\(\int (e x)^{5/2} \sqrt {c+d x} (a+b x^2) \, dx\) [742]

Optimal result

Integrand size = 24, antiderivative size = 272 \[ \int (e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {c^3 \left (21 b c^2+40 a d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x}}{512 d^5}-\frac {c^2 \left (21 b c^2+40 a d^2\right ) e (e x)^{3/2} \sqrt {c+d x}}{768 d^4}+\frac {c \left (21 b c^2+40 a d^2\right ) (e x)^{5/2} \sqrt {c+d x}}{960 d^3}+\frac {\left (21 b c^2+40 a d^2\right ) (e x)^{7/2} \sqrt {c+d x}}{160 d^2 e}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{20 d^2 e}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}-\frac {c^4 \left (21 b c^2+40 a d^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{512 d^{11/2}} \] Output:

1/512*c^3*(40*a*d^2+21*b*c^2)*e^2*(e*x)^(1/2)*(d*x+c)^(1/2)/d^5-1/768*c^2* 
(40*a*d^2+21*b*c^2)*e*(e*x)^(3/2)*(d*x+c)^(1/2)/d^4+1/960*c*(40*a*d^2+21*b 
*c^2)*(e*x)^(5/2)*(d*x+c)^(1/2)/d^3+1/160*(40*a*d^2+21*b*c^2)*(e*x)^(7/2)* 
(d*x+c)^(1/2)/d^2/e-3/20*b*c*(e*x)^(7/2)*(d*x+c)^(3/2)/d^2/e+1/6*b*(e*x)^( 
9/2)*(d*x+c)^(3/2)/d/e^2-1/512*c^4*(40*a*d^2+21*b*c^2)*e^(5/2)*arctanh(d^( 
1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(11/2)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.67 \[ \int (e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {(e x)^{5/2} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (40 a d^2 \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )+b \left (315 c^5-210 c^4 d x+168 c^3 d^2 x^2-144 c^2 d^3 x^3+128 c d^4 x^4+1280 d^5 x^5\right )\right )+30 c^4 \left (21 b c^2+40 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )\right )}{7680 d^{11/2} x^{5/2}} \] Input:

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

Output:

((e*x)^(5/2)*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(40*a*d^2*(15*c^3 - 10*c^2*d*x 
 + 8*c*d^2*x^2 + 48*d^3*x^3) + b*(315*c^5 - 210*c^4*d*x + 168*c^3*d^2*x^2 
- 144*c^2*d^3*x^3 + 128*c*d^4*x^4 + 1280*d^5*x^5)) + 30*c^4*(21*b*c^2 + 40 
*a*d^2)*ArcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sqrt[c + d*x])]))/(7680*d^(11 
/2)*x^(5/2))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {521, 27, 90, 60, 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.

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

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {3}{2} e^2 (e x)^{5/2} (4 a d-3 b c x) \sqrt {c+d x}dx}{6 d e^2}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{5/2} (4 a d-3 b c x) \sqrt {c+d x}dx}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \int (e x)^{5/2} \sqrt {c+d x}dx}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \left (\frac {1}{8} c \int \frac {(e x)^{5/2}}{\sqrt {c+d x}}dx+\frac {(e x)^{7/2} \sqrt {c+d x}}{4 e}\right )}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \left (\frac {1}{8} c \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )+\frac {(e x)^{7/2} \sqrt {c+d x}}{4 e}\right )}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \left (\frac {1}{8} c \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )+\frac {(e x)^{7/2} \sqrt {c+d x}}{4 e}\right )}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \left (\frac {1}{8} c \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )+\frac {(e x)^{7/2} \sqrt {c+d x}}{4 e}\right )}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \left (\frac {1}{8} c \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )+\frac {(e x)^{7/2} \sqrt {c+d x}}{4 e}\right )}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\left (40 a d^2+21 b c^2\right ) \left (\frac {1}{8} c \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )}{6 d}\right )+\frac {(e x)^{7/2} \sqrt {c+d x}}{4 e}\right )}{10 d}-\frac {3 b c (e x)^{7/2} (c+d x)^{3/2}}{5 d e}}{4 d}+\frac {b (e x)^{9/2} (c+d x)^{3/2}}{6 d e^2}\)

