\(\int \frac {(c+d x)^{3/2} (a+b x^2)^2}{(e x)^{13/2}} \, dx\) [807]

Optimal result

Integrand size = 26, antiderivative size = 238 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 b^2 c \sqrt {c+d x}}{3 e^5 (e x)^{3/2}}-\frac {8 b^2 d \sqrt {c+d x}}{3 e^6 \sqrt {e x}}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}+\frac {4 a^2 d (c+d x)^{5/2}}{33 c^2 e^2 (e x)^{9/2}}-\frac {4 a \left (33 b c^2+4 a d^2\right ) (c+d x)^{5/2}}{231 c^3 e^3 (e x)^{7/2}}+\frac {8 a d \left (33 b c^2+4 a d^2\right ) (c+d x)^{5/2}}{1155 c^4 e^4 (e x)^{5/2}}+\frac {2 b^2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{e^{13/2}} \] Output:

-2/3*b^2*c*(d*x+c)^(1/2)/e^5/(e*x)^(3/2)-8/3*b^2*d*(d*x+c)^(1/2)/e^6/(e*x) 
^(1/2)-2/11*a^2*(d*x+c)^(5/2)/c/e/(e*x)^(11/2)+4/33*a^2*d*(d*x+c)^(5/2)/c^ 
2/e^2/(e*x)^(9/2)-4/231*a*(4*a*d^2+33*b*c^2)*(d*x+c)^(5/2)/c^3/e^3/(e*x)^( 
7/2)+8/1155*a*d*(4*a*d^2+33*b*c^2)*(d*x+c)^(5/2)/c^4/e^4/(e*x)^(5/2)+2*b^2 
*d^(3/2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/e^(13/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 \sqrt {e x} \left (\sqrt {c+d x} \left (66 a b c^2 x^2 (5 c-2 d x) (c+d x)^2+385 b^2 c^4 x^4 (c+4 d x)+a^2 (c+d x)^2 \left (105 c^3-70 c^2 d x+40 c d^2 x^2-16 d^3 x^3\right )\right )+1155 b^2 c^4 d^{3/2} x^{11/2} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )\right )}{1155 c^4 e^7 x^6} \] Input:

Integrate[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(13/2),x]
 

Output:

(-2*Sqrt[e*x]*(Sqrt[c + d*x]*(66*a*b*c^2*x^2*(5*c - 2*d*x)*(c + d*x)^2 + 3 
85*b^2*c^4*x^4*(c + 4*d*x) + a^2*(c + d*x)^2*(105*c^3 - 70*c^2*d*x + 40*c* 
d^2*x^2 - 16*d^3*x^3)) + 1155*b^2*c^4*d^(3/2)*x^(11/2)*Log[-(Sqrt[d]*Sqrt[ 
x]) + Sqrt[c + d*x]]))/(1155*c^4*e^7*x^6)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.16, 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, 57, 57, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^{3/2}}{(e x)^{13/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(c+d x)^{3/2} \left (-11 b^2 c x^3-22 a b c x+6 a^2 d\right )}{2 (e x)^{11/2}}dx}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x)^{3/2} \left (-11 b^2 c x^3-22 a b c x+6 a^2 d\right )}{(e x)^{11/2}}dx}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {3 (c+d x)^{3/2} \left (33 b^2 c^2 x^2+2 a \left (33 b c^2+4 a d^2\right )\right )}{2 (e x)^{9/2}}dx}{9 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^{3/2} \left (33 b^2 c^2 x^2+2 a \left (33 b c^2+4 a d^2\right )\right )}{(e x)^{9/2}}dx}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {\left (4 a d \left (33 b c^2+4 a d^2\right )-231 b^2 c^3 x\right ) (c+d x)^{3/2}}{2 (e x)^{7/2}}dx}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (4 a d \left (33 b c^2+4 a d^2\right )-231 b^2 c^3 x\right ) (c+d x)^{3/2}}{(e x)^{7/2}}dx}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {231 b^2 c^3 \int \frac {(c+d x)^{3/2}}{(e x)^{5/2}}dx}{e}-\frac {8 a d (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 57

