\(\int \frac {\sqrt {c+d x} (a+b x^2)^2}{(e x)^{3/2}} \, dx\) [791]

Optimal result

Integrand size = 26, antiderivative size = 234 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=-\frac {2 a^2 \sqrt {c+d x}}{e \sqrt {e x}}-\frac {b c \left (5 b c^2+32 a d^2\right ) \sqrt {e x} \sqrt {c+d x}}{64 d^3 e^2}+\frac {b \left (5 b c^2+32 a d^2\right ) \sqrt {e x} (c+d x)^{3/2}}{32 d^3 e^2}-\frac {5 b^2 c (e x)^{3/2} (c+d x)^{3/2}}{24 d^2 e^3}+\frac {b^2 (e x)^{5/2} (c+d x)^{3/2}}{4 d e^4}-\frac {\left (5 b^2 c^4+32 a b c^2 d^2-128 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{64 d^{7/2} e^{3/2}} \] Output:

-2*a^2*(d*x+c)^(1/2)/e/(e*x)^(1/2)-1/64*b*c*(32*a*d^2+5*b*c^2)*(e*x)^(1/2) 
*(d*x+c)^(1/2)/d^3/e^2+1/32*b*(32*a*d^2+5*b*c^2)*(e*x)^(1/2)*(d*x+c)^(3/2) 
/d^3/e^2-5/24*b^2*c*(e*x)^(3/2)*(d*x+c)^(3/2)/d^2/e^3+1/4*b^2*(e*x)^(5/2)* 
(d*x+c)^(3/2)/d/e^4-1/64*(-128*a^2*d^4+32*a*b*c^2*d^2+5*b^2*c^4)*arctanh(d 
^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(7/2)/e^(3/2)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {x \left (\sqrt {d} \sqrt {c+d x} \left (-384 a^2 d^3+96 a b d^2 x (c+2 d x)+b^2 x \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )\right )+6 \left (5 b^2 c^4+32 a b c^2 d^2-128 a^2 d^4\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )\right )}{192 d^{7/2} (e x)^{3/2}} \] Input:

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

Output:

(x*(Sqrt[d]*Sqrt[c + d*x]*(-384*a^2*d^3 + 96*a*b*d^2*x*(c + 2*d*x) + b^2*x 
*(15*c^3 - 10*c^2*d*x + 8*c*d^2*x^2 + 48*d^3*x^3)) + 6*(5*b^2*c^4 + 32*a*b 
*c^2*d^2 - 128*a^2*d^4)*Sqrt[x]*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sqrt[ 
c + d*x])]))/(192*d^(7/2)*(e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {520, 27, 2125, 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.

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

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2125

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\frac {\frac {\int \frac {3 e^5 \sqrt {c+d x} \left (32 a^2 d^3+b c \left (5 b c^2+32 a d^2\right ) x\right )}{2 \sqrt {e x}}dx}{3 d e^2}-\frac {5 b^2 c^2 e (e x)^{3/2} (c+d x)^{3/2}}{3 d}}{8 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{3/2}}{4 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{3/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {e^3 \int \frac {\sqrt {c+d x} \left (32 a^2 d^3+b c \left (5 b c^2+32 a d^2\right ) x\right )}{\sqrt {e x}}dx}{2 d}-\frac {5 b^2 c^2 e (e x)^{3/2} (c+d x)^{3/2}}{3 d}}{8 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{3/2}}{4 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{3/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {e^3 \left (\frac {b c \sqrt {e x} (c+d x)^{3/2} \left (32 a d^2+5 b c^2\right )}{2 d e}-\frac {\left (-128 a^2 d^4+32 a b c^2 d^2+5 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x}}dx}{4 d}\right )}{2 d}-\frac {5 b^2 c^2 e (e x)^{3/2} (c+d x)^{3/2}}{3 d}}{8 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{3/2}}{4 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{3/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {e^3 \left (\frac {b c \sqrt {e x} (c+d x)^{3/2} \left (32 a d^2+5 b c^2\right )}{2 d e}-\frac {\left (-128 a^2 d^4+32 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {1}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{4 d}\right )}{2 d}-\frac {5 b^2 c^2 e (e x)^{3/2} (c+d x)^{3/2}}{3 d}}{8 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{3/2}}{4 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{3/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {\frac {e^3 \left (\frac {b c \sqrt {e x} (c+d x)^{3/2} \left (32 a d^2+5 b c^2\right )}{2 d e}-\frac {\left (-128 a^2 d^4+32 a b c^2 d^2+5 b^2 c^4\right ) \left (c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{4 d}\right )}{2 d}-\frac {5 b^2 c^2 e (e x)^{3/2} (c+d x)^{3/2}}{3 d}}{8 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{3/2}}{4 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{3/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {e^3 \left (\frac {b c \sqrt {e x} (c+d x)^{3/2} \left (32 a d^2+5 b c^2\right )}{2 d e}-\frac {\left (-128 a^2 d^4+32 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )}{4 d}\right )}{2 d}-\frac {5 b^2 c^2 e (e x)^{3/2} (c+d x)^{3/2}}{3 d}}{8 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{3/2}}{4 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{3/2}}{c e \sqrt {e x}}\)

