\(\int \frac {(c+d x)^{3/2} (a+b x^2)^2}{(e x)^{5/2}} \, dx\) [803]

Optimal result

Integrand size = 26, antiderivative size = 286 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=-\frac {\left (b^2 c^4+32 a b c^2 d^2+64 a^2 d^4\right ) \sqrt {c+d x}}{32 d^3 e^2 \sqrt {e x}}+\frac {b c \left (b c^2+32 a d^2\right ) \sqrt {e x} \sqrt {c+d x}}{64 d^2 e^3}-\frac {2 a^2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}+\frac {b \left (b c^2+32 a d^2\right ) (c+d x)^{5/2}}{32 d^3 e^2 \sqrt {e x}}-\frac {b^2 c \sqrt {e x} (c+d x)^{5/2}}{8 d^2 e^3}+\frac {b^2 (e x)^{3/2} (c+d x)^{5/2}}{4 d e^4}+\frac {\left (3 b^2 c^4+96 a b c^2 d^2+128 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{64 d^{5/2} e^{5/2}} \] Output:

-1/32*(64*a^2*d^4+32*a*b*c^2*d^2+b^2*c^4)*(d*x+c)^(1/2)/d^3/e^2/(e*x)^(1/2 
)+1/64*b*c*(32*a*d^2+b*c^2)*(e*x)^(1/2)*(d*x+c)^(1/2)/d^2/e^3-2/3*a^2*(d*x 
+c)^(3/2)/e/(e*x)^(3/2)+1/32*b*(32*a*d^2+b*c^2)*(d*x+c)^(5/2)/d^3/e^2/(e*x 
)^(1/2)-1/8*b^2*c*(e*x)^(1/2)*(d*x+c)^(5/2)/d^2/e^3+1/4*b^2*(e*x)^(3/2)*(d 
*x+c)^(5/2)/d/e^4+1/64*(128*a^2*d^4+96*a*b*c^2*d^2+3*b^2*c^4)*arctanh(d^(1 
/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(5/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.59 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=\frac {x \left (\sqrt {d} \sqrt {c+d x} \left (96 a b d^2 x^2 (5 c+2 d x)-128 a^2 d^2 (c+4 d x)+3 b^2 x^2 \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )\right )+6 \left (3 b^2 c^4+96 a b c^2 d^2+128 a^2 d^4\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )\right )}{192 d^{5/2} (e x)^{5/2}} \] Input:

Integrate[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(5/2),x]
 

Output:

(x*(Sqrt[d]*Sqrt[c + d*x]*(96*a*b*d^2*x^2*(5*c + 2*d*x) - 128*a^2*d^2*(c + 
 4*d*x) + 3*b^2*x^2*(-3*c^3 + 2*c^2*d*x + 24*c*d^2*x^2 + 16*d^3*x^3)) + 6* 
(3*b^2*c^4 + 96*a*b*c^2*d^2 + 128*a^2*d^4)*x^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[x 
])/(-Sqrt[c] + Sqrt[c + d*x])]))/(192*d^(5/2)*(e*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 271, 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 = {520, 27, 2124, 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.