\(\displaystyle -\frac {-\frac {-\frac {-\frac {231 b^2 c^3 \left (\frac {d \int \frac {\sqrt {c+d x}}{(e x)^{3/2}}dx}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {8 a d (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 57

\(\displaystyle -\frac {-\frac {-\frac {-\frac {231 b^2 c^3 \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{e}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {8 a d (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 65

\(\displaystyle -\frac {-\frac {-\frac {-\frac {231 b^2 c^3 \left (\frac {d \left (\frac {2 d \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{e}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {8 a d (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {4 a^2 d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}-\frac {-\frac {-\frac {8 a d (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{5 c e (e x)^{5/2}}-\frac {231 b^2 c^3 \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{e^{3/2}}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}}{7 c e}-\frac {4 a (c+d x)^{5/2} \left (4 a d^2+33 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}}{11 c e}-\frac {2 a^2 (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\)

Input:

Int[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(13/2),x]
 

Output:

(-2*a^2*(c + d*x)^(5/2))/(11*c*e*(e*x)^(11/2)) - ((-4*a^2*d*(c + d*x)^(5/2 
))/(3*c*e*(e*x)^(9/2)) - ((-4*a*(33*b*c^2 + 4*a*d^2)*(c + d*x)^(5/2))/(7*c 
*e*(e*x)^(7/2)) - ((-8*a*d*(33*b*c^2 + 4*a*d^2)*(c + d*x)^(5/2))/(5*c*e*(e 
*x)^(5/2)) - (231*b^2*c^3*((-2*(c + d*x)^(3/2))/(3*e*(e*x)^(3/2)) + (d*((- 
2*Sqrt[c + d*x])/(e*Sqrt[e*x]) + (2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(S 
qrt[e]*Sqrt[c + d*x])])/e^(3/2)))/e))/e)/(7*c*e))/(3*c*e))/(11*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 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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 = 0.28 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-16 a^{2} d^{5} x^{5}-132 a b \,c^{2} d^{3} x^{5}+1540 b^{2} d \,x^{5} c^{4}+8 a^{2} c \,d^{4} x^{4}+66 a b \,c^{3} d^{2} x^{4}+385 b^{2} c^{5} x^{4}-6 a^{2} c^{2} d^{3} x^{3}+528 a b \,c^{4} d \,x^{3}+5 a^{2} c^{3} d^{2} x^{2}+330 a b \,c^{5} x^{2}+140 a^{2} c^{4} d x +105 a^{2} c^{5}\right )}{1155 x^{5} c^{4} e^{6} \sqrt {e x}}+\frac {b^{2} d^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{\sqrt {d e}\, e^{6} \sqrt {e x}\, \sqrt {d x +c}}\)	\(232\)
default	\(\frac {\sqrt {d x +c}\, \left (1155 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} d^{2} e \,x^{6}+32 a^{2} d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+264 a b \,c^{2} d^{3} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-3080 b^{2} c^{4} d \,x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-16 a^{2} c \,d^{4} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-132 a b \,c^{3} d^{2} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-770 b^{2} c^{5} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+12 a^{2} c^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-1056 a b \,c^{4} d \,x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-10 a^{2} c^{3} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-660 a b \,c^{5} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-280 a^{2} c^{4} d x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-210 a^{2} c^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\right )}{1155 e^{6} x^{5} c^{4} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(408\)

Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(13/2),x,method=_RETURNVERBOSE)
 

Output:

-2/1155*(d*x+c)^(1/2)*(-16*a^2*d^5*x^5-132*a*b*c^2*d^3*x^5+1540*b^2*c^4*d* 
x^5+8*a^2*c*d^4*x^4+66*a*b*c^3*d^2*x^4+385*b^2*c^5*x^4-6*a^2*c^2*d^3*x^3+5 
28*a*b*c^4*d*x^3+5*a^2*c^3*d^2*x^2+330*a*b*c^5*x^2+140*a^2*c^4*d*x+105*a^2 
*c^5)/x^5/c^4/e^6/(e*x)^(1/2)+b^2*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^6*((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 = 407, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\left [\frac {1155 \, b^{2} c^{4} d e x^{6} \sqrt {\frac {d}{e}} \log \left (2 \, d x + 2 \, \sqrt {d x + c} \sqrt {e x} \sqrt {\frac {d}{e}} + c\right ) - 2 \, {\left (140 \, a^{2} c^{4} d x + 105 \, a^{2} c^{5} + 4 \, {\left (385 \, b^{2} c^{4} d - 33 \, a b c^{2} d^{3} - 4 \, a^{2} d^{5}\right )} x^{5} + {\left (385 \, b^{2} c^{5} + 66 \, a b c^{3} d^{2} + 8 \, a^{2} c d^{4}\right )} x^{4} + 6 \, {\left (88 \, a b c^{4} d - a^{2} c^{2} d^{3}\right )} x^{3} + 5 \, {\left (66 \, a b c^{5} + a^{2} c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{1155 \, c^{4} e^{7} x^{6}}, -\frac {2 \, {\left (1155 \, b^{2} c^{4} d e x^{6} \sqrt {-\frac {d}{e}} \arctan \left (\frac {\sqrt {e x} \sqrt {-\frac {d}{e}}}{\sqrt {d x + c}}\right ) + {\left (140 \, a^{2} c^{4} d x + 105 \, a^{2} c^{5} + 4 \, {\left (385 \, b^{2} c^{4} d - 33 \, a b c^{2} d^{3} - 4 \, a^{2} d^{5}\right )} x^{5} + {\left (385 \, b^{2} c^{5} + 66 \, a b c^{3} d^{2} + 8 \, a^{2} c d^{4}\right )} x^{4} + 6 \, {\left (88 \, a b c^{4} d - a^{2} c^{2} d^{3}\right )} x^{3} + 5 \, {\left (66 \, a b c^{5} + a^{2} c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}\right )}}{1155 \, c^{4} e^{7} x^{6}}\right ] \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(13/2),x, algorithm="fricas")
 

Output:

[1/1155*(1155*b^2*c^4*d*e*x^6*sqrt(d/e)*log(2*d*x + 2*sqrt(d*x + c)*sqrt(e 
*x)*sqrt(d/e) + c) - 2*(140*a^2*c^4*d*x + 105*a^2*c^5 + 4*(385*b^2*c^4*d - 
 33*a*b*c^2*d^3 - 4*a^2*d^5)*x^5 + (385*b^2*c^5 + 66*a*b*c^3*d^2 + 8*a^2*c 
*d^4)*x^4 + 6*(88*a*b*c^4*d - a^2*c^2*d^3)*x^3 + 5*(66*a*b*c^5 + a^2*c^3*d 
^2)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c^4*e^7*x^6), -2/1155*(1155*b^2*c^4*d*e 
*x^6*sqrt(-d/e)*arctan(sqrt(e*x)*sqrt(-d/e)/sqrt(d*x + c)) + (140*a^2*c^4* 
d*x + 105*a^2*c^5 + 4*(385*b^2*c^4*d - 33*a*b*c^2*d^3 - 4*a^2*d^5)*x^5 + ( 
385*b^2*c^5 + 66*a*b*c^3*d^2 + 8*a^2*c*d^4)*x^4 + 6*(88*a*b*c^4*d - a^2*c^ 
2*d^3)*x^3 + 5*(66*a*b*c^5 + a^2*c^3*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c 
^4*e^7*x^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(3/2)*(b*x**2+a)**2/(e*x)**(13/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(13/2),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 = 0.21 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.43 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=-\frac {2 \, d^{7} {\left (\frac {1155 \, b^{2} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}} - \frac {{\left (1155 \, b^{2} c^{5} d e^{5} - {\left (6160 \, b^{2} c^{4} d e^{5} + {\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {4 \, {\left (385 \, b^{2} c^{5} d^{5} e^{5} - 33 \, a b c^{3} d^{7} e^{5} - 4 \, a^{2} c d^{9} e^{5}\right )} {\left (d x + c\right )}}{c^{5} d^{4}} - \frac {11 \, {\left (665 \, b^{2} c^{6} d^{5} e^{5} - 66 \, a b c^{4} d^{7} e^{5} - 8 \, a^{2} c^{2} d^{9} e^{5}\right )}}{c^{5} d^{4}}\right )} + \frac {66 \, {\left (210 \, b^{2} c^{7} d^{5} e^{5} - 16 \, a b c^{5} d^{7} e^{5} - 3 \, a^{2} c^{3} d^{9} e^{5}\right )}}{c^{5} d^{4}}\right )} - \frac {77 \, {\left (170 \, b^{2} c^{8} d^{5} e^{5} - 6 \, a b c^{6} d^{7} e^{5} - 3 \, a^{2} c^{4} d^{9} e^{5}\right )}}{c^{5} d^{4}}\right )} {\left (d x + c\right )}\right )} {\left (d x + c\right )}\right )} \sqrt {d x + c}}{{\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {11}{2}}}\right )}}{1155 \, e^{6} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(13/2),x, algorithm="giac")
 

