\(\int \sqrt {e x} \sqrt {c+d x} (a+b x^2) \, dx\) [744]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {c \left (5 b c^2+16 a d^2\right ) \sqrt {e x} \sqrt {c+d x}}{64 d^3}+\frac {\left (5 b c^2+16 a d^2\right ) (e x)^{3/2} \sqrt {c+d x}}{32 d^2 e}-\frac {5 b c (e x)^{3/2} (c+d x)^{3/2}}{24 d^2 e}+\frac {b (e x)^{5/2} (c+d x)^{3/2}}{4 d e^2}-\frac {c^2 \left (5 b c^2+16 a d^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{64 d^{7/2}} \] Output:

1/64*c*(16*a*d^2+5*b*c^2)*(e*x)^(1/2)*(d*x+c)^(1/2)/d^3+1/32*(16*a*d^2+5*b 
*c^2)*(e*x)^(3/2)*(d*x+c)^(1/2)/d^2/e-5/24*b*c*(e*x)^(3/2)*(d*x+c)^(3/2)/d 
^2/e+1/4*b*(e*x)^(5/2)*(d*x+c)^(3/2)/d/e^2-1/64*c^2*(16*a*d^2+5*b*c^2)*e^( 
1/2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.72 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\sqrt {e x} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (48 a d^2 (c+2 d x)+b \left (15 c^3-10 c^2 d x+8 c d^2 x^2+48 d^3 x^3\right )\right )+6 c^2 \left (5 b c^2+16 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}-\sqrt {c+d x}}\right )\right )}{192 d^{7/2} \sqrt {x}} \] Input:

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

Output:

(Sqrt[e*x]*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(48*a*d^2*(c + 2*d*x) + b*(15*c^ 
3 - 10*c^2*d*x + 8*c*d^2*x^2 + 48*d^3*x^3)) + 6*c^2*(5*b*c^2 + 16*a*d^2)*A 
rcTanh[(Sqrt[d]*Sqrt[x])/(Sqrt[c] - Sqrt[c + d*x])]))/(192*d^(7/2)*Sqrt[x] 
)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {521, 27, 90, 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.

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

\(\Big \downarrow \) 521

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(b*(e*x)^(5/2)*(c + d*x)^(3/2))/(4*d*e^2) + ((-5*b*c*(e*x)^(3/2)*(c + d*x) 
^(3/2))/(3*d*e) + ((5*b*c^2 + 16*a*d^2)*(((e*x)^(3/2)*Sqrt[c + d*x])/(2*e) 
 + (c*((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))/(2*d))/(8*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 = 0.24 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (48 b \,d^{3} x^{3}+8 b c \,d^{2} x^{2}+96 a x \,d^{3}-10 b \,c^{2} d x +48 a \,d^{2} c +15 b \,c^{3}\right ) x \sqrt {d x +c}\, e}{192 d^{3} \sqrt {e x}}-\frac {c^{2} \left (16 a \,d^{2}+5 b \,c^{2}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e \sqrt {\left (d x +c \right ) e x}}{128 d^{3} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(148\)
default	\(-\frac {\sqrt {e x}\, \sqrt {d x +c}\, \left (-96 b \,d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-16 b c \,d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+48 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a \,c^{2} d^{2} e +15 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{4} e -192 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a \,d^{3} x +20 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b \,c^{2} d x -96 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a c \,d^{2}-30 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b \,c^{3}\right )}{384 \sqrt {\left (d x +c \right ) e x}\, d^{3} \sqrt {d e}}\)	\(256\)

Input:

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

Output:

1/192*(48*b*d^3*x^3+8*b*c*d^2*x^2+96*a*d^3*x-10*b*c^2*d*x+48*a*c*d^2+15*b* 
c^3)*x*(d*x+c)^(1/2)/d^3*e/(e*x)^(1/2)-1/128*c^2*(16*a*d^2+5*b*c^2)/d^3*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 = 256, normalized size of antiderivative = 1.35 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\left [\frac {3 \, {\left (5 \, b c^{4} + 16 \, a c^{2} d^{2}\right )} \sqrt {\frac {e}{d}} \log \left (2 \, d e x - 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (48 \, b d^{3} x^{3} + 8 \, b c d^{2} x^{2} + 15 \, b c^{3} + 48 \, a c d^{2} - 2 \, {\left (5 \, b c^{2} d - 48 \, a d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{384 \, d^{3}}, \frac {3 \, {\left (5 \, b c^{4} + 16 \, a c^{2} d^{2}\right )} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) + {\left (48 \, b d^{3} x^{3} + 8 \, b c d^{2} x^{2} + 15 \, b c^{3} + 48 \, a c d^{2} - 2 \, {\left (5 \, b c^{2} d - 48 \, a d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{192 \, d^{3}}\right ] \] Input:

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

Output:

[1/384*(3*(5*b*c^4 + 16*a*c^2*d^2)*sqrt(e/d)*log(2*d*e*x - 2*sqrt(d*x + c) 
*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(48*b*d^3*x^3 + 8*b*c*d^2*x^2 + 15*b*c^3 
 + 48*a*c*d^2 - 2*(5*b*c^2*d - 48*a*d^3)*x)*sqrt(d*x + c)*sqrt(e*x))/d^3, 
1/192*(3*(5*b*c^4 + 16*a*c^2*d^2)*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x 
)*d*sqrt(-e/d)/(d*e*x + c*e)) + (48*b*d^3*x^3 + 8*b*c*d^2*x^2 + 15*b*c^3 + 
 48*a*c*d^2 - 2*(5*b*c^2*d - 48*a*d^3)*x)*sqrt(d*x + c)*sqrt(e*x))/d^3]
 

Sympy [A] (verification not implemented)

Time = 23.43 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.68 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {a c^{\frac {3}{2}} \sqrt {e} \sqrt {x}}{4 d \sqrt {1 + \frac {d x}{c}}} + \frac {3 a \sqrt {c} \sqrt {e} x^{\frac {3}{2}}}{4 \sqrt {1 + \frac {d x}{c}}} - \frac {a c^{2} \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{4 d^{\frac {3}{2}}} + \frac {a d \sqrt {e} x^{\frac {5}{2}}}{2 \sqrt {c} \sqrt {1 + \frac {d x}{c}}} + \frac {5 b c^{\frac {7}{2}} \sqrt {e} \sqrt {x}}{64 d^{3} \sqrt {1 + \frac {d x}{c}}} + \frac {5 b c^{\frac {5}{2}} \sqrt {e} x^{\frac {3}{2}}}{192 d^{2} \sqrt {1 + \frac {d x}{c}}} - \frac {b c^{\frac {3}{2}} \sqrt {e} x^{\frac {5}{2}}}{96 d \sqrt {1 + \frac {d x}{c}}} + \frac {7 b \sqrt {c} \sqrt {e} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {d x}{c}}} - \frac {5 b c^{4} \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{64 d^{\frac {7}{2}}} + \frac {b d \sqrt {e} x^{\frac {9}{2}}}{4 \sqrt {c} \sqrt {1 + \frac {d x}{c}}} \] Input:

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

Output:

a*c**(3/2)*sqrt(e)*sqrt(x)/(4*d*sqrt(1 + d*x/c)) + 3*a*sqrt(c)*sqrt(e)*x** 
(3/2)/(4*sqrt(1 + d*x/c)) - a*c**2*sqrt(e)*asinh(sqrt(d)*sqrt(x)/sqrt(c))/ 
(4*d**(3/2)) + a*d*sqrt(e)*x**(5/2)/(2*sqrt(c)*sqrt(1 + d*x/c)) + 5*b*c**( 
7/2)*sqrt(e)*sqrt(x)/(64*d**3*sqrt(1 + d*x/c)) + 5*b*c**(5/2)*sqrt(e)*x**( 
3/2)/(192*d**2*sqrt(1 + d*x/c)) - b*c**(3/2)*sqrt(e)*x**(5/2)/(96*d*sqrt(1 
 + d*x/c)) + 7*b*sqrt(c)*sqrt(e)*x**(7/2)/(24*sqrt(1 + d*x/c)) - 5*b*c**4* 
sqrt(e)*asinh(sqrt(d)*sqrt(x)/sqrt(c))/(64*d**(7/2)) + b*d*sqrt(e)*x**(9/2 
)/(4*sqrt(c)*sqrt(1 + d*x/c))
 

