\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{3/2}} \, dx\) [255]

Optimal result

Integrand size = 28, antiderivative size = 316 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{12 c^3 d^3}+\frac {5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt {b d+2 c d x}}+\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{12 c^4 d^{3/2} \sqrt {a+b x+c x^2}} \] Output:

-1/12*(-4*a*c+b^2)*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^3/d^3+5/18*(2 
*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(3/2)/c^2/d^3-(c*x^2+b*x+a)^(5/2)/c/d/(2*c 
*d*x+b*d)^(1/2)+1/12*(-4*a*c+b^2)^(11/4)*(-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^4/d^(3/ 
2)/(c*x^2+b*x+a)^(1/2)-1/12*(-4*a*c+b^2)^(11/4)*(-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^ 
4/d^(3/2)/(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 = 10.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=-\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {1}{4},\frac {3}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d \sqrt {d (b+2 c x)} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2),x]
 

Output:

-1/32*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -1/4, 
 3/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d*Sqrt[d*(b + 2*c*x)]*Sqrt[(c*(a 
+ x*(b + c*x)))/(-b^2 + 4*a*c)])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1108, 1109, 1109, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 1108

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

\(\Big \downarrow \) 1109

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

\(\Big \downarrow \) 1109

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

\(\Big \downarrow \) 1115

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

\(\Big \downarrow \) 1114

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2),x]
 

Output:

-((a + b*x + c*x^2)^(5/2)/(c*d*Sqrt[b*d + 2*c*d*x])) + (5*(((b*d + 2*c*d*x 
)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c*d) - ((b^2 - 4*a*c)*(((b*d + 2*c*d*x 
)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c*d) - ((b^2 - 4*a*c)*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[S 
qrt[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^2*d*Sqrt[a + b*x + c*x^2])))/(6*c)))/(2*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 1109

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 + 2*p + 1))), x 
] - Simp[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1)))   Int[(d + e*x)^m*(a + b*x 
 + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b* 
e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1 
)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && 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 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. \(699\) vs. \(2(268)=536\).

Time = 5.62 (sec) , antiderivative size = 700, normalized size of antiderivative = 2.22

method	result	size
default	\(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (8 x^{6} c^{6}+24 x^{5} b \,c^{5}+192 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{3} c^{3}-144 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{2} b^{2} c^{2}+36 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,b^{4} c -3 \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{6}+40 a \,c^{5} x^{4}+20 x^{4} b^{2} c^{4}+80 a b \,c^{4} x^{3}-40 a^{2} c^{4} x^{2}+80 a \,b^{2} c^{3} x^{2}-10 c^{2} x^{2} b^{4}-40 a^{2} b \,c^{3} x +40 x a \,b^{3} c^{2}-6 x c \,b^{5}-72 a^{3} c^{3}+44 a^{2} b^{2} c^{2}-6 a \,b^{4} c \right )}{72 d^{2} c^{4} \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right )}\)	\(700\)
elliptic	\(\text {Expression too large to display}\)	\(1774\)
risch	\(\text {Expression too large to display}\)	\(3848\)

Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/72*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(8*x^6*c^6+24*x^5*b*c^5+192*( 
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^3-144*(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+36*(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)*Ell 
ipticE(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-3*(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+40*a*c^5*x^4+20*x^4*b^2*c^4+ 
80*a*b*c^4*x^3-40*a^2*c^4*x^2+80*a*b^2*c^3*x^2-10*c^2*x^2*b^4-40*a^2*b*c^3 
*x+40*x*a*b^3*c^2-6*x*c*b^5-72*a^3*c^3+44*a^2*b^2*c^2-6*a*b^4*c)/d^2/c^4/( 
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 = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (4 \, c^{5} x^{4} + 8 \, b c^{4} x^{3} - 3 \, b^{4} c + 22 \, a b^{2} c^{2} - 36 \, a^{2} c^{3} + 2 \, {\left (b^{2} c^{3} + 8 \, a c^{4}\right )} x^{2} - 2 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{36 \, {\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(3/2),x, algorithm="fricas")
 

Output:

-1/36*(3*sqrt(2)*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2 + 2*(b^4*c - 8*a*b^2*c^2 
+ 16*a^2*c^3)*x)*sqrt(c^2*d)*weierstrassZeta((b^2 - 4*a*c)/c^2, 0, weierst 
rassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) - (4*c^5*x^4 + 8*b* 
c^4*x^3 - 3*b^4*c + 22*a*b^2*c^2 - 36*a^2*c^3 + 2*(b^2*c^3 + 8*a*c^4)*x^2 
- 2*(b^3*c^2 - 8*a*b*c^3)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(2 
*c^5*d^2*x + b*c^4*d^2)
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(3/2),x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d*(b + 2*c*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(3/2),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(3/2),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2),x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx=\text {too large to display} \] Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(3/2),x)
 

Output:

(sqrt(d)*( - 72*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**3*c**2 + 32*sqrt 
(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b**2*c + 32*sqrt(b + 2*c*x)*sqrt(a 
 + b*x + c*x**2)*a**2*b*c**2*x + 32*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2) 
*a**2*c**3*x**2 - 4*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**4 - 20*sqr 
t(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**3*c*x - 12*sqrt(b + 2*c*x)*sqrt(a 
 + b*x + c*x**2)*a*b**2*c**2*x**2 + 16*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x* 
*2)*a*b*c**3*x**3 + 8*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*c**4*x**4 + 
 2*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**5*x - 2*sqrt(b + 2*c*x)*sqrt( 
a + b*x + c*x**2)*b**4*c*x**2 - 8*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b 
**3*c**2*x**3 - 4*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**2*c**3*x**4 + 
384*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b**2*c + 8*a 
**2*b*c**2*x + 8*a**2*c**3*x**2 - a*b**4 - 2*a*b**3*c*x + 6*a*b**2*c**2*x* 
*2 + 16*a*b*c**3*x**3 + 8*a*c**4*x**4 - b**5*x - 5*b**4*c*x**2 - 8*b**3*c* 
*2*x**3 - 4*b**2*c**3*x**4),x)*a**4*b*c**5 + 768*int((sqrt(b + 2*c*x)*sqrt 
(a + b*x + c*x**2)*x**2)/(2*a**2*b**2*c + 8*a**2*b*c**2*x + 8*a**2*c**3*x* 
*2 - a*b**4 - 2*a*b**3*c*x + 6*a*b**2*c**2*x**2 + 16*a*b*c**3*x**3 + 8*a*c 
**4*x**4 - b**5*x - 5*b**4*c*x**2 - 8*b**3*c**2*x**3 - 4*b**2*c**3*x**4),x 
)*a**4*c**6*x - 480*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a 
**2*b**2*c + 8*a**2*b*c**2*x + 8*a**2*c**3*x**2 - a*b**4 - 2*a*b**3*c*x + 
6*a*b**2*c**2*x**2 + 16*a*b*c**3*x**3 + 8*a*c**4*x**4 - b**5*x - 5*b**4...