\(\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx\) [232]

Optimal result

Integrand size = 28, antiderivative size = 283 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 \sqrt [4]{b^2-4 a c} d^{7/2} \sqrt {a+b x+c x^2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{5 c^2 \sqrt [4]{b^2-4 a c} d^{7/2} \sqrt {a+b x+c x^2}} \] Output:

-1/5*(c*x^2+b*x+a)^(1/2)/c/d/(2*c*d*x+b*d)^(5/2)+2/5*(c*x^2+b*x+a)^(1/2)/c 
/(-4*a*c+b^2)/d^3/(2*c*d*x+b*d)^(1/2)-1/5*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^ 
(1/2)*EllipticE((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)/c^2/(-4* 
a*c+b^2)^(1/4)/d^(7/2)/(c*x^2+b*x+a)^(1/2)+1/5*(-c*(c*x^2+b*x+a)/(-4*a*c+b 
^2))^(1/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)/c^2 
/(-4*a*c+b^2)^(1/4)/d^(7/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=-\frac {\sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{10 c d (d (b+2 c x))^{5/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:

Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(7/2),x]
 

Output:

-1/10*(Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/4, -1/2, -1/4, (b + 2*c* 
x)^2/(b^2 - 4*a*c)])/(c*d*(d*(b + 2*c*x))^(5/2)*Sqrt[(c*(a + x*(b + c*x))) 
/(-b^2 + 4*a*c)])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1108, 1117, 1115, 1114, 836, 27, 762, 1389, 327}

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.

\(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx\)

\(\Big \downarrow \) 1108

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

\(\Big \downarrow \) 1117

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

\(\Big \downarrow \) 1115

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

\(\Big \downarrow \) 1114

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{10 c d^2}-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {\sqrt {\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}+1}}{\sqrt {1-\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{10 c d^2}-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {4 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d^{3/2} \left (b^2-4 a c\right )^{3/4} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{c d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{10 c d^2}-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}\)

Input:

Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(7/2),x]
 

Output:

-1/5*Sqrt[a + b*x + c*x^2]/(c*d*(b*d + 2*c*d*x)^(5/2)) + ((4*Sqrt[a + b*x 
+ c*x^2])/((b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]) - (2*Sqrt[-((c*(a + b*x + 
c*x^2))/(b^2 - 4*a*c))]*((b^2 - 4*a*c)^(3/4)*d^(3/2)*EllipticE[ArcSin[Sqrt 
[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1] - (b^2 - 4*a*c)^(3/4)* 
d^(3/2)*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d]) 
], -1]))/(c*(b^2 - 4*a*c)*d^3*Sqrt[a + b*x + c*x^2]))/(10*c*d^2)
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1108

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[b*(p/(d*e*(m + 1)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x 
], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 
3, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0] 
) && IntegerQ[2*p]
 

rule 1114

Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symb 
ol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)]   Subst[Int[x^2/Sqrt[Simp[1 - b^2* 
(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
 

rule 1115

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym 
bol] :> Simp[Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/Sqrt[a + b*x + c* 
x^2]   Int[(d + e*x)^m/Sqrt[(-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) 
- c^2*(x^2/(b^2 - 4*a*c))], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c* 
d - b*e, 0] && EqQ[m^2, 1/4]
 

rule 1117

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m 
+ 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* 
c)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & 
& (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) 
/2])
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(873\) vs. \(2(239)=478\).

Time = 2.09 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.09

method	result	size
default	\(\frac {\left (16 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,c^{3} x^{2}-4 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{2} c^{2} x^{2}+16 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a b \,c^{2} x -4 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{3} c x +4 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,b^{2} c -\sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{4}-16 c^{4} x^{4}-32 b \,c^{3} x^{3}-24 a \,c^{3} x^{2}-18 b^{2} c^{2} x^{2}-24 a b \,c^{2} x -2 b^{3} c x -8 a^{2} c^{2}-2 c a \,b^{2}\right ) \sqrt {d \left (2 c x +b \right )}}{10 d^{4} \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right ) \left (2 c x +b \right )^{3} c^{2}}\)	\(874\)
elliptic	\(\text {Expression too large to display}\)	\(1094\)

