\(\int \frac {(c+d x)^{5/2} (a+b x^2)^2}{(e x)^{3/2}} \, dx\) [814]

Optimal result

Integrand size = 26, antiderivative size = 343 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {25 c \left (384 a^2 d^4-b c^2 \left (b c^2+16 a d^2\right )\right ) \sqrt {e x} \sqrt {c+d x}}{1536 d^3 e^2}+\frac {5 \left (384 a^2 d^4-b c^2 \left (b c^2+16 a d^2\right )\right ) (e x)^{3/2} \sqrt {c+d x}}{768 d^2 e^3}-\frac {2 a^2 (c+d x)^{5/2}}{e \sqrt {e x}}+\frac {5 b c \left (b c^2+16 a d^2\right ) \sqrt {e x} (c+d x)^{5/2}}{192 d^3 e^2}-\frac {b \left (5 b c^2-48 a d^2\right ) (e x)^{3/2} (c+d x)^{5/2}}{96 d^2 e^3}-\frac {b^2 c (e x)^{5/2} (c+d x)^{5/2}}{12 d e^4}+\frac {b^2 (e x)^{5/2} (c+d x)^{7/2}}{6 d e^4}+\frac {5 c^2 \left (384 a^2 d^4-b c^2 \left (b c^2+16 a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{512 d^{7/2} e^{3/2}} \] Output:

25/1536*c*(384*a^2*d^4-b*c^2*(16*a*d^2+b*c^2))*(e*x)^(1/2)*(d*x+c)^(1/2)/d 
^3/e^2+5/768*(384*a^2*d^4-b*c^2*(16*a*d^2+b*c^2))*(e*x)^(3/2)*(d*x+c)^(1/2 
)/d^2/e^3-2*a^2*(d*x+c)^(5/2)/e/(e*x)^(1/2)+5/192*b*c*(16*a*d^2+b*c^2)*(e* 
x)^(1/2)*(d*x+c)^(5/2)/d^3/e^2-1/96*b*(-48*a*d^2+5*b*c^2)*(e*x)^(3/2)*(d*x 
+c)^(5/2)/d^2/e^3-1/12*b^2*c*(e*x)^(5/2)*(d*x+c)^(5/2)/d/e^4+1/6*b^2*(e*x) 
^(5/2)*(d*x+c)^(7/2)/d/e^4+5/512*c^2*(384*a^2*d^4-b*c^2*(16*a*d^2+b*c^2))* 
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 = 1.23 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.75 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {x \sqrt {c+d x} \left (-3072 a^2 c^2 d^3+15 b^2 c^5 x+240 a b c^3 d^2 x+3456 a^2 c d^4 x-10 b^2 c^4 d x^2+1888 a b c^2 d^3 x^2+768 a^2 d^5 x^2+8 b^2 c^3 d^2 x^3+2176 a b c d^4 x^3+432 b^2 c^2 d^3 x^4+768 a b d^5 x^4+640 b^2 c d^4 x^5+256 b^2 d^5 x^6\right )}{1536 d^3 (e x)^{3/2}}+\frac {5 \left (-b^2 c^6-16 a b c^4 d^2+384 a^2 c^2 d^4\right ) x^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )}{256 d^{7/2} (e x)^{3/2}} \] Input:

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

Output:

(x*Sqrt[c + d*x]*(-3072*a^2*c^2*d^3 + 15*b^2*c^5*x + 240*a*b*c^3*d^2*x + 3 
456*a^2*c*d^4*x - 10*b^2*c^4*d*x^2 + 1888*a*b*c^2*d^3*x^2 + 768*a^2*d^5*x^ 
2 + 8*b^2*c^3*d^2*x^3 + 2176*a*b*c*d^4*x^3 + 432*b^2*c^2*d^3*x^4 + 768*a*b 
*d^5*x^4 + 640*b^2*c*d^4*x^5 + 256*b^2*d^5*x^6))/(1536*d^3*(e*x)^(3/2)) + 
(5*(-(b^2*c^6) - 16*a*b*c^4*d^2 + 384*a^2*c^2*d^4)*x^(3/2)*ArcTanh[(Sqrt[d 
]*Sqrt[x])/(-Sqrt[c] + Sqrt[c + d*x])])/(256*d^(7/2)*(e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.90, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {520, 27, 2125, 27, 1194, 27, 90, 60, 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.