Input:

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

Output:

(-2*a^2*(c + d*x)^(3/2))/(c*e*Sqrt[e*x]) + ((b^2*c*(e*x)^(5/2)*(c + d*x)^( 
3/2))/(4*d*e^3) + ((-5*b^2*c^2*e*(e*x)^(3/2)*(c + d*x)^(3/2))/(3*d) + (e^3 
*((b*c*(5*b*c^2 + 32*a*d^2)*Sqrt[e*x]*(c + d*x)^(3/2))/(2*d*e) - ((5*b^2*c 
^4 + 32*a*b*c^2*d^2 - 128*a^2*d^4)*((Sqrt[e*x]*Sqrt[c + d*x])/e + (c*ArcTa 
nh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(Sqrt[d]*Sqrt[e])))/(4*d) 
))/(2*d))/(8*d*e^3))/(c*e)
 

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 520

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

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 = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-48 b^{2} x^{4} d^{3}-8 b^{2} c \,x^{3} d^{2}-192 a b \,d^{3} x^{2}+10 b^{2} c^{2} d \,x^{2}-96 a b c \,d^{2} x -15 c^{3} b^{2} x +384 a^{2} d^{3}\right )}{192 d^{3} e \sqrt {e x}}+\frac {\left (128 a^{2} d^{4}-32 b \,c^{2} d^{2} a -5 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{128 d^{3} \sqrt {d e}\, e \sqrt {e x}\, \sqrt {d x +c}}\)	\(186\)
default	\(\frac {\sqrt {d x +c}\, \left (96 b^{2} d^{3} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+16 x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c \,d^{2}+384 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} d^{4} e x -96 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{2} d^{2} e x -15 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} e x +384 a b \,d^{3} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-20 x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{2} d +192 a b c \,d^{2} x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+30 b^{2} c^{3} x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-768 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a^{2} d^{3}\right )}{384 e \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, d^{3}}\)	\(347\)

Input:

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

Output:

-1/192*(d*x+c)^(1/2)*(-48*b^2*d^3*x^4-8*b^2*c*d^2*x^3-192*a*b*d^3*x^2+10*b 
^2*c^2*d*x^2-96*a*b*c*d^2*x-15*b^2*c^3*x+384*a^2*d^3)/d^3/e/(e*x)^(1/2)+1/ 
128*(128*a^2*d^4-32*a*b*c^2*d^2-5*b^2*c^4)/d^3*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 = 0.11 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{2} c^{4} + 32 \, a b c^{2} d^{2} - 128 \, a^{2} d^{4}\right )} \sqrt {d e} x \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) - 2 \, {\left (48 \, b^{2} d^{4} x^{4} + 8 \, b^{2} c d^{3} x^{3} - 384 \, a^{2} d^{4} - 2 \, {\left (5 \, b^{2} c^{2} d^{2} - 96 \, a b d^{4}\right )} x^{2} + 3 \, {\left (5 \, b^{2} c^{3} d + 32 \, a b c d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{384 \, d^{4} e^{2} x}, \frac {3 \, {\left (5 \, b^{2} c^{4} + 32 \, a b c^{2} d^{2} - 128 \, a^{2} d^{4}\right )} \sqrt {-d e} x \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) + {\left (48 \, b^{2} d^{4} x^{4} + 8 \, b^{2} c d^{3} x^{3} - 384 \, a^{2} d^{4} - 2 \, {\left (5 \, b^{2} c^{2} d^{2} - 96 \, a b d^{4}\right )} x^{2} + 3 \, {\left (5 \, b^{2} c^{3} d + 32 \, a b c d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{192 \, d^{4} e^{2} x}\right ] \] Input:

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

Output:

[-1/384*(3*(5*b^2*c^4 + 32*a*b*c^2*d^2 - 128*a^2*d^4)*sqrt(d*e)*x*log(2*d* 
e*x + c*e + 2*sqrt(d*e)*sqrt(d*x + c)*sqrt(e*x)) - 2*(48*b^2*d^4*x^4 + 8*b 
^2*c*d^3*x^3 - 384*a^2*d^4 - 2*(5*b^2*c^2*d^2 - 96*a*b*d^4)*x^2 + 3*(5*b^2 
*c^3*d + 32*a*b*c*d^3)*x)*sqrt(d*x + c)*sqrt(e*x))/(d^4*e^2*x), 1/192*(3*( 
5*b^2*c^4 + 32*a*b*c^2*d^2 - 128*a^2*d^4)*sqrt(-d*e)*x*arctan(sqrt(-d*e)*s 
qrt(d*x + c)*sqrt(e*x)/(d*e*x + c*e)) + (48*b^2*d^4*x^4 + 8*b^2*c*d^3*x^3 
- 384*a^2*d^4 - 2*(5*b^2*c^2*d^2 - 96*a*b*d^4)*x^2 + 3*(5*b^2*c^3*d + 32*a 
*b*c*d^3)*x)*sqrt(d*x + c)*sqrt(e*x))/(d^4*e^2*x)]
 

Sympy [A] (verification not implemented)

Time = 27.29 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=- \frac {2 a^{2} \sqrt {c}}{e^{\frac {3}{2}} \sqrt {x} \sqrt {1 + \frac {d x}{c}}} + \frac {2 a^{2} \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{e^{\frac {3}{2}}} - \frac {2 a^{2} d \sqrt {x}}{\sqrt {c} e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {a b c^{\frac {3}{2}} \sqrt {x}}{2 d e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {3 a b \sqrt {c} x^{\frac {3}{2}}}{2 e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} - \frac {a b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{2 d^{\frac {3}{2}} e^{\frac {3}{2}}} + \frac {a b d x^{\frac {5}{2}}}{\sqrt {c} e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {5 b^{2} c^{\frac {7}{2}} \sqrt {x}}{64 d^{3} e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {5 b^{2} c^{\frac {5}{2}} x^{\frac {3}{2}}}{192 d^{2} e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} - \frac {b^{2} c^{\frac {3}{2}} x^{\frac {5}{2}}}{96 d e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {7 b^{2} \sqrt {c} x^{\frac {7}{2}}}{24 e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} - \frac {5 b^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{64 d^{\frac {7}{2}} e^{\frac {3}{2}}} + \frac {b^{2} d x^{\frac {9}{2}}}{4 \sqrt {c} e^{\frac {3}{2}} \sqrt {1 + \frac {d x}{c}}} \] Input:

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

Output:

-2*a**2*sqrt(c)/(e**(3/2)*sqrt(x)*sqrt(1 + d*x/c)) + 2*a**2*sqrt(d)*asinh( 
sqrt(d)*sqrt(x)/sqrt(c))/e**(3/2) - 2*a**2*d*sqrt(x)/(sqrt(c)*e**(3/2)*sqr 
t(1 + d*x/c)) + a*b*c**(3/2)*sqrt(x)/(2*d*e**(3/2)*sqrt(1 + d*x/c)) + 3*a* 
b*sqrt(c)*x**(3/2)/(2*e**(3/2)*sqrt(1 + d*x/c)) - a*b*c**2*asinh(sqrt(d)*s 
qrt(x)/sqrt(c))/(2*d**(3/2)*e**(3/2)) + a*b*d*x**(5/2)/(sqrt(c)*e**(3/2)*s 
qrt(1 + d*x/c)) + 5*b**2*c**(7/2)*sqrt(x)/(64*d**3*e**(3/2)*sqrt(1 + d*x/c 
)) + 5*b**2*c**(5/2)*x**(3/2)/(192*d**2*e**(3/2)*sqrt(1 + d*x/c)) - b**2*c 
**(3/2)*x**(5/2)/(96*d*e**(3/2)*sqrt(1 + d*x/c)) + 7*b**2*sqrt(c)*x**(7/2) 
/(24*e**(3/2)*sqrt(1 + d*x/c)) - 5*b**2*c**4*asinh(sqrt(d)*sqrt(x)/sqrt(c) 
)/(64*d**(7/2)*e**(3/2)) + b**2*d*x**(9/2)/(4*sqrt(c)*e**(3/2)*sqrt(1 + d* 
x/c))
 

Maxima [F(-2)]

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

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(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 = 0.18 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {{\left (\frac {{\left ({\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )} b^{2}}{d^{4}} - \frac {23 \, b^{2} c}{d^{4}}\right )} + \frac {127 \, b^{2} c^{2} d^{16} + 96 \, a b d^{18}}{d^{20}}\right )} - \frac {133 \, b^{2} c^{3} d^{16} + 288 \, a b c d^{18}}{d^{20}}\right )} {\left (d x + c\right )} + \frac {3 \, {\left (5 \, b^{2} c^{4} d^{16} + 32 \, a b c^{2} d^{18} - 128 \, a^{2} d^{20}\right )}}{d^{20}}\right )} \sqrt {d x + c}}{\sqrt {{\left (d x + c\right )} d e - c d e}} + \frac {3 \, {\left (5 \, b^{2} c^{4} + 32 \, a b c^{2} d^{2} - 128 \, a^{2} d^{4}\right )} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}}\right )} d^{2}}{192 \, e {\left | d \right |}} \] Input:

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

Output:

1/192*(((2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)*b^2/d^4 - 23*b^2*c/d^4) + ( 
127*b^2*c^2*d^16 + 96*a*b*d^18)/d^20) - (133*b^2*c^3*d^16 + 288*a*b*c*d^18 
)/d^20)*(d*x + c) + 3*(5*b^2*c^4*d^16 + 32*a*b*c^2*d^18 - 128*a^2*d^20)/d^ 
20)*sqrt(d*x + c)/sqrt((d*x + c)*d*e - c*d*e) + 3*(5*b^2*c^4 + 32*a*b*c^2* 
d^2 - 128*a^2*d^4)*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - 
 c*d*e)))/(sqrt(d*e)*d^4))*d^2/(e*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-384 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{4}+96 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{3} x +192 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{4} x^{2}+15 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d x -10 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{2} x^{2}+8 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{3} x^{3}+48 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{4} x^{4}+384 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} d^{4} x -96 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{2} d^{2} x -15 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} x -384 \sqrt {d}\, a^{2} d^{4} x -3 \sqrt {d}\, b^{2} c^{4} x \right )}{192 d^{4} e^{2} x} \] Input:

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

Output:

(sqrt(e)*( - 384*sqrt(x)*sqrt(c + d*x)*a**2*d**4 + 96*sqrt(x)*sqrt(c + d*x 
)*a*b*c*d**3*x + 192*sqrt(x)*sqrt(c + d*x)*a*b*d**4*x**2 + 15*sqrt(x)*sqrt 
(c + d*x)*b**2*c**3*d*x - 10*sqrt(x)*sqrt(c + d*x)*b**2*c**2*d**2*x**2 + 8 
*sqrt(x)*sqrt(c + d*x)*b**2*c*d**3*x**3 + 48*sqrt(x)*sqrt(c + d*x)*b**2*d* 
*4*x**4 + 384*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a**2* 
d**4*x - 96*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c** 
2*d**2*x - 15*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2* 
c**4*x - 384*sqrt(d)*a**2*d**4*x - 3*sqrt(d)*b**2*c**4*x))/(192*d**4*e**2* 
x)