\(\int \frac {(c+d x)^{3/2} (a+b x^2)^2}{(e x)^{9/2}} \, dx\) [805]

Optimal result

Integrand size = 26, antiderivative size = 236 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=-\frac {b \left (b c^2+8 a d^2\right ) \sqrt {c+d x}}{2 d e^4 \sqrt {e x}}+\frac {b^2 c \sqrt {e x} \sqrt {c+d x}}{4 e^5}-\frac {4 a b (c+d x)^{3/2}}{3 e^3 (e x)^{3/2}}-\frac {2 a^2 (c+d x)^{5/2}}{7 c e (e x)^{7/2}}+\frac {4 a^2 d (c+d x)^{5/2}}{35 c^2 e^2 (e x)^{5/2}}+\frac {b^2 (c+d x)^{5/2}}{2 d e^4 \sqrt {e x}}+\frac {b \left (3 b c^2+16 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{4 \sqrt {d} e^{9/2}} \] Output:

-1/2*b*(8*a*d^2+b*c^2)*(d*x+c)^(1/2)/d/e^4/(e*x)^(1/2)+1/4*b^2*c*(e*x)^(1/ 
2)*(d*x+c)^(1/2)/e^5-4/3*a*b*(d*x+c)^(3/2)/e^3/(e*x)^(3/2)-2/7*a^2*(d*x+c) 
^(5/2)/c/e/(e*x)^(7/2)+4/35*a^2*d*(d*x+c)^(5/2)/c^2/e^2/(e*x)^(5/2)+1/2*b^ 
2*(d*x+c)^(5/2)/d/e^4/(e*x)^(1/2)+1/4*b*(16*a*d^2+3*b*c^2)*arctanh(d^(1/2) 
*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(1/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=\frac {\sqrt {e x} \left (\sqrt {d} \sqrt {c+d x} \left (-24 a^2 (5 c-2 d x) (c+d x)^2+105 b^2 c^2 x^4 (5 c+2 d x)-560 a b c^2 x^2 (c+4 d x)\right )-105 b c^2 \left (3 b c^2+16 a d^2\right ) x^{7/2} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )\right )}{420 c^2 \sqrt {d} e^5 x^4} \] Input:

Integrate[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(9/2),x]
 

Output:

(Sqrt[e*x]*(Sqrt[d]*Sqrt[c + d*x]*(-24*a^2*(5*c - 2*d*x)*(c + d*x)^2 + 105 
*b^2*c^2*x^4*(5*c + 2*d*x) - 560*a*b*c^2*x^2*(c + 4*d*x)) - 105*b*c^2*(3*b 
*c^2 + 16*a*d^2)*x^(7/2)*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c + d*x]]))/(420*c^ 
2*Sqrt[d]*e^5*x^4)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.07, 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, 520, 27, 87, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^{3/2}}{(e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(c+d x)^{3/2} \left (-7 b^2 c x^3-14 a b c x+2 a^2 d\right )}{2 (e x)^{7/2}}dx}{7 c e}-\frac {2 a^2 (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x)^{3/2} \left (-7 b^2 c x^3-14 a b c x+2 a^2 d\right )}{(e x)^{7/2}}dx}{7 c e}-\frac {2 a^2 (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {35 b c^2 (c+d x)^{3/2} \left (b x^2+2 a\right )}{2 (e x)^{5/2}}dx}{5 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a^2 (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(9/2),x]
 

Output:

(-2*a^2*(c + d*x)^(5/2))/(7*c*e*(e*x)^(7/2)) - ((-4*a^2*d*(c + d*x)^(5/2)) 
/(5*c*e*(e*x)^(5/2)) - (7*b*c*((-4*a*(c + d*x)^(5/2))/(3*c*e*(e*x)^(3/2)) 
+ ((-8*a*d*(c + d*x)^(5/2))/(c*e*Sqrt[e*x]) + ((3*b*c^2 + 16*a*d^2)*((Sqrt 
[e*x]*(c + d*x)^(3/2))/(2*e) + (3*c*((Sqrt[e*x]*Sqrt[c + d*x])/e + (c*ArcT 
anh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(Sqrt[d]*Sqrt[e])))/4))/ 
(c*e))/(3*c*e)))/e)/(7*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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 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 = 187, normalized size of antiderivative = 0.79

