\(\int (e x)^{3/2} \sqrt {c+d x} (a+b x^2) \, dx\) [743]

Optimal result

Integrand size = 24, antiderivative size = 230 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\frac {c^2 \left (7 b c^2+16 a d^2\right ) e \sqrt {e x} \sqrt {c+d x}}{128 d^4}+\frac {c \left (7 b c^2+16 a d^2\right ) (e x)^{3/2} \sqrt {c+d x}}{192 d^3}+\frac {\left (7 b c^2+16 a d^2\right ) (e x)^{5/2} \sqrt {c+d x}}{48 d^2 e}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{40 d^2 e}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}+\frac {c^3 \left (7 b c^2+16 a d^2\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{128 d^{9/2}} \] Output:

-1/128*c^2*(16*a*d^2+7*b*c^2)*e*(e*x)^(1/2)*(d*x+c)^(1/2)/d^4+1/192*c*(16* 
a*d^2+7*b*c^2)*(e*x)^(3/2)*(d*x+c)^(1/2)/d^3+1/48*(16*a*d^2+7*b*c^2)*(e*x) 
^(5/2)*(d*x+c)^(1/2)/d^2/e-7/40*b*c*(e*x)^(5/2)*(d*x+c)^(3/2)/d^2/e+1/5*b* 
(e*x)^(7/2)*(d*x+c)^(3/2)/d/e^2+1/128*c^3*(16*a*d^2+7*b*c^2)*e^(3/2)*arcta 
nh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.70 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {(e x)^{3/2} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (80 a d^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )+b \left (-105 c^4+70 c^3 d x-56 c^2 d^2 x^2+48 c d^3 x^3+384 d^4 x^4\right )\right )+30 c^3 \left (7 b c^2+16 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )\right )}{1920 d^{9/2} x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(80*a*d^2*(-3*c^2 + 2*c*d*x + 
8*d^2*x^2) + b*(-105*c^4 + 70*c^3*d*x - 56*c^2*d^2*x^2 + 48*c*d^3*x^3 + 38 
4*d^4*x^4)) + 30*c^3*(7*b*c^2 + 16*a*d^2)*ArcTanh[(Sqrt[d]*Sqrt[x])/(-Sqrt 
[c] + Sqrt[c + d*x])]))/(1920*d^(9/2)*x^(3/2))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {521, 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.

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

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {1}{2} e^2 (e x)^{3/2} (10 a d-7 b c x) \sqrt {c+d x}dx}{5 d e^2}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{3/2} (10 a d-7 b c x) \sqrt {c+d x}dx}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {5 \left (16 a d^2+7 b c^2\right ) \int (e x)^{3/2} \sqrt {c+d x}dx}{8 d}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{4 d e}}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {5 \left (16 a d^2+7 b c^2\right ) \left (\frac {1}{6} c \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{4 d e}}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {5 \left (16 a d^2+7 b c^2\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{4 d e}}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {5 \left (16 a d^2+7 b c^2\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{4 d e}}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {5 \left (16 a d^2+7 b c^2\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{4 d e}}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {5 \left (16 a d^2+7 b c^2\right ) \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )}{8 d}-\frac {7 b c (e x)^{5/2} (c+d x)^{3/2}}{4 d e}}{10 d}+\frac {b (e x)^{7/2} (c+d x)^{3/2}}{5 d e^2}\)

Input:

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

Output:

(b*(e*x)^(7/2)*(c + d*x)^(3/2))/(5*d*e^2) + ((-7*b*c*(e*x)^(5/2)*(c + d*x) 
^(3/2))/(4*d*e) + (5*(7*b*c^2 + 16*a*d^2)*(((e*x)^(5/2)*Sqrt[c + d*x])/(3* 
e) + (c*(((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[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/ 
d^(3/2)))/(4*d)))/6))/(8*d))/(10*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 = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {\left (-384 b \,x^{4} d^{4}-48 c b \,d^{3} x^{3}-640 a \,d^{4} x^{2}+56 b \,c^{2} d^{2} x^{2}-160 a c \,d^{3} x -70 b \,c^{3} d x +240 a \,c^{2} d^{2}+105 b \,c^{4}\right ) x \sqrt {d x +c}\, e^{2}}{1920 d^{4} \sqrt {e x}}+\frac {c^{3} \left (16 a \,d^{2}+7 b \,c^{2}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e^{2} \sqrt {\left (d x +c \right ) e x}}{256 d^{4} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(176\)
default	\(\frac {\sqrt {e x}\, \sqrt {d x +c}\, e \left (768 b \,d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+96 b c \,d^{3} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+1280 a \,d^{4} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}-112 b \,c^{2} d^{2} x^{2} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+240 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a \,c^{3} d^{2} e +105 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{5} e +320 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a c \,d^{3} x +140 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b \,c^{3} d x -480 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a \,c^{2} d^{2}-210 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b \,c^{4}\right )}{3840 \sqrt {\left (d x +c \right ) e x}\, d^{4} \sqrt {d e}}\)	\(311\)