Input:

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

Output:

(b*(e*x)^(9/2)*(c + d*x)^(3/2))/(6*d*e^2) + ((-3*b*c*(e*x)^(7/2)*(c + d*x) 
^(3/2))/(5*d*e) + ((21*b*c^2 + 40*a*d^2)*(((e*x)^(7/2)*Sqrt[c + d*x])/(4*e 
) + (c*(((e*x)^(5/2)*Sqrt[c + d*x])/(3*d) - (5*c*e*(((e*x)^(3/2)*Sqrt[c + 
d*x])/(2*d) - (3*c*e*((Sqrt[e*x]*Sqrt[c + d*x])/d - (c*Sqrt[e]*ArcTanh[(Sq 
rt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/d^(3/2)))/(4*d)))/(6*d)))/8))/( 
10*d))/(4*d)
 

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 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]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {\left (1280 b \,d^{5} x^{5}+128 b c \,d^{4} x^{4}+1920 a \,d^{5} x^{3}-144 b \,c^{2} d^{3} x^{3}+320 a c \,d^{4} x^{2}+168 b \,c^{3} d^{2} x^{2}-400 a \,c^{2} d^{3} x -210 b \,c^{4} d x +600 a \,c^{3} d^{2}+315 b \,c^{5}\right ) x \sqrt {d x +c}\, e^{3}}{7680 d^{5} \sqrt {e x}}-\frac {c^{4} \left (40 a \,d^{2}+21 b \,c^{2}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e^{3} \sqrt {\left (d x +c \right ) e x}}{1024 d^{5} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(200\)
default	\(-\frac {e^{2} \sqrt {e x}\, \sqrt {d x +c}\, \left (-2560 b \,d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-256 b c \,d^{4} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-3840 a \,d^{5} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+288 b \,c^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-640 a c \,d^{4} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-336 b \,c^{3} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+600 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a \,c^{4} d^{2} e +315 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{6} e +800 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a \,c^{2} d^{3} x +420 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b \,c^{4} d x -1200 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a \,c^{3} d^{2}-630 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b \,c^{5}\right )}{15360 \sqrt {\left (d x +c \right ) e x}\, d^{5} \sqrt {d e}}\)	\(367\)

Input:

int((e*x)^(5/2)*(d*x+c)^(1/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

1/7680*(1280*b*d^5*x^5+128*b*c*d^4*x^4+1920*a*d^5*x^3-144*b*c^2*d^3*x^3+32 
0*a*c*d^4*x^2+168*b*c^3*d^2*x^2-400*a*c^2*d^3*x-210*b*c^4*d*x+600*a*c^3*d^ 
2+315*b*c^5)*x*(d*x+c)^(1/2)/d^5*e^3/(e*x)^(1/2)-1/1024*c^4*(40*a*d^2+21*b 
*c^2)/d^5*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^3*((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 = 400, normalized size of antiderivative = 1.47 \[ \int (e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\left [\frac {15 \, {\left (21 \, b c^{6} + 40 \, a c^{4} d^{2}\right )} e^{2} \sqrt {\frac {e}{d}} \log \left (2 \, d e x - 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (1280 \, b d^{5} e^{2} x^{5} + 128 \, b c d^{4} e^{2} x^{4} - 48 \, {\left (3 \, b c^{2} d^{3} - 40 \, a d^{5}\right )} e^{2} x^{3} + 8 \, {\left (21 \, b c^{3} d^{2} + 40 \, a c d^{4}\right )} e^{2} x^{2} - 10 \, {\left (21 \, b c^{4} d + 40 \, a c^{2} d^{3}\right )} e^{2} x + 15 \, {\left (21 \, b c^{5} + 40 \, a c^{3} d^{2}\right )} e^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{15360 \, d^{5}}, \frac {15 \, {\left (21 \, b c^{6} + 40 \, a c^{4} d^{2}\right )} e^{2} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) + {\left (1280 \, b d^{5} e^{2} x^{5} + 128 \, b c d^{4} e^{2} x^{4} - 48 \, {\left (3 \, b c^{2} d^{3} - 40 \, a d^{5}\right )} e^{2} x^{3} + 8 \, {\left (21 \, b c^{3} d^{2} + 40 \, a c d^{4}\right )} e^{2} x^{2} - 10 \, {\left (21 \, b c^{4} d + 40 \, a c^{2} d^{3}\right )} e^{2} x + 15 \, {\left (21 \, b c^{5} + 40 \, a c^{3} d^{2}\right )} e^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{7680 \, d^{5}}\right ] \] Input:

integrate((e*x)^(5/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="fricas")
 

Output:

[1/15360*(15*(21*b*c^6 + 40*a*c^4*d^2)*e^2*sqrt(e/d)*log(2*d*e*x - 2*sqrt( 
d*x + c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(1280*b*d^5*e^2*x^5 + 128*b*c*d^ 
4*e^2*x^4 - 48*(3*b*c^2*d^3 - 40*a*d^5)*e^2*x^3 + 8*(21*b*c^3*d^2 + 40*a*c 
*d^4)*e^2*x^2 - 10*(21*b*c^4*d + 40*a*c^2*d^3)*e^2*x + 15*(21*b*c^5 + 40*a 
*c^3*d^2)*e^2)*sqrt(d*x + c)*sqrt(e*x))/d^5, 1/7680*(15*(21*b*c^6 + 40*a*c 
^4*d^2)*e^2*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d*sqrt(-e/d)/(d*e*x 
+ c*e)) + (1280*b*d^5*e^2*x^5 + 128*b*c*d^4*e^2*x^4 - 48*(3*b*c^2*d^3 - 40 
*a*d^5)*e^2*x^3 + 8*(21*b*c^3*d^2 + 40*a*c*d^4)*e^2*x^2 - 10*(21*b*c^4*d + 
 40*a*c^2*d^3)*e^2*x + 15*(21*b*c^5 + 40*a*c^3*d^2)*e^2)*sqrt(d*x + c)*sqr 
t(e*x))/d^5]
 

Sympy [F(-1)]

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

integrate((e*x)**(5/2)*(d*x+c)**(1/2)*(b*x**2+a),x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(5/2)*(d*x+c)^(1/2)*(b*x^2+a),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (222) = 444\).

Time = 0.26 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.97 \[ \int (e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\frac {\frac {40 \, {\left (\frac {105 \, c^{4} e \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^{2}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{3}} - \frac {25 \, c}{d^{3}}\right )} + \frac {163 \, c^{2}}{d^{3}}\right )} - \frac {279 \, c^{3}}{d^{3}}\right )} \sqrt {d x + c}\right )} a e^{2} {\left | d \right |}}{d} + \frac {{\left (\frac {3465 \, c^{6} e \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}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (d x + c\right )} {\left (8 \, {\left (d x + c\right )} {\left (\frac {10 \, {\left (d x + c\right )}}{d^{5}} - \frac {61 \, c}{d^{5}}\right )} + \frac {1251 \, c^{2}}{d^{5}}\right )} - \frac {3481 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {11395 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {11895 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c}\right )} b e^{2} {\left | d \right |}}{d} - \frac {320 \, {\left (\frac {15 \, c^{3} d e \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}} + \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, d x - 9 \, c\right )} {\left (d x + c\right )} + 33 \, c^{2}\right )} \sqrt {d x + c}\right )} a c e^{2} {\left | d \right |}}{d^{4}} - \frac {12 \, {\left (\frac {315 \, c^{5} d e \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}} + {\left (965 \, c^{4} - 2 \, {\left (745 \, c^{3} - 4 \, {\left (2 \, {\left (8 \, d x - 33 \, c\right )} {\left (d x + c\right )} + 171 \, c^{2}\right )} {\left (d x + c\right )}\right )} {\left (d x + c\right )}\right )} \sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {d x + c}\right )} b c e^{2} {\left | d \right |}}{d^{6}}}{7680 \, d} \] Input:

integrate((e*x)^(5/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="giac")
 