Maxima [F(-2)]

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

integrate((e*x)^(1/2)*(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. 399 vs. \(2 (152) = 304\).

Time = 0.23 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.10 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\frac {192 \, {\left (\frac {c d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} + \sqrt {{\left (d x + c\right )} d e - c d e} \sqrt {d x + c}\right )} a c {\left | d \right |}}{d^{2}} - \frac {{\left (\frac {105 \, c^{4} e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{2}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )}}{d^{3}} - \frac {25 \, c}{d^{3}}\right )} + \frac {163 \, c^{2}}{d^{3}}\right )} - \frac {279 \, c^{3}}{d^{3}}\right )} \sqrt {d x + c}\right )} b {\left | d \right |}}{d} - \frac {48 \, {\left (\frac {3 \, c^{2} d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} - \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, d x - 3 \, c\right )} \sqrt {d x + c}\right )} a {\left | d \right |}}{d^{2}} + \frac {8 \, {\left (\frac {15 \, c^{3} d e \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} + \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, d x - 9 \, c\right )} {\left (d x + c\right )} + 33 \, c^{2}\right )} \sqrt {d x + c}\right )} b c {\left | d \right |}}{d^{4}}}{192 \, d} \] Input:

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

Output:

1/192*(192*(c*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)*sqrt(d*x + c))*a*c*abs(d) 
/d^2 - (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)*sqrt(d*x 
 + c))*b*abs(d)/d - 48*(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*abs(d)/d^2 + 8*(15*c^3*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*(4*d*x - 9*c)*(d*x + c) + 33*c^2)*sqrt(d*x + c))*b*c*abs(d)/d^4)/ 
d
 

Mupad [B] (verification not implemented)

Time = 33.44 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.99 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\frac {2789\,b\,c^4\,e^5\,{\left (e\,x\right )}^{7/2}}{96\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}+\frac {2789\,b\,c^4\,d\,e^4\,{\left (e\,x\right )}^{9/2}}{96\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}+\frac {5\,b\,c^4\,d^4\,e\,{\left (e\,x\right )}^{15/2}}{32\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{15}}+\frac {5\,b\,c^4\,e^8\,\sqrt {e\,x}}{32\,d^3\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {115\,b\,c^4\,e^7\,{\left (e\,x\right )}^{3/2}}{96\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {383\,b\,c^4\,e^6\,{\left (e\,x\right )}^{5/2}}{96\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {383\,b\,c^4\,d^2\,e^3\,{\left (e\,x\right )}^{11/2}}{96\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}-\frac {115\,b\,c^4\,d^3\,e^2\,{\left (e\,x\right )}^{13/2}}{96\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{13}}}{e^8-\frac {8\,d\,e^8\,x}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {28\,d^2\,e^8\,x^2}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {56\,d^3\,e^8\,x^3}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {70\,d^4\,e^8\,x^4}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {56\,d^5\,e^8\,x^5}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}+\frac {28\,d^6\,e^8\,x^6}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}-\frac {8\,d^7\,e^8\,x^7}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{14}}+\frac {d^8\,e^8\,x^8}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{16}}}+a\,\sqrt {e\,x}\,\left (\frac {x}{2}+\frac {c}{4\,d}\right )\,\sqrt {c+d\,x}-\frac {5\,b\,c^4\,\sqrt {e}\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {e\,x}}{\sqrt {e}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{32\,d^{7/2}}-\frac {a\,c^2\,\sqrt {e}\,\ln \left (c\,\sqrt {e}+2\,\sqrt {d}\,\sqrt {e\,x}\,\sqrt {c+d\,x}+2\,d\,\sqrt {e}\,x\right )}{8\,d^{3/2}} \] Input:

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

Output:

((2789*b*c^4*e^5*(e*x)^(7/2))/(96*((c + d*x)^(1/2) - c^(1/2))^7) + (2789*b 
*c^4*d*e^4*(e*x)^(9/2))/(96*((c + d*x)^(1/2) - c^(1/2))^9) + (5*b*c^4*d^4* 
e*(e*x)^(15/2))/(32*((c + d*x)^(1/2) - c^(1/2))^15) + (5*b*c^4*e^8*(e*x)^( 
1/2))/(32*d^3*((c + d*x)^(1/2) - c^(1/2))) - (115*b*c^4*e^7*(e*x)^(3/2))/( 
96*d^2*((c + d*x)^(1/2) - c^(1/2))^3) + (383*b*c^4*e^6*(e*x)^(5/2))/(96*d* 
((c + d*x)^(1/2) - c^(1/2))^5) + (383*b*c^4*d^2*e^3*(e*x)^(11/2))/(96*((c 
+ d*x)^(1/2) - c^(1/2))^11) - (115*b*c^4*d^3*e^2*(e*x)^(13/2))/(96*((c + d 
*x)^(1/2) - c^(1/2))^13))/(e^8 - (8*d*e^8*x)/((c + d*x)^(1/2) - c^(1/2))^2 
 + (28*d^2*e^8*x^2)/((c + d*x)^(1/2) - c^(1/2))^4 - (56*d^3*e^8*x^3)/((c + 
 d*x)^(1/2) - c^(1/2))^6 + (70*d^4*e^8*x^4)/((c + d*x)^(1/2) - c^(1/2))^8 
- (56*d^5*e^8*x^5)/((c + d*x)^(1/2) - c^(1/2))^10 + (28*d^6*e^8*x^6)/((c + 
 d*x)^(1/2) - c^(1/2))^12 - (8*d^7*e^8*x^7)/((c + d*x)^(1/2) - c^(1/2))^14 
 + (d^8*e^8*x^8)/((c + d*x)^(1/2) - c^(1/2))^16) + a*(e*x)^(1/2)*(x/2 + c/ 
(4*d))*(c + d*x)^(1/2) - (5*b*c^4*e^(1/2)*atanh((d^(1/2)*(e*x)^(1/2))/(e^( 
1/2)*((c + d*x)^(1/2) - c^(1/2)))))/(32*d^(7/2)) - (a*c^2*e^(1/2)*log(c*e^ 
(1/2) + 2*d^(1/2)*(e*x)^(1/2)*(c + d*x)^(1/2) + 2*d*e^(1/2)*x))/(8*d^(3/2) 
)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85 \[ \int \sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx=\frac {\sqrt {e}\, \left (48 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{3}+96 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{4} x +15 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} d -10 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d^{2} x +8 \sqrt {x}\, \sqrt {d x +c}\, b c \,d^{3} x^{2}+48 \sqrt {x}\, \sqrt {d x +c}\, b \,d^{4} x^{3}-48 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a \,c^{2} d^{2}-15 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b \,c^{4}\right )}{192 d^{4}} \] Input:

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

Output:

(sqrt(e)*(48*sqrt(x)*sqrt(c + d*x)*a*c*d**3 + 96*sqrt(x)*sqrt(c + d*x)*a*d 
**4*x + 15*sqrt(x)*sqrt(c + d*x)*b*c**3*d - 10*sqrt(x)*sqrt(c + d*x)*b*c** 
2*d**2*x + 8*sqrt(x)*sqrt(c + d*x)*b*c*d**3*x**2 + 48*sqrt(x)*sqrt(c + d*x 
)*b*d**4*x**3 - 48*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))* 
a*c**2*d**2 - 15*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b* 
c**4))/(192*d**4)