Input:

int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/10*(16*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c* 
x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2) 
^(1/2))^(1/2)*EllipticE(1/2*2^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b 
^2)^(1/2)+b))^(1/2),2^(1/2))*a*c^3*x^2-4*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4* 
a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+(- 
4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*2^(1/2)*(1/(-4 
*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2),2^(1/2))*b^2*c^2*x^2+1 
6*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(- 
4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2)) 
^(1/2)*EllipticE(1/2*2^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/ 
2)+b))^(1/2),2^(1/2))*a*b*c^2*x-4*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2 
)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+(-4*a*c+b 
^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*2^(1/2)*(1/(-4*a*c+b^ 
2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2),2^(1/2))*b^3*c*x+4*(1/(-4*a*c 
+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^( 
1/2))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*Ellip 
ticE(1/2*2^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2) 
,2^(1/2))*a*b^2*c-(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2 
)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4 
*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*2^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=\frac {\sqrt {2} {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (8 \, c^{3} x^{2} + 8 \, b c^{2} x + b^{2} c + 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{5 \, {\left (8 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{4} x + {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{4}\right )}} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x, algorithm="fricas")
 

Output:

1/5*(sqrt(2)*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*sqrt(c^2*d)*weie 
rstrassZeta((b^2 - 4*a*c)/c^2, 0, weierstrassPInverse((b^2 - 4*a*c)/c^2, 0 
, 1/2*(2*c*x + b)/c)) + (8*c^3*x^2 + 8*b*c^2*x + b^2*c + 4*a*c^2)*sqrt(2*c 
*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(8*(b^2*c^5 - 4*a*c^6)*d^4*x^3 + 12*(b^ 
3*c^4 - 4*a*b*c^5)*d^4*x^2 + 6*(b^4*c^3 - 4*a*b^2*c^4)*d^4*x + (b^5*c^2 - 
4*a*b^3*c^3)*d^4)
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**(7/2),x)
 

Output:

Integral(sqrt(a + b*x + c*x**2)/(d*(b + 2*c*x))**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (b\,d+2\,c\,d\,x\right )}^{7/2}} \,d x \] Input:

int((a + b*x + c*x^2)^(1/2)/(b*d + 2*c*d*x)^(7/2),x)
 

Output:

int((a + b*x + c*x^2)^(1/2)/(b*d + 2*c*d*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx=\text {too large to display} \] Input:

int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x)
 

Output:

(sqrt(d)*( - 12*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*c + 6*sqrt(b + 
 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2 - 8*sqrt(b + 2*c*x)*sqrt(a + b*x + c 
*x**2)*a*b*c*x + 2*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**3*x - 160*int 
((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**3)/(20*a**3*b**4*c**2 + 160*a* 
*3*b**3*c**3*x + 480*a**3*b**2*c**4*x**2 + 640*a**3*b*c**5*x**3 + 320*a**3 
*c**6*x**4 - 12*a**2*b**6*c - 76*a**2*b**5*c**2*x - 108*a**2*b**4*c**3*x** 
2 + 256*a**2*b**3*c**4*x**3 + 928*a**2*b**2*c**5*x**4 + 960*a**2*b*c**6*x* 
*5 + 320*a**2*c**7*x**6 + a*b**8 - 4*a*b**7*c*x - 84*a*b**6*c**2*x**2 - 35 
2*a*b**5*c**3*x**3 - 656*a*b**4*c**4*x**4 - 576*a*b**3*c**5*x**5 - 192*a*b 
**2*c**6*x**6 + b**9*x + 9*b**8*c*x**2 + 32*b**7*c**2*x**3 + 56*b**6*c**3* 
x**4 + 48*b**5*c**4*x**5 + 16*b**4*c**5*x**6),x)*a**3*b**4*c**5 - 960*int( 
(sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**3)/(20*a**3*b**4*c**2 + 160*a** 
3*b**3*c**3*x + 480*a**3*b**2*c**4*x**2 + 640*a**3*b*c**5*x**3 + 320*a**3* 
c**6*x**4 - 12*a**2*b**6*c - 76*a**2*b**5*c**2*x - 108*a**2*b**4*c**3*x**2 
 + 256*a**2*b**3*c**4*x**3 + 928*a**2*b**2*c**5*x**4 + 960*a**2*b*c**6*x** 
5 + 320*a**2*c**7*x**6 + a*b**8 - 4*a*b**7*c*x - 84*a*b**6*c**2*x**2 - 352 
*a*b**5*c**3*x**3 - 656*a*b**4*c**4*x**4 - 576*a*b**3*c**5*x**5 - 192*a*b* 
*2*c**6*x**6 + b**9*x + 9*b**8*c*x**2 + 32*b**7*c**2*x**3 + 56*b**6*c**3*x 
**4 + 48*b**5*c**4*x**5 + 16*b**4*c**5*x**6),x)*a**3*b**3*c**6*x - 1920*in 
t((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**3)/(20*a**3*b**4*c**2 + 16...