Output:

-2/1155*d^7*(1155*b^2*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d* 
e - c*d*e)))/(sqrt(d*e)*d^4) - (1155*b^2*c^5*d*e^5 - (6160*b^2*c^4*d*e^5 + 
 ((d*x + c)*((d*x + c)*(4*(385*b^2*c^5*d^5*e^5 - 33*a*b*c^3*d^7*e^5 - 4*a^ 
2*c*d^9*e^5)*(d*x + c)/(c^5*d^4) - 11*(665*b^2*c^6*d^5*e^5 - 66*a*b*c^4*d^ 
7*e^5 - 8*a^2*c^2*d^9*e^5)/(c^5*d^4)) + 66*(210*b^2*c^7*d^5*e^5 - 16*a*b*c 
^5*d^7*e^5 - 3*a^2*c^3*d^9*e^5)/(c^5*d^4)) - 77*(170*b^2*c^8*d^5*e^5 - 6*a 
*b*c^6*d^7*e^5 - 3*a^2*c^4*d^9*e^5)/(c^5*d^4))*(d*x + c))*(d*x + c))*sqrt( 
d*x + c)/((d*x + c)*d*e - c*d*e)^(11/2))/(e^6*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{13/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(13/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(13/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{13/2}} \, dx=\frac {2 \sqrt {e}\, \left (-105 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5}-140 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d x -5 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{2} x^{2}+6 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{3} x^{3}-8 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{4} x^{4}+16 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{5} x^{5}-330 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} x^{2}-528 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d \,x^{3}-66 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{2} x^{4}+132 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{3} x^{5}-385 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} x^{4}-1540 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d \,x^{5}+1155 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} d \,x^{6}-16 \sqrt {d}\, a^{2} d^{5} x^{6}-132 \sqrt {d}\, a b \,c^{2} d^{3} x^{6}+1120 \sqrt {d}\, b^{2} c^{4} d \,x^{6}\right )}{1155 c^{4} e^{7} x^{6}} \] Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(13/2),x)
 

Output:

(2*sqrt(e)*( - 105*sqrt(x)*sqrt(c + d*x)*a**2*c**5 - 140*sqrt(x)*sqrt(c + 
d*x)*a**2*c**4*d*x - 5*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**2*x**2 + 6*sqrt( 
x)*sqrt(c + d*x)*a**2*c**2*d**3*x**3 - 8*sqrt(x)*sqrt(c + d*x)*a**2*c*d**4 
*x**4 + 16*sqrt(x)*sqrt(c + d*x)*a**2*d**5*x**5 - 330*sqrt(x)*sqrt(c + d*x 
)*a*b*c**5*x**2 - 528*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d*x**3 - 66*sqrt(x)*s 
qrt(c + d*x)*a*b*c**3*d**2*x**4 + 132*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**3* 
x**5 - 385*sqrt(x)*sqrt(c + d*x)*b**2*c**5*x**4 - 1540*sqrt(x)*sqrt(c + d* 
x)*b**2*c**4*d*x**5 + 1155*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/s 
qrt(c))*b**2*c**4*d*x**6 - 16*sqrt(d)*a**2*d**5*x**6 - 132*sqrt(d)*a*b*c** 
2*d**3*x**6 + 1120*sqrt(d)*b**2*c**4*d*x**6))/(1155*c**4*e**7*x**6)