Output:

-1/7680*(40*(105*c^4*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d 
*e - c*d*e)))/(sqrt(d*e)*d^2) - sqrt((d*x + c)*d*e - c*d*e)*(2*(d*x + c)*( 
4*(d*x + c)*(6*(d*x + c)/d^3 - 25*c/d^3) + 163*c^2/d^3) - 279*c^3/d^3)*sqr 
t(d*x + c))*a*e^2*abs(d)/d + (3465*c^6*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) 
+ sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^4) - sqrt((d*x + c)*d*e - c*d 
*e)*(2*(4*(2*(d*x + c)*(8*(d*x + c)*(10*(d*x + c)/d^5 - 61*c/d^5) + 1251*c 
^2/d^5) - 3481*c^3/d^5)*(d*x + c) + 11395*c^4/d^5)*(d*x + c) - 11895*c^5/d 
^5)*sqrt(d*x + c))*b*e^2*abs(d)/d - 320*(15*c^3*d*e*log(abs(-sqrt(d*e)*sqr 
t(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/sqrt(d*e) + sqrt((d*x + c)*d*e 
- c*d*e)*(2*(4*d*x - 9*c)*(d*x + c) + 33*c^2)*sqrt(d*x + c))*a*c*e^2*abs(d 
)/d^4 - 12*(315*c^5*d*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)* 
d*e - c*d*e)))/sqrt(d*e) + (965*c^4 - 2*(745*c^3 - 4*(2*(8*d*x - 33*c)*(d* 
x + c) + 171*c^2)*(d*x + c))*(d*x + c))*sqrt((d*x + c)*d*e - c*d*e)*sqrt(d 
*x + c))*b*c*e^2*abs(d)/d^6)/d
 

Mupad [F(-1)]

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

int((e*x)^(5/2)*(a + b*x^2)*(c + d*x)^(1/2),x)
 

Output:

int((e*x)^(5/2)*(a + b*x^2)*(c + d*x)^(1/2), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.90 \[ \int (e x)^{5/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\sqrt {e}\, e^{2} \left (600 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{3} d^{3}-400 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{2} d^{4} x +320 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{5} x^{2}+1920 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{6} x^{3}+315 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{5} d -210 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{4} d^{2} x +168 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} d^{3} x^{2}-144 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d^{4} x^{3}+128 \sqrt {x}\, \sqrt {d x +c}\, b c \,d^{5} x^{4}+1280 \sqrt {x}\, \sqrt {d x +c}\, b \,d^{6} x^{5}-600 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a \,c^{4} d^{2}-315 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b \,c^{6}\right )}{7680 d^{6}} \] Input:

int((e*x)^(5/2)*(d*x+c)^(1/2)*(b*x^2+a),x)
 

Output:

(sqrt(e)*e**2*(600*sqrt(x)*sqrt(c + d*x)*a*c**3*d**3 - 400*sqrt(x)*sqrt(c 
+ d*x)*a*c**2*d**4*x + 320*sqrt(x)*sqrt(c + d*x)*a*c*d**5*x**2 + 1920*sqrt 
(x)*sqrt(c + d*x)*a*d**6*x**3 + 315*sqrt(x)*sqrt(c + d*x)*b*c**5*d - 210*s 
qrt(x)*sqrt(c + d*x)*b*c**4*d**2*x + 168*sqrt(x)*sqrt(c + d*x)*b*c**3*d**3 
*x**2 - 144*sqrt(x)*sqrt(c + d*x)*b*c**2*d**4*x**3 + 128*sqrt(x)*sqrt(c + 
d*x)*b*c*d**5*x**4 + 1280*sqrt(x)*sqrt(c + d*x)*b*d**6*x**5 - 600*sqrt(d)* 
log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*c**4*d**2 - 315*sqrt(d)*l 
og((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b*c**6))/(7680*d**6)