\(\int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx\) [264]

Optimal result

Integrand size = 28, antiderivative size = 273 \[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {28}{45} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {4}{9} d (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}+\frac {14 \left (b^2-4 a c\right )^{11/4} d^{9/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{15 c \sqrt {a+b x+c x^2}}-\frac {14 \left (b^2-4 a c\right )^{11/4} d^{9/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{15 c \sqrt {a+b x+c x^2}} \] Output:

28/45*(-4*a*c+b^2)*d^3*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2)+4/9*d*(2*c* 
d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2)+14/15*(-4*a*c+b^2)^(11/4)*d^(9/2)*(-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/(c*x^2+b*x+a)^(1/2)-14/15*(-4*a*c+b^2)^(11/4)*d^(9/2) 
*(-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/(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 = 10.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.61 \[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 d^3 (d (b+2 c x))^{3/2} \left (8 c \left (-7 a^2 c+a \left (3 b^2-2 b c x-2 c^2 x^2\right )+x \left (3 b^3+8 b^2 c x+10 b c^2 x^2+5 c^3 x^3\right )\right )+7 \left (b^2-4 a c\right )^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{45 c \sqrt {a+x (b+c x)}} \] Input:

Integrate[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]
 

Output:

(2*d^3*(d*(b + 2*c*x))^(3/2)*(8*c*(-7*a^2*c + a*(3*b^2 - 2*b*c*x - 2*c^2*x 
^2) + x*(3*b^3 + 8*b^2*c*x + 10*b*c^2*x^2 + 5*c^3*x^3)) + 7*(b^2 - 4*a*c)^ 
2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/2, 3/4, 7 
/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(45*c*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1116, 1116, 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.

\(\displaystyle \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1115

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

\(\Big \downarrow \) 1114

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {7}{9} d^2 \left (b^2-4 a c\right ) \left (\frac {6 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\sqrt {b^2-4 a c} \int \frac {d+\frac {b d+2 c x d}{\sqrt {b^2-4 a c}}}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{5 c \sqrt {a+b x+c x^2}}+\frac {4}{5} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}\right )+\frac {4}{9} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {7}{9} d^2 \left (b^2-4 a c\right ) \left (\frac {6 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d \sqrt {b^2-4 a c} \int \frac {\sqrt {\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}+1}}{\sqrt {1-\frac {b d+2 c x d}{\sqrt {b^2-4 a c} d}}}d\sqrt {b d+2 c x d}-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{5 c \sqrt {a+b x+c x^2}}+\frac {4}{5} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}\right )+\frac {4}{9} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {7}{9} d^2 \left (b^2-4 a c\right ) \left (\frac {6 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (d^{3/2} \left (b^2-4 a c\right )^{3/4} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )-d^{3/2} \left (b^2-4 a c\right )^{3/4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )\right )}{5 c \sqrt {a+b x+c x^2}}+\frac {4}{5} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}\right )+\frac {4}{9} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}\)

Input:

Int[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]
 

Output:

(4*d*(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/9 + (7*(b^2 - 4*a*c)*d^2 
*((4*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/5 + (6*(b^2 - 4*a*c)*d 
*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]))/(5*c*Sqrt[a + b*x + c*x^2])))/9
 

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 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 1116

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

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. \(702\) vs. \(2(229)=458\).

Time = 4.03 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.58

method	result	size
default	\(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{4} \left (320 x^{6} c^{6}+960 x^{5} b \,c^{5}+1344 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{3} c^{3}-1008 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{2} b^{2} c^{2}+252 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,b^{4} c -21 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{6}-128 a \,c^{5} x^{4}+1232 x^{4} b^{2} c^{4}-256 a b \,c^{4} x^{3}+864 b^{3} c^{3} x^{3}-448 a^{2} c^{4} x^{2}+32 a \,b^{2} c^{3} x^{2}+320 c^{2} x^{2} b^{4}-448 a^{2} b \,c^{3} x +160 x a \,b^{3} c^{2}+48 x c \,b^{5}-112 a^{2} b^{2} c^{2}+48 a \,b^{4} c \right )}{45 c \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right )}\)	\(703\)
elliptic	\(\text {Expression too large to display}\)	\(1549\)
risch	\(\text {Expression too large to display}\)	\(1775\)