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-210 b^{2} c^{2} d \,x^{5}-525 b^{2} x^{4} c^{3}-48 a^{2} d^{3} x^{3}+2240 a b d \,x^{3} c^{2}+24 a^{2} c \,d^{2} x^{2}+560 a b \,c^{3} x^{2}+192 a^{2} c^{2} d x +120 a^{2} c^{3}\right )}{420 x^{3} c^{2} e^{4} \sqrt {e x}}+\frac {b \left (16 a \,d^{2}+3 b \,c^{2}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{8 \sqrt {d e}\, e^{4} \sqrt {e x}\, \sqrt {d x +c}}\)	\(187\)
default	\(\frac {\sqrt {d x +c}\, \left (1680 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{2} d^{2} e \,x^{4}+315 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} e \,x^{4}+420 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{2} d \,x^{5}+1050 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{3} x^{4}+96 a^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-4480 a b \,c^{2} d \,x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-48 a^{2} c \,d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-1120 a b \,c^{3} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-384 a^{2} c^{2} d x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-240 a^{2} c^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\right )}{840 e^{4} x^{3} c^{2} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(339\)

Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-1/420*(d*x+c)^(1/2)*(-210*b^2*c^2*d*x^5-525*b^2*c^3*x^4-48*a^2*d^3*x^3+22 
40*a*b*c^2*d*x^3+24*a^2*c*d^2*x^2+560*a*b*c^3*x^2+192*a^2*c^2*d*x+120*a^2* 
c^3)/x^3/c^2/e^4/(e*x)^(1/2)+1/8*b*(16*a*d^2+3*b*c^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^4*((d*x+c)*e*x)^(1/2)/(e*x 
)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.56 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=\left [\frac {105 \, {\left (3 \, b^{2} c^{4} + 16 \, a b c^{2} d^{2}\right )} \sqrt {d e} x^{4} \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) + 2 \, {\left (210 \, b^{2} c^{2} d^{2} x^{5} + 525 \, b^{2} c^{3} d x^{4} - 192 \, a^{2} c^{2} d^{2} x - 120 \, a^{2} c^{3} d - 16 \, {\left (140 \, a b c^{2} d^{2} - 3 \, a^{2} d^{4}\right )} x^{3} - 8 \, {\left (70 \, a b c^{3} d + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{840 \, c^{2} d e^{5} x^{4}}, -\frac {105 \, {\left (3 \, b^{2} c^{4} + 16 \, a b c^{2} d^{2}\right )} \sqrt {-d e} x^{4} \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) - {\left (210 \, b^{2} c^{2} d^{2} x^{5} + 525 \, b^{2} c^{3} d x^{4} - 192 \, a^{2} c^{2} d^{2} x - 120 \, a^{2} c^{3} d - 16 \, {\left (140 \, a b c^{2} d^{2} - 3 \, a^{2} d^{4}\right )} x^{3} - 8 \, {\left (70 \, a b c^{3} d + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{420 \, c^{2} d e^{5} x^{4}}\right ] \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(9/2),x, algorithm="fricas")
 

Output:

[1/840*(105*(3*b^2*c^4 + 16*a*b*c^2*d^2)*sqrt(d*e)*x^4*log(2*d*e*x + c*e + 
 2*sqrt(d*e)*sqrt(d*x + c)*sqrt(e*x)) + 2*(210*b^2*c^2*d^2*x^5 + 525*b^2*c 
^3*d*x^4 - 192*a^2*c^2*d^2*x - 120*a^2*c^3*d - 16*(140*a*b*c^2*d^2 - 3*a^2 
*d^4)*x^3 - 8*(70*a*b*c^3*d + 3*a^2*c*d^3)*x^2)*sqrt(d*x + c)*sqrt(e*x))/( 
c^2*d*e^5*x^4), -1/420*(105*(3*b^2*c^4 + 16*a*b*c^2*d^2)*sqrt(-d*e)*x^4*ar 
ctan(sqrt(-d*e)*sqrt(d*x + c)*sqrt(e*x)/(d*e*x + c*e)) - (210*b^2*c^2*d^2* 
x^5 + 525*b^2*c^3*d*x^4 - 192*a^2*c^2*d^2*x - 120*a^2*c^3*d - 16*(140*a*b* 
c^2*d^2 - 3*a^2*d^4)*x^3 - 8*(70*a*b*c^3*d + 3*a^2*c*d^3)*x^2)*sqrt(d*x + 
c)*sqrt(e*x))/(c^2*d*e^5*x^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (221) = 442\).

Time = 89.85 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.70 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**(3/2)*(b*x**2+a)**2/(e*x)**(9/2),x)
 

Output:

-30*a**2*c**6*d**(9/2)*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 21 
0*c**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) - 66*a**2*c**5*d* 
*(11/2)*x*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e 
**(9/2)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) - 34*a**2*c**4*d**(13/2)*x**2* 
sqrt(c/(d*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x* 
*4 + 105*c**3*d**6*e**(9/2)*x**5) - 6*a**2*c**3*d**(15/2)*x**3*sqrt(c/(d*x 
) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x**4 + 105*c* 
*3*d**6*e**(9/2)*x**5) - 24*a**2*c**2*d**(17/2)*x**4*sqrt(c/(d*x) + 1)/(10 
5*c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e* 
*(9/2)*x**5) - 16*a**2*c*d**(19/2)*x**5*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e 
**(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) 
- 2*a**2*d**(3/2)*sqrt(c/(d*x) + 1)/(5*e**(9/2)*x**2) - 2*a**2*d**(5/2)*sq 
rt(c/(d*x) + 1)/(15*c*e**(9/2)*x) + 4*a**2*d**(7/2)*sqrt(c/(d*x) + 1)/(15* 
c**2*e**(9/2)) - 4*a*b*sqrt(c)*d/(e**(9/2)*sqrt(x)*sqrt(1 + d*x/c)) - 4*a* 
b*c*sqrt(d)*sqrt(c/(d*x) + 1)/(3*e**(9/2)*x) - 4*a*b*d**(3/2)*sqrt(c/(d*x) 
 + 1)/(3*e**(9/2)) + 4*a*b*d**(3/2)*asinh(sqrt(d)*sqrt(x)/sqrt(c))/e**(9/2 
) - 4*a*b*d**2*sqrt(x)/(sqrt(c)*e**(9/2)*sqrt(1 + d*x/c)) + b**2*c**(3/2)* 
sqrt(x)*sqrt(1 + d*x/c)/e**(9/2) + b**2*c**(3/2)*sqrt(x)/(4*e**(9/2)*sqrt( 
1 + d*x/c)) + 3*b**2*sqrt(c)*d*x**(3/2)/(4*e**(9/2)*sqrt(1 + d*x/c)) + 3*b 
**2*c**2*asinh(sqrt(d)*sqrt(x)/sqrt(c))/(4*sqrt(d)*e**(9/2)) + b**2*d**...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(9/2),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 = 322, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=\frac {d^{5} {\left (\frac {{\left ({\left ({\left ({\left (105 \, {\left (\frac {2 \, {\left (d x + c\right )} b^{2} e^{3}}{d} - \frac {5 \, b^{2} c e^{3}}{d}\right )} {\left (d x + c\right )} - \frac {16 \, {\left (140 \, a b c^{3} d^{9} e^{6} - 3 \, a^{2} c d^{11} e^{6}\right )}}{c^{3} d^{8} e^{3}}\right )} {\left (d x + c\right )} + \frac {14 \, {\left (75 \, b^{2} c^{6} d^{7} e^{6} + 440 \, a b c^{4} d^{9} e^{6} - 12 \, a^{2} c^{2} d^{11} e^{6}\right )}}{c^{3} d^{8} e^{3}}\right )} {\left (d x + c\right )} - \frac {350 \, {\left (3 \, b^{2} c^{7} d^{7} e^{6} + 16 \, a b c^{5} d^{9} e^{6}\right )}}{c^{3} d^{8} e^{3}}\right )} {\left (d x + c\right )} + \frac {105 \, {\left (3 \, b^{2} c^{8} d^{7} e^{6} + 16 \, a b c^{6} d^{9} e^{6}\right )}}{c^{3} d^{8} e^{3}}\right )} \sqrt {d x + c}}{{\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {7}{2}}} - \frac {105 \, {\left (3 \, b^{2} c^{2} + 16 \, a b d^{2}\right )} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}}\right )}}{420 \, e^{4} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(9/2),x, algorithm="giac")
 