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

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 521

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {\frac {\left (128 a^2 d^4+96 a b c^2 d^2+3 b^2 c^4\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {e x}}dx}{2 d}-\frac {3 b^2 c^3 \sqrt {e x} (c+d x)^{5/2}}{d e}}{8 d}+\frac {3 b^2 c^2 (e x)^{3/2} (c+d x)^{5/2}}{4 d e^2}}{c e}-\frac {4 a^2 d (c+d x)^{5/2}}{c e \sqrt {e x}}}{3 c e}-\frac {2 a^2 (c+d x)^{5/2}}{3 c e (e x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {\frac {\left (128 a^2 d^4+96 a b c^2 d^2+3 b^2 c^4\right ) \left (\frac {3}{4} c \int \frac {\sqrt {c+d x}}{\sqrt {e x}}dx+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{2 d}-\frac {3 b^2 c^3 \sqrt {e x} (c+d x)^{5/2}}{d e}}{8 d}+\frac {3 b^2 c^2 (e x)^{3/2} (c+d x)^{5/2}}{4 d e^2}}{c e}-\frac {4 a^2 d (c+d x)^{5/2}}{c e \sqrt {e x}}}{3 c e}-\frac {2 a^2 (c+d x)^{5/2}}{3 c e (e x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {\frac {\left (128 a^2 d^4+96 a b c^2 d^2+3 b^2 c^4\right ) \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{2 d}-\frac {3 b^2 c^3 \sqrt {e x} (c+d x)^{5/2}}{d e}}{8 d}+\frac {3 b^2 c^2 (e x)^{3/2} (c+d x)^{5/2}}{4 d e^2}}{c e}-\frac {4 a^2 d (c+d x)^{5/2}}{c e \sqrt {e x}}}{3 c e}-\frac {2 a^2 (c+d x)^{5/2}}{3 c e (e x)^{3/2}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {\frac {\frac {\left (128 a^2 d^4+96 a b c^2 d^2+3 b^2 c^4\right ) \left (\frac {3}{4} c \left (c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{2 d}-\frac {3 b^2 c^3 \sqrt {e x} (c+d x)^{5/2}}{d e}}{8 d}+\frac {3 b^2 c^2 (e x)^{3/2} (c+d x)^{5/2}}{4 d e^2}}{c e}-\frac {4 a^2 d (c+d x)^{5/2}}{c e \sqrt {e x}}}{3 c e}-\frac {2 a^2 (c+d x)^{5/2}}{3 c e (e x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {\left (128 a^2 d^4+96 a b c^2 d^2+3 b^2 c^4\right ) \left (\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{2 d}-\frac {3 b^2 c^3 \sqrt {e x} (c+d x)^{5/2}}{d e}}{8 d}+\frac {3 b^2 c^2 (e x)^{3/2} (c+d x)^{5/2}}{4 d e^2}}{c e}-\frac {4 a^2 d (c+d x)^{5/2}}{c e \sqrt {e x}}}{3 c e}-\frac {2 a^2 (c+d x)^{5/2}}{3 c e (e x)^{3/2}}\)

Input:

Int[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(5/2),x]
 

Output:

(-2*a^2*(c + d*x)^(5/2))/(3*c*e*(e*x)^(3/2)) + ((-4*a^2*d*(c + d*x)^(5/2)) 
/(c*e*Sqrt[e*x]) + ((3*b^2*c^2*(e*x)^(3/2)*(c + d*x)^(5/2))/(4*d*e^2) + (( 
-3*b^2*c^3*Sqrt[e*x]*(c + d*x)^(5/2))/(d*e) + ((3*b^2*c^4 + 96*a*b*c^2*d^2 
 + 128*a^2*d^4)*((Sqrt[e*x]*(c + d*x)^(3/2))/(2*e) + (3*c*((Sqrt[e*x]*Sqrt 
[c + d*x])/e + (c*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(S 
qrt[d]*Sqrt[e])))/4))/(2*d))/(8*d))/(c*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 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 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 = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-48 b^{2} d^{3} x^{5}-72 b^{2} c \,d^{2} x^{4}-192 a b \,d^{3} x^{3}-6 b^{2} c^{2} d \,x^{3}-480 a b c \,d^{2} x^{2}+9 b^{2} c^{3} x^{2}+512 a^{2} d^{3} x +128 a^{2} c \,d^{2}\right )}{192 x \,d^{2} e^{2} \sqrt {e x}}+\frac {\left (128 a^{2} d^{4}+96 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{128 d^{2} \sqrt {d e}\, e^{2} \sqrt {e x}\, \sqrt {d x +c}}\)	\(203\)
default	\(\frac {\sqrt {d x +c}\, \left (96 b^{2} d^{3} x^{5} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+144 b^{2} c \,d^{2} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+384 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} d^{4} e \,x^{2}+288 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{2} d^{2} e \,x^{2}+9 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} e \,x^{2}+384 a b \,d^{3} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+12 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{2} d \,x^{3}+960 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a b c \,d^{2} x^{2}-18 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{3} x^{2}-1024 a^{2} d^{3} x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-256 a^{2} c \,d^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\right )}{384 e^{2} x \,d^{2} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(385\)

Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/192*(d*x+c)^(1/2)*(-48*b^2*d^3*x^5-72*b^2*c*d^2*x^4-192*a*b*d^3*x^3-6*b 
^2*c^2*d*x^3-480*a*b*c*d^2*x^2+9*b^2*c^3*x^2+512*a^2*d^3*x+128*a^2*c*d^2)/ 
x/d^2/e^2/(e*x)^(1/2)+1/128*(128*a^2*d^4+96*a*b*c^2*d^2+3*b^2*c^4)/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^2*((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 = 363, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=\left [\frac {3 \, {\left (3 \, b^{2} c^{4} + 96 \, a b c^{2} d^{2} + 128 \, a^{2} d^{4}\right )} \sqrt {d e} x^{2} \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) + 2 \, {\left (48 \, b^{2} d^{4} x^{5} + 72 \, b^{2} c d^{3} x^{4} - 512 \, a^{2} d^{4} x - 128 \, a^{2} c d^{3} + 6 \, {\left (b^{2} c^{2} d^{2} + 32 \, a b d^{4}\right )} x^{3} - 3 \, {\left (3 \, b^{2} c^{3} d - 160 \, a b c d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{384 \, d^{3} e^{3} x^{2}}, -\frac {3 \, {\left (3 \, b^{2} c^{4} + 96 \, a b c^{2} d^{2} + 128 \, a^{2} d^{4}\right )} \sqrt {-d e} x^{2} \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) - {\left (48 \, b^{2} d^{4} x^{5} + 72 \, b^{2} c d^{3} x^{4} - 512 \, a^{2} d^{4} x - 128 \, a^{2} c d^{3} + 6 \, {\left (b^{2} c^{2} d^{2} + 32 \, a b d^{4}\right )} x^{3} - 3 \, {\left (3 \, b^{2} c^{3} d - 160 \, a b c d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{192 \, d^{3} e^{3} x^{2}}\right ] \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(5/2),x, algorithm="fricas")
 

Output:

[1/384*(3*(3*b^2*c^4 + 96*a*b*c^2*d^2 + 128*a^2*d^4)*sqrt(d*e)*x^2*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^5 + 72 
*b^2*c*d^3*x^4 - 512*a^2*d^4*x - 128*a^2*c*d^3 + 6*(b^2*c^2*d^2 + 32*a*b*d 
^4)*x^3 - 3*(3*b^2*c^3*d - 160*a*b*c*d^3)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(d 
^3*e^3*x^2), -1/192*(3*(3*b^2*c^4 + 96*a*b*c^2*d^2 + 128*a^2*d^4)*sqrt(-d* 
e)*x^2*arctan(sqrt(-d*e)*sqrt(d*x + c)*sqrt(e*x)/(d*e*x + c*e)) - (48*b^2* 
d^4*x^5 + 72*b^2*c*d^3*x^4 - 512*a^2*d^4*x - 128*a^2*c*d^3 + 6*(b^2*c^2*d^ 
2 + 32*a*b*d^4)*x^3 - 3*(3*b^2*c^3*d - 160*a*b*c*d^3)*x^2)*sqrt(d*x + c)*s 
qrt(e*x))/(d^3*e^3*x^2)]
 

Sympy [A] (verification not implemented)

