\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{5/2}} \, dx\) [248]

Optimal result

Integrand size = 28, antiderivative size = 219 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{5/2}} \, dx=-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^3}+\frac {5 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{84 c^4 d^{5/2} \sqrt {a+b x+c x^2}} \] Output:

-5/84*(-4*a*c+b^2)*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3/d^3+5/42*(2 
*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(3/2)/c^2/d^3-1/3*(c*x^2+b*x+a)^(5/2)/c/d/ 
(2*c*d*x+b*d)^(3/2)+5/84*(-4*a*c+b^2)^(9/4)*(-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^ 
(5/2)/(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 = 101, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{5/2}} \, dx=-\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {3}{4},\frac {1}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{96 c^3 d (d (b+2 c x))^{3/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:

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

Output:

-1/96*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -3/4, 
 1/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d*(d*(b + 2*c*x))^(3/2)*Sqrt[(c*( 
a + x*(b + c*x)))/(-b^2 + 4*a*c)])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1108, 1109, 1109, 1115, 1113, 762}

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.

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

\(\Big \downarrow \) 1108

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

\(\Big \downarrow \) 1109

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b d+2 c x d}}dx}{14 c}\right )}{6 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx}{6 c}\right )}{14 c}\right )}{6 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}\)

\(\Big \downarrow \) 1115

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{6 c \sqrt {a+b x+c x^2}}\right )}{14 c}\right )}{6 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}\)

\(\Big \downarrow \) 1113

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c^2 d \sqrt {a+b x+c x^2}}\right )}{14 c}\right )}{6 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{7 c d}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}}\right )}{14 c}\right )}{6 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}\)

Input:

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

Output:

-1/3*(a + b*x + c*x^2)^(5/2)/(c*d*(b*d + 2*c*d*x)^(3/2)) + (5*((Sqrt[b*d + 
 2*c*d*x]*(a + b*x + c*x^2)^(3/2))/(7*c*d) - (3*(b^2 - 4*a*c)*((Sqrt[b*d + 
 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(3*c*d) - ((b^2 - 4*a*c)^(5/4)*Sqrt[-((c* 
(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/(( 
b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(3*c^2*Sqrt[d]*Sqrt[a + b*x + c*x^2]))) 
/(14*c)))/(6*c*d^2)
 

Defintions of rubi rules used

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 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 1113

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ 
Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)]   Subst[Int[1/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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(999\) vs. \(2(185)=370\).

Time = 5.32 (sec) , antiderivative size = 1000, normalized size of antiderivative = 4.57

method	result	size
default	\(\frac {\left (24 x^{6} c^{6}+160 \sqrt {-4 a c +b^{2}}\, \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 {EllipticF}\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} c^{3} x -80 \sqrt {-4 a c +b^{2}}\, \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 {EllipticF}\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^{2} x +10 \sqrt {-4 a c +b^{2}}\, \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 {EllipticF}\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} c x +72 x^{5} b \,c^{5}+80 \sqrt {-4 a c +b^{2}}\, \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 {EllipticF}\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 \