Input:

int((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/1920*(-384*b*d^4*x^4-48*b*c*d^3*x^3-640*a*d^4*x^2+56*b*c^2*d^2*x^2-160* 
a*c*d^3*x-70*b*c^3*d*x+240*a*c^2*d^2+105*b*c^4)*x*(d*x+c)^(1/2)/d^4*e^2/(e 
*x)^(1/2)+1/256*c^3*(16*a*d^2+7*b*c^2)/d^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^2*((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 = 323, normalized size of antiderivative = 1.40 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\left [\frac {15 \, {\left (7 \, b c^{5} + 16 \, a c^{3} d^{2}\right )} e \sqrt {\frac {e}{d}} \log \left (2 \, d e x + 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (384 \, b d^{4} e x^{4} + 48 \, b c d^{3} e x^{3} - 8 \, {\left (7 \, b c^{2} d^{2} - 80 \, a d^{4}\right )} e x^{2} + 10 \, {\left (7 \, b c^{3} d + 16 \, a c d^{3}\right )} e x - 15 \, {\left (7 \, b c^{4} + 16 \, a c^{2} d^{2}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{3840 \, d^{4}}, -\frac {15 \, {\left (7 \, b c^{5} + 16 \, a c^{3} d^{2}\right )} e \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) - {\left (384 \, b d^{4} e x^{4} + 48 \, b c d^{3} e x^{3} - 8 \, {\left (7 \, b c^{2} d^{2} - 80 \, a d^{4}\right )} e x^{2} + 10 \, {\left (7 \, b c^{3} d + 16 \, a c d^{3}\right )} e x - 15 \, {\left (7 \, b c^{4} + 16 \, a c^{2} d^{2}\right )} e\right )} \sqrt {d x + c} \sqrt {e x}}{1920 \, d^{4}}\right ] \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="fricas")
 

Output:

[1/3840*(15*(7*b*c^5 + 16*a*c^3*d^2)*e*sqrt(e/d)*log(2*d*e*x + 2*sqrt(d*x 
+ c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(384*b*d^4*e*x^4 + 48*b*c*d^3*e*x^3 
- 8*(7*b*c^2*d^2 - 80*a*d^4)*e*x^2 + 10*(7*b*c^3*d + 16*a*c*d^3)*e*x - 15* 
(7*b*c^4 + 16*a*c^2*d^2)*e)*sqrt(d*x + c)*sqrt(e*x))/d^4, -1/1920*(15*(7*b 
*c^5 + 16*a*c^3*d^2)*e*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d*sqrt(-e 
/d)/(d*e*x + c*e)) - (384*b*d^4*e*x^4 + 48*b*c*d^3*e*x^3 - 8*(7*b*c^2*d^2 
- 80*a*d^4)*e*x^2 + 10*(7*b*c^3*d + 16*a*c*d^3)*e*x - 15*(7*b*c^4 + 16*a*c 
^2*d^2)*e)*sqrt(d*x + c)*sqrt(e*x))/d^4]
 

Sympy [A] (verification not implemented)

Time = 128.18 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.68 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=- \frac {a c^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x}}{8 d^{2} \sqrt {1 + \frac {d x}{c}}} - \frac {a c^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {3}{2}}}{24 d \sqrt {1 + \frac {d x}{c}}} + \frac {5 a \sqrt {c} e^{\frac {3}{2}} x^{\frac {5}{2}}}{12 \sqrt {1 + \frac {d x}{c}}} + \frac {a c^{3} e^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{8 d^{\frac {5}{2}}} + \frac {a d e^{\frac {3}{2}} x^{\frac {7}{2}}}{3 \sqrt {c} \sqrt {1 + \frac {d x}{c}}} - \frac {7 b c^{\frac {9}{2}} e^{\frac {3}{2}} \sqrt {x}}{128 d^{4} \sqrt {1 + \frac {d x}{c}}} - \frac {7 b c^{\frac {7}{2}} e^{\frac {3}{2}} x^{\frac {3}{2}}}{384 d^{3} \sqrt {1 + \frac {d x}{c}}} + \frac {7 b c^{\frac {5}{2}} e^{\frac {3}{2}} x^{\frac {5}{2}}}{960 d^{2} \sqrt {1 + \frac {d x}{c}}} - \frac {b c^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {7}{2}}}{240 d \sqrt {1 + \frac {d x}{c}}} + \frac {9 b \sqrt {c} e^{\frac {3}{2}} x^{\frac {9}{2}}}{40 \sqrt {1 + \frac {d x}{c}}} + \frac {7 b c^{5} e^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{128 d^{\frac {9}{2}}} + \frac {b d e^{\frac {3}{2}} x^{\frac {11}{2}}}{5 \sqrt {c} \sqrt {1 + \frac {d x}{c}}} \] Input:

integrate((e*x)**(3/2)*(d*x+c)**(1/2)*(b*x**2+a),x)
 

Output:

-a*c**(5/2)*e**(3/2)*sqrt(x)/(8*d**2*sqrt(1 + d*x/c)) - a*c**(3/2)*e**(3/2 
)*x**(3/2)/(24*d*sqrt(1 + d*x/c)) + 5*a*sqrt(c)*e**(3/2)*x**(5/2)/(12*sqrt 
(1 + d*x/c)) + a*c**3*e**(3/2)*asinh(sqrt(d)*sqrt(x)/sqrt(c))/(8*d**(5/2)) 
 + a*d*e**(3/2)*x**(7/2)/(3*sqrt(c)*sqrt(1 + d*x/c)) - 7*b*c**(9/2)*e**(3/ 
2)*sqrt(x)/(128*d**4*sqrt(1 + d*x/c)) - 7*b*c**(7/2)*e**(3/2)*x**(3/2)/(38 
4*d**3*sqrt(1 + d*x/c)) + 7*b*c**(5/2)*e**(3/2)*x**(5/2)/(960*d**2*sqrt(1 
+ d*x/c)) - b*c**(3/2)*e**(3/2)*x**(7/2)/(240*d*sqrt(1 + d*x/c)) + 9*b*sqr 
t(c)*e**(3/2)*x**(9/2)/(40*sqrt(1 + d*x/c)) + 7*b*c**5*e**(3/2)*asinh(sqrt 
(d)*sqrt(x)/sqrt(c))/(128*d**(9/2)) + b*d*e**(3/2)*x**(11/2)/(5*sqrt(c)*sq 
rt(1 + d*x/c))
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)*(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. 450 vs. \(2 (186) = 372\).

Time = 0.24 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.96 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=-\frac {{\left (\frac {10 \, {\left (\frac {105 \, c^{4} e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{2}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{3}} - \frac {25 \, c}{d^{3}}\right )} + \frac {163 \, c^{2}}{d^{3}}\right )} - \frac {279 \, c^{3}}{d^{3}}\right )} \sqrt {d x + c}\right )} b c {\left | d \right |}}{d^{2}} + \frac {480 \, {\left (\frac {3 \, c^{2} d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, d x - 3 \, c\right )} \sqrt {d x + c}\right )} a c {\left | d \right |}}{d^{3}} - \frac {80 \, {\left (\frac {15 \, c^{3} d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} + \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, d x - 9 \, c\right )} {\left (d x + c\right )} + 33 \, c^{2}\right )} \sqrt {d x + c}\right )} a {\left | d \right |}}{d^{3}} - \frac {3 \, {\left (\frac {315 \, c^{5} d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} + {\left (965 \, c^{4} - 2 \, {\left (745 \, c^{3} - 4 \, {\left (2 \, {\left (8 \, d x - 33 \, c\right )} {\left (d x + c\right )} + 171 \, c^{2}\right )} {\left (d x + c\right )}\right )} {\left (d x + c\right )}\right )} \sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {d x + c}\right )} b {\left | d \right |}}{d^{5}}\right )} e}{1920 \, d} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a),x, algorithm="giac")
 

Output:

-1/1920*(10*(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))*b*c*abs(d)/d^2 + 480*(3*c^2*d*e*log(abs(-sqrt(d*e)*sqrt(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*d*x - 3*c)*sqrt(d*x + c))*a*c*abs(d)/d^3 - 80*(15*c^3*d*e*log(abs(-sqr 
t(d*e)*sqrt(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*a 
bs(d)/d^3 - 3*(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)*sqr 
t(d*x + c))*b*abs(d)/d^5)*e/d
 

Mupad [F(-1)]

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

int((e*x)^(3/2)*(a + b*x^2)*(c + d*x)^(1/2),x)
 

Output:

int((e*x)^(3/2)*(a + b*x^2)*(c + d*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.88 \[ \int (e x)^{3/2} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\sqrt {e}\, e \left (-240 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{2} d^{3}+160 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{4} x +640 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{5} x^{2}-105 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{4} d +70 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} d^{2} x -56 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d^{3} x^{2}+48 \sqrt {x}\, \sqrt {d x +c}\, b c \,d^{4} x^{3}+384 \sqrt {x}\, \sqrt {d x +c}\, b \,d^{5} x^{4}+240 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a \,c^{3} d^{2}+105 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b \,c^{5}\right )}{1920 d^{5}} \] Input:

int((e*x)^(3/2)*(d*x+c)^(1/2)*(b*x^2+a),x)
 

Output:

(sqrt(e)*e*( - 240*sqrt(x)*sqrt(c + d*x)*a*c**2*d**3 + 160*sqrt(x)*sqrt(c 
+ d*x)*a*c*d**4*x + 640*sqrt(x)*sqrt(c + d*x)*a*d**5*x**2 - 105*sqrt(x)*sq 
rt(c + d*x)*b*c**4*d + 70*sqrt(x)*sqrt(c + d*x)*b*c**3*d**2*x - 56*sqrt(x) 
*sqrt(c + d*x)*b*c**2*d**3*x**2 + 48*sqrt(x)*sqrt(c + d*x)*b*c*d**4*x**3 + 
 384*sqrt(x)*sqrt(c + d*x)*b*d**5*x**4 + 240*sqrt(d)*log((sqrt(c + d*x) + 
sqrt(x)*sqrt(d))/sqrt(c))*a*c**3*d**2 + 105*sqrt(d)*log((sqrt(c + d*x) + s 
qrt(x)*sqrt(d))/sqrt(c))*b*c**5))/(1920*d**5)