Time = 35.65 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.84 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=- \frac {2 a^{2} \sqrt {c} d}{e^{\frac {5}{2}} \sqrt {x} \sqrt {1 + \frac {d x}{c}}} - \frac {2 a^{2} c \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{3 e^{\frac {5}{2}} x} - \frac {2 a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}}{3 e^{\frac {5}{2}}} + \frac {2 a^{2} d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{e^{\frac {5}{2}}} - \frac {2 a^{2} d^{2} \sqrt {x}}{\sqrt {c} e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {2 a b c^{\frac {3}{2}} \sqrt {x} \sqrt {1 + \frac {d x}{c}}}{e^{\frac {5}{2}}} + \frac {a b c^{\frac {3}{2}} \sqrt {x}}{2 e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {3 a b \sqrt {c} d x^{\frac {3}{2}}}{2 e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {3 a b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{2 \sqrt {d} e^{\frac {5}{2}}} + \frac {a b d^{2} x^{\frac {5}{2}}}{\sqrt {c} e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} - \frac {3 b^{2} c^{\frac {7}{2}} \sqrt {x}}{64 d^{2} e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} - \frac {b^{2} c^{\frac {5}{2}} x^{\frac {3}{2}}}{64 d e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {13 b^{2} c^{\frac {3}{2}} x^{\frac {5}{2}}}{32 e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {5 b^{2} \sqrt {c} d x^{\frac {7}{2}}}{8 e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {3 b^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{64 d^{\frac {5}{2}} e^{\frac {5}{2}}} + \frac {b^{2} d^{2} x^{\frac {9}{2}}}{4 \sqrt {c} e^{\frac {5}{2}} \sqrt {1 + \frac {d x}{c}}} \] Input:

integrate((d*x+c)**(3/2)*(b*x**2+a)**2/(e*x)**(5/2),x)
 

Output:

-2*a**2*sqrt(c)*d/(e**(5/2)*sqrt(x)*sqrt(1 + d*x/c)) - 2*a**2*c*sqrt(d)*sq 
rt(c/(d*x) + 1)/(3*e**(5/2)*x) - 2*a**2*d**(3/2)*sqrt(c/(d*x) + 1)/(3*e**( 
5/2)) + 2*a**2*d**(3/2)*asinh(sqrt(d)*sqrt(x)/sqrt(c))/e**(5/2) - 2*a**2*d 
**2*sqrt(x)/(sqrt(c)*e**(5/2)*sqrt(1 + d*x/c)) + 2*a*b*c**(3/2)*sqrt(x)*sq 
rt(1 + d*x/c)/e**(5/2) + a*b*c**(3/2)*sqrt(x)/(2*e**(5/2)*sqrt(1 + d*x/c)) 
 + 3*a*b*sqrt(c)*d*x**(3/2)/(2*e**(5/2)*sqrt(1 + d*x/c)) + 3*a*b*c**2*asin 
h(sqrt(d)*sqrt(x)/sqrt(c))/(2*sqrt(d)*e**(5/2)) + a*b*d**2*x**(5/2)/(sqrt( 
c)*e**(5/2)*sqrt(1 + d*x/c)) - 3*b**2*c**(7/2)*sqrt(x)/(64*d**2*e**(5/2)*s 
qrt(1 + d*x/c)) - b**2*c**(5/2)*x**(3/2)/(64*d*e**(5/2)*sqrt(1 + d*x/c)) + 
 13*b**2*c**(3/2)*x**(5/2)/(32*e**(5/2)*sqrt(1 + d*x/c)) + 5*b**2*sqrt(c)* 
d*x**(7/2)/(8*e**(5/2)*sqrt(1 + d*x/c)) + 3*b**2*c**4*asinh(sqrt(d)*sqrt(x 
)/sqrt(c))/(64*d**(5/2)*e**(5/2)) + b**2*d**2*x**(9/2)/(4*sqrt(c)*e**(5/2) 
*sqrt(1 + d*x/c))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Time = 0.20 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=\frac {d^{3} {\left (\frac {{\left ({\left (3 \, {\left (2 \, {\left (4 \, {\left (d x + c\right )} {\left (\frac {2 \, {\left (d x + c\right )} b^{2} e}{d^{3}} - \frac {7 \, b^{2} c e}{d^{3}}\right )} + \frac {33 \, b^{2} c^{3} d^{13} e^{2} + 32 \, a b c d^{15} e^{2}}{c d^{16} e}\right )} {\left (d x + c\right )} - \frac {25 \, b^{2} c^{4} d^{13} e^{2} + 32 \, a b c^{2} d^{15} e^{2}}{c d^{16} e}\right )} {\left (d x + c\right )} - \frac {4 \, {\left (3 \, b^{2} c^{5} d^{13} e^{2} + 96 \, a b c^{3} d^{15} e^{2} + 128 \, a^{2} c d^{17} e^{2}\right )}}{c d^{16} e}\right )} {\left (d x + c\right )} + \frac {3 \, {\left (3 \, b^{2} c^{6} d^{13} e^{2} + 96 \, a b c^{4} d^{15} e^{2} + 128 \, a^{2} c^{2} d^{17} e^{2}\right )}}{c d^{16} e}\right )} \sqrt {d x + c}}{{\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {3}{2}}} - \frac {3 \, {\left (3 \, b^{2} c^{4} + 96 \, a b c^{2} d^{2} + 128 \, a^{2} d^{4}\right )} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}}\right )}}{192 \, e^{2} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(5/2),x, algorithm="giac")
 