Input:

int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/45*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^4*(320*x^6*c^6+960*x^5*b*c^ 
5+1344*(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^3*c^3-1008*(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^2*b^2*c^2+25 
2*(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^4*c-21*(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^6-128*a*c^5*x^4+123 
2*x^4*b^2*c^4-256*a*b*c^4*x^3+864*b^3*c^3*x^3-448*a^2*c^4*x^2+32*a*b^2*c^3 
*x^2+320*c^2*x^2*b^4-448*a^2*b*c^3*x+160*x*a*b^3*c^2+48*x*c*b^5-112*a^2*b^ 
2*c^2+48*a*b^4*c)/c/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.61 \[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left (21 \, \sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} d^{4} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - 8 \, {\left (10 \, c^{4} d^{4} x^{3} + 15 \, b c^{3} d^{4} x^{2} + {\left (11 \, b^{2} c^{2} - 14 \, a c^{3}\right )} d^{4} x + {\left (3 \, b^{3} c - 7 \, a b c^{2}\right )} d^{4}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{45 \, c} \] Input:

integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

-2/45*(21*sqrt(2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c^2*d)*d^4*weierstra 
ssZeta((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*(10*c^4*d^4*x^3 + 15*b*c^3*d^4*x^2 + (11*b^2*c^2 - 14 
*a*c^3)*d^4*x + (3*b^3*c - 7*a*b*c^2)*d^4)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 
+ b*x + a))/c
 

Sympy [F]

\[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d \left (b + 2 c x\right )\right )^{\frac {9}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral((d*(b + 2*c*x))**(9/2)/sqrt(a + b*x + c*x**2), x)
 

Maxima [F]

\[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:

integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a), x)
 

Giac [F]

\[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:

integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(1/2),x)
 

Output:

int((b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(b d+2 c d x)^{9/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {d}\, d^{4} \left (112 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2}-112 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c -112 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} x +31 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b^{4}+88 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b^{3} c x +120 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} x^{2}+80 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} x^{3}-336 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) a^{2} c^{4}+168 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) a \,b^{2} c^{3}-21 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) b^{4} c^{2}-112 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) a^{3} c^{3}+168 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) a^{2} b^{2} c^{2}-63 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) a \,b^{4} c +7 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}}{2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b}d x \right ) b^{6}\right )}{45 b} \] Input:

int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

(2*sqrt(d)*d**4*(112*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*c**2 - 11 
2*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2*c - 112*sqrt(b + 2*c*x)*sq 
rt(a + b*x + c*x**2)*a*b*c**2*x + 31*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2 
)*b**4 + 88*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**3*c*x + 120*sqrt(b + 
 2*c*x)*sqrt(a + b*x + c*x**2)*b**2*c**2*x**2 + 80*sqrt(b + 2*c*x)*sqrt(a 
+ b*x + c*x**2)*b*c**3*x**3 - 336*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x* 
*2)*x**2)/(a*b + 2*a*c*x + b**2*x + 3*b*c*x**2 + 2*c**2*x**3),x)*a**2*c**4 
 + 168*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(a*b + 2*a*c*x + 
b**2*x + 3*b*c*x**2 + 2*c**2*x**3),x)*a*b**2*c**3 - 21*int((sqrt(b + 2*c*x 
)*sqrt(a + b*x + c*x**2)*x**2)/(a*b + 2*a*c*x + b**2*x + 3*b*c*x**2 + 2*c* 
*2*x**3),x)*b**4*c**2 - 112*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2))/( 
a*b + 2*a*c*x + b**2*x + 3*b*c*x**2 + 2*c**2*x**3),x)*a**3*c**3 + 168*int( 
(sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2))/(a*b + 2*a*c*x + b**2*x + 3*b*c*x 
**2 + 2*c**2*x**3),x)*a**2*b**2*c**2 - 63*int((sqrt(b + 2*c*x)*sqrt(a + b* 
x + c*x**2))/(a*b + 2*a*c*x + b**2*x + 3*b*c*x**2 + 2*c**2*x**3),x)*a*b**4 
*c + 7*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2))/(a*b + 2*a*c*x + b**2* 
x + 3*b*c*x**2 + 2*c**2*x**3),x)*b**6))/(45*b)