Output:

1/420*d^5*(((((105*(2*(d*x + c)*b^2*e^3/d - 5*b^2*c*e^3/d)*(d*x + c) - 16* 
(140*a*b*c^3*d^9*e^6 - 3*a^2*c*d^11*e^6)/(c^3*d^8*e^3))*(d*x + c) + 14*(75 
*b^2*c^6*d^7*e^6 + 440*a*b*c^4*d^9*e^6 - 12*a^2*c^2*d^11*e^6)/(c^3*d^8*e^3 
))*(d*x + c) - 350*(3*b^2*c^7*d^7*e^6 + 16*a*b*c^5*d^9*e^6)/(c^3*d^8*e^3)) 
*(d*x + c) + 105*(3*b^2*c^8*d^7*e^6 + 16*a*b*c^6*d^9*e^6)/(c^3*d^8*e^3))*s 
qrt(d*x + c)/((d*x + c)*d*e - c*d*e)^(7/2) - 105*(3*b^2*c^2 + 16*a*b*d^2)* 
log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e 
)*d^4))/(e^4*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(9/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{9/2}} \, dx=\frac {\sqrt {e}\, \left (-960 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d -1536 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{2} x -192 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{3} x^{2}+384 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{4} x^{3}-4480 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d \,x^{2}-17920 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{2} x^{3}+4200 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d \,x^{4}+1680 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{2} x^{5}+13440 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{2} d^{2} x^{4}+2520 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} x^{4}-384 \sqrt {d}\, a^{2} d^{4} x^{4}+10240 \sqrt {d}\, a b \,c^{2} d^{2} x^{4}+1875 \sqrt {d}\, b^{2} c^{4} x^{4}\right )}{3360 c^{2} d \,e^{5} x^{4}} \] Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(9/2),x)
 

Output:

(sqrt(e)*( - 960*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d - 1536*sqrt(x)*sqrt(c + 
 d*x)*a**2*c**2*d**2*x - 192*sqrt(x)*sqrt(c + d*x)*a**2*c*d**3*x**2 + 384* 
sqrt(x)*sqrt(c + d*x)*a**2*d**4*x**3 - 4480*sqrt(x)*sqrt(c + d*x)*a*b*c**3 
*d*x**2 - 17920*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**2*x**3 + 4200*sqrt(x)*sq 
rt(c + d*x)*b**2*c**3*d*x**4 + 1680*sqrt(x)*sqrt(c + d*x)*b**2*c**2*d**2*x 
**5 + 13440*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c** 
2*d**2*x**4 + 2520*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))* 
b**2*c**4*x**4 - 384*sqrt(d)*a**2*d**4*x**4 + 10240*sqrt(d)*a*b*c**2*d**2* 
x**4 + 1875*sqrt(d)*b**2*c**4*x**4))/(3360*c**2*d*e**5*x**4)