Output:

1/192*d^3*(((3*(2*(4*(d*x + c)*(2*(d*x + c)*b^2*e/d^3 - 7*b^2*c*e/d^3) + ( 
33*b^2*c^3*d^13*e^2 + 32*a*b*c*d^15*e^2)/(c*d^16*e))*(d*x + c) - (25*b^2*c 
^4*d^13*e^2 + 32*a*b*c^2*d^15*e^2)/(c*d^16*e))*(d*x + c) - 4*(3*b^2*c^5*d^ 
13*e^2 + 96*a*b*c^3*d^15*e^2 + 128*a^2*c*d^17*e^2)/(c*d^16*e))*(d*x + c) + 
 3*(3*b^2*c^6*d^13*e^2 + 96*a*b*c^4*d^15*e^2 + 128*a^2*c^2*d^17*e^2)/(c*d^ 
16*e))*sqrt(d*x + c)/((d*x + c)*d*e - c*d*e)^(3/2) - 3*(3*b^2*c^4 + 96*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))/(e^2*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(5/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{5/2}} \, dx=\frac {\sqrt {e}\, \left (-256 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{3}-1024 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{4} x +960 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{3} x^{2}+384 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{4} x^{3}-18 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d \,x^{2}+12 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{2} x^{3}+144 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{3} x^{4}+96 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{4} x^{5}+768 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} d^{4} x^{2}+576 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{2} d^{2} x^{2}+18 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} x^{2}+120 \sqrt {d}\, a b \,c^{2} d^{2} x^{2}+\sqrt {d}\, b^{2} c^{4} x^{2}\right )}{384 d^{3} e^{3} x^{2}} \] Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(5/2),x)
 

Output:

(sqrt(e)*( - 256*sqrt(x)*sqrt(c + d*x)*a**2*c*d**3 - 1024*sqrt(x)*sqrt(c + 
 d*x)*a**2*d**4*x + 960*sqrt(x)*sqrt(c + d*x)*a*b*c*d**3*x**2 + 384*sqrt(x 
)*sqrt(c + d*x)*a*b*d**4*x**3 - 18*sqrt(x)*sqrt(c + d*x)*b**2*c**3*d*x**2 
+ 12*sqrt(x)*sqrt(c + d*x)*b**2*c**2*d**2*x**3 + 144*sqrt(x)*sqrt(c + d*x) 
*b**2*c*d**3*x**4 + 96*sqrt(x)*sqrt(c + d*x)*b**2*d**4*x**5 + 768*sqrt(d)* 
log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a**2*d**4*x**2 + 576*sqrt(d 
)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c**2*d**2*x**2 + 18*s 
qrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**4*x**2 + 120 
*sqrt(d)*a*b*c**2*d**2*x**2 + sqrt(d)*b**2*c**4*x**2))/(384*d**3*e**3*x**2 
)