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

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int -\frac {(c+d x)^{5/2} \left (b^2 c x^3+2 a b c x+6 a^2 d\right )}{2 \sqrt {e x}}dx}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (b^2 c x^3+2 a b c x+6 a^2 d\right )}{\sqrt {e x}}dx}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int \frac {(c+d x)^{5/2} \left (72 a^2 d^2 e^3-5 b^2 c^2 x^2 e^3+24 a b c d x e^3\right )}{2 \sqrt {e x}}dx}{6 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(c+d x)^{5/2} \left (72 a^2 d^2 e^3-5 b^2 c^2 x^2 e^3+24 a b c d x e^3\right )}{\sqrt {e x}}dx}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\frac {\frac {\int \frac {15 e^5 (c+d x)^{5/2} \left (48 a^2 d^3+b c \left (b c^2+16 a d^2\right ) x\right )}{2 \sqrt {e x}}dx}{5 d e^2}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 e^3 \int \frac {(c+d x)^{5/2} \left (48 a^2 d^3+b c \left (b c^2+16 a d^2\right ) x\right )}{\sqrt {e x}}dx}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {3 e^3 \left (\frac {\left (384 a^2 d^4-b c^2 \left (16 a d^2+b c^2\right )\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {e x}}dx}{8 d}+\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (16 a d^2+b c^2\right )}{4 d e}\right )}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {3 e^3 \left (\frac {\left (384 a^2 d^4-b c^2 \left (16 a d^2+b c^2\right )\right ) \left (\frac {5}{6} c \int \frac {(c+d x)^{3/2}}{\sqrt {e x}}dx+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}+\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (16 a d^2+b c^2\right )}{4 d e}\right )}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {3 e^3 \left (\frac {\left (384 a^2 d^4-b c^2 \left (16 a d^2+b c^2\right )\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \int \frac {\sqrt {c+d x}}{\sqrt {e x}}dx+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}+\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (16 a d^2+b c^2\right )}{4 d e}\right )}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {3 e^3 \left (\frac {\left (384 a^2 d^4-b c^2 \left (16 a d^2+b c^2\right )\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}+\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (16 a d^2+b c^2\right )}{4 d e}\right )}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {\frac {3 e^3 \left (\frac {\left (384 a^2 d^4-b c^2 \left (16 a d^2+b c^2\right )\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \left (c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}+\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (16 a d^2+b c^2\right )}{4 d e}\right )}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {3 e^3 \left (\frac {\left (384 a^2 d^4-b c^2 \left (16 a d^2+b c^2\right )\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}+\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (16 a d^2+b c^2\right )}{4 d e}\right )}{2 d}-\frac {b^2 c^2 e (e x)^{3/2} (c+d x)^{7/2}}{d}}{12 d e^3}+\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{6 d e^3}}{c e}-\frac {2 a^2 (c+d x)^{7/2}}{c e \sqrt {e x}}\)

Input:

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

Output:

(-2*a^2*(c + d*x)^(7/2))/(c*e*Sqrt[e*x]) + ((b^2*c*(e*x)^(5/2)*(c + d*x)^( 
7/2))/(6*d*e^3) + (-((b^2*c^2*e*(e*x)^(3/2)*(c + d*x)^(7/2))/d) + (3*e^3*( 
(b*c*(b*c^2 + 16*a*d^2)*Sqrt[e*x]*(c + d*x)^(7/2))/(4*d*e) + ((384*a^2*d^4 
 - b*c^2*(b*c^2 + 16*a*d^2))*((Sqrt[e*x]*(c + d*x)^(5/2))/(3*e) + (5*c*((S 
qrt[e*x]*(c + d*x)^(3/2))/(2*e) + (3*c*((Sqrt[e*x]*Sqrt[c + d*x])/e + (c*A 
rcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(Sqrt[d]*Sqrt[e])))/4 
))/6))/(8*d)))/(2*d))/(12*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 = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-256 b^{2} x^{6} d^{5}-640 b^{2} c \,x^{5} d^{4}-768 a b \,d^{5} x^{4}-432 c^{2} d^{3} x^{4} b^{2}-2176 a b c \,d^{4} x^{3}-8 b^{2} c^{3} d^{2} x^{3}-768 a^{2} x^{2} d^{5}-1888 a b \,c^{2} d^{3} x^{2}+10 b^{2} c^{4} d \,x^{2}-3456 a^{2} c \,d^{4} x -240 a \,c^{3} d^{2} b x -15 c^{5} b^{2} x +3072 d^{3} a^{2} c^{2}\right )}{1536 d^{3} e \sqrt {e x}}+\frac {5 c^{2} \left (384 a^{2} d^{4}-16 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{1024 d^{3} \sqrt {d e}\, e \sqrt {e x}\, \sqrt {d x +c}}\)	\(267\)
default	\(\frac {\sqrt {d x +c}\, \left (512 b^{2} d^{5} x^{6} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+1280 b^{2} c \,d^{4} x^{5} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+1536 a b \,d^{5} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+864 b^{2} c^{2} d^{3} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+4352 a b c \,d^{4} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+16 b^{2} c^{3} d^{2} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+5760 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{2} d^{4} e x -240 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{4} d^{2} e x -15 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{6} e x +1536 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} d^{5} x^{2}+3776 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{2} d^{3} x^{2}-20 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{4} d \,x^{2}+6912 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{4} x +480 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{2} x +30 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5} x -6144 a^{2} c^{2} d^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\right )}{3072 e \,d^{3} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(518\)

