2} c^{2} d^{4} x^{3}+64 \sqrt {x}\, b^{2} c \,d^{5} x^{4}\right )}{192 \sqrt {d x +c}\, c \,d^{6} \left (d x +c \right )} \] Input:

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

Output:

(sqrt(e)*( - 1920*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) + sqrt(x)*sqrt( 
d))/sqrt(c))*a*b*c**3*d**2 - 1920*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) 
 + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c**2*d**3*x - 2520*sqrt(d)*sqrt(c + d*x)* 
log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**5 - 2520*sqrt(d)*sq 
rt(c + d*x)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**4*d*x + 
 128*sqrt(d)*sqrt(c + d*x)*a**2*c*d**4 + 128*sqrt(d)*sqrt(c + d*x)*a**2*d* 
*5*x - 320*sqrt(d)*sqrt(c + d*x)*a*b*c**3*d**2 - 320*sqrt(d)*sqrt(c + d*x) 
*a*b*c**2*d**3*x - 567*sqrt(d)*sqrt(c + d*x)*b**2*c**5 - 567*sqrt(d)*sqrt( 
c + d*x)*b**2*c**4*d*x + 128*sqrt(x)*a**2*d**6*x + 1920*sqrt(x)*a*b*c**3*d 
**3 + 2560*sqrt(x)*a*b*c**2*d**4*x + 384*sqrt(x)*a*b*c*d**5*x**2 + 2520*sq 
rt(x)*b**2*c**5*d + 3360*sqrt(x)*b**2*c**4*d**2*x + 504*sqrt(x)*b**2*c**3* 
d**3*x**2 - 144*sqrt(x)*b**2*c**2*d**4*x**3 + 64*sqrt(x)*b**2*c*d**5*x**4) 
)/(192*sqrt(c + d*x)*c*d**6*(c + d*x))