Input:

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

Output:

-1/1536*(d*x+c)^(1/2)*(-256*b^2*d^5*x^6-640*b^2*c*d^4*x^5-768*a*b*d^5*x^4- 
432*b^2*c^2*d^3*x^4-2176*a*b*c*d^4*x^3-8*b^2*c^3*d^2*x^3-768*a^2*d^5*x^2-1 
888*a*b*c^2*d^3*x^2+10*b^2*c^4*d*x^2-3456*a^2*c*d^4*x-240*a*b*c^3*d^2*x-15 
*b^2*c^5*x+3072*a^2*c^2*d^3)/d^3/e/(e*x)^(1/2)+5/1024*c^2/d^3*(384*a^2*d^4 
-16*a*b*c^2*d^2-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)/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 = 486, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (b^{2} c^{6} + 16 \, a b c^{4} d^{2} - 384 \, a^{2} c^{2} d^{4}\right )} \sqrt {d e} x \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) - 2 \, {\left (256 \, b^{2} d^{6} x^{6} + 640 \, b^{2} c d^{5} x^{5} - 3072 \, a^{2} c^{2} d^{4} + 48 \, {\left (9 \, b^{2} c^{2} d^{4} + 16 \, a b d^{6}\right )} x^{4} + 8 \, {\left (b^{2} c^{3} d^{3} + 272 \, a b c d^{5}\right )} x^{3} - 2 \, {\left (5 \, b^{2} c^{4} d^{2} - 944 \, a b c^{2} d^{4} - 384 \, a^{2} d^{6}\right )} x^{2} + 3 \, {\left (5 \, b^{2} c^{5} d + 80 \, a b c^{3} d^{3} + 1152 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{3072 \, d^{4} e^{2} x}, \frac {15 \, {\left (b^{2} c^{6} + 16 \, a b c^{4} d^{2} - 384 \, a^{2} c^{2} d^{4}\right )} \sqrt {-d e} x \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) + {\left (256 \, b^{2} d^{6} x^{6} + 640 \, b^{2} c d^{5} x^{5} - 3072 \, a^{2} c^{2} d^{4} + 48 \, {\left (9 \, b^{2} c^{2} d^{4} + 16 \, a b d^{6}\right )} x^{4} + 8 \, {\left (b^{2} c^{3} d^{3} + 272 \, a b c d^{5}\right )} x^{3} - 2 \, {\left (5 \, b^{2} c^{4} d^{2} - 944 \, a b c^{2} d^{4} - 384 \, a^{2} d^{6}\right )} x^{2} + 3 \, {\left (5 \, b^{2} c^{5} d + 80 \, a b c^{3} d^{3} + 1152 \, a^{2} c d^{5}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{1536 \, d^{4} e^{2} x}\right ] \] Input:

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

Output:

[-1/3072*(15*(b^2*c^6 + 16*a*b*c^4*d^2 - 384*a^2*c^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*(256*b^2*d^6*x^6 
+ 640*b^2*c*d^5*x^5 - 3072*a^2*c^2*d^4 + 48*(9*b^2*c^2*d^4 + 16*a*b*d^6)*x 
^4 + 8*(b^2*c^3*d^3 + 272*a*b*c*d^5)*x^3 - 2*(5*b^2*c^4*d^2 - 944*a*b*c^2* 
d^4 - 384*a^2*d^6)*x^2 + 3*(5*b^2*c^5*d + 80*a*b*c^3*d^3 + 1152*a^2*c*d^5) 
*x)*sqrt(d*x + c)*sqrt(e*x))/(d^4*e^2*x), 1/1536*(15*(b^2*c^6 + 16*a*b*c^4 
*d^2 - 384*a^2*c^2*d^4)*sqrt(-d*e)*x*arctan(sqrt(-d*e)*sqrt(d*x + c)*sqrt( 
e*x)/(d*e*x + c*e)) + (256*b^2*d^6*x^6 + 640*b^2*c*d^5*x^5 - 3072*a^2*c^2* 
d^4 + 48*(9*b^2*c^2*d^4 + 16*a*b*d^6)*x^4 + 8*(b^2*c^3*d^3 + 272*a*b*c*d^5 
)*x^3 - 2*(5*b^2*c^4*d^2 - 944*a*b*c^2*d^4 - 384*a^2*d^6)*x^2 + 3*(5*b^2*c 
^5*d + 80*a*b*c^3*d^3 + 1152*a^2*c*d^5)*x)*sqrt(d*x + c)*sqrt(e*x))/(d^4*e 
^2*x)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)^(5/2)*(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 = 0.22 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {{\left (\frac {{\left ({\left (2 \, {\left (4 \, {\left (2 \, {\left (d x + c\right )} {\left (8 \, {\left (d x + c\right )} {\left (\frac {2 \, {\left (d x + c\right )} b^{2}}{d^{4}} - \frac {7 \, b^{2} c}{d^{4}}\right )} + \frac {67 \, b^{2} c^{2} d^{16} + 48 \, a b d^{18}}{d^{20}}\right )} - \frac {55 \, b^{2} c^{3} d^{16} + 112 \, a b c d^{18}}{d^{20}}\right )} {\left (d x + c\right )} - \frac {b^{2} c^{4} d^{16} + 16 \, a b c^{2} d^{18} - 384 \, a^{2} d^{20}}{d^{20}}\right )} {\left (d x + c\right )} - \frac {5 \, {\left (b^{2} c^{5} d^{16} + 16 \, a b c^{3} d^{18} - 384 \, a^{2} c d^{20}\right )}}{d^{20}}\right )} {\left (d x + c\right )} + \frac {15 \, {\left (b^{2} c^{6} d^{16} + 16 \, a b c^{4} d^{18} - 384 \, a^{2} c^{2} d^{20}\right )}}{d^{20}}\right )} \sqrt {d x + c}}{\sqrt {{\left (d x + c\right )} d e - c d e}} + \frac {15 \, {\left (b^{2} c^{6} + 16 \, a b c^{4} d^{2} - 384 \, a^{2} c^{2} d^{4}\right )} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}}\right )} d^{2}}{1536 \, e {\left | d \right |}} \] Input:

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

Output:

1/1536*(((2*(4*(2*(d*x + c)*(8*(d*x + c)*(2*(d*x + c)*b^2/d^4 - 7*b^2*c/d^ 
4) + (67*b^2*c^2*d^16 + 48*a*b*d^18)/d^20) - (55*b^2*c^3*d^16 + 112*a*b*c* 
d^18)/d^20)*(d*x + c) - (b^2*c^4*d^16 + 16*a*b*c^2*d^18 - 384*a^2*d^20)/d^ 
20)*(d*x + c) - 5*(b^2*c^5*d^16 + 16*a*b*c^3*d^18 - 384*a^2*c*d^20)/d^20)* 
(d*x + c) + 15*(b^2*c^6*d^16 + 16*a*b*c^4*d^18 - 384*a^2*c^2*d^20)/d^20)*s 
qrt(d*x + c)/sqrt((d*x + c)*d*e - c*d*e) + 15*(b^2*c^6 + 16*a*b*c^4*d^2 - 
384*a^2*c^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. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{5/2}}{{\left (e\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.13 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-3072 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{4}+3456 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{5} x +768 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{6} x^{2}+240 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{3} x +1888 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{4} x^{2}+2176 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{5} x^{3}+768 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{6} x^{4}+15 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d x -10 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{2} x^{2}+8 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{3} x^{3}+432 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{4} x^{4}+640 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{5} x^{5}+256 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{6} x^{6}+5760 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{2} d^{4} x -240 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{4} d^{2} x -15 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{6} x -3840 \sqrt {d}\, a^{2} c^{2} d^{4} x +48 \sqrt {d}\, a b \,c^{4} d^{2} x \right )}{1536 d^{4} e^{2} x} \] Input:

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

Output:

(sqrt(e)*( - 3072*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**4 + 3456*sqrt(x)*sqrt 
(c + d*x)*a**2*c*d**5*x + 768*sqrt(x)*sqrt(c + d*x)*a**2*d**6*x**2 + 240*s 
qrt(x)*sqrt(c + d*x)*a*b*c**3*d**3*x + 1888*sqrt(x)*sqrt(c + d*x)*a*b*c**2 
*d**4*x**2 + 2176*sqrt(x)*sqrt(c + d*x)*a*b*c*d**5*x**3 + 768*sqrt(x)*sqrt 
(c + d*x)*a*b*d**6*x**4 + 15*sqrt(x)*sqrt(c + d*x)*b**2*c**5*d*x - 10*sqrt 
(x)*sqrt(c + d*x)*b**2*c**4*d**2*x**2 + 8*sqrt(x)*sqrt(c + d*x)*b**2*c**3* 
d**3*x**3 + 432*sqrt(x)*sqrt(c + d*x)*b**2*c**2*d**4*x**4 + 640*sqrt(x)*sq 
rt(c + d*x)*b**2*c*d**5*x**5 + 256*sqrt(x)*sqrt(c + d*x)*b**2*d**6*x**6 + 
5760*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a**2*c**2*d**4 
*x - 240*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c**4*d 
**2*x - 15*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c** 
6*x - 3840*sqrt(d)*a**2*c**2*d**4*x + 48*sqrt(d)*a*b*c**4*d**2*x))/(1536*d 
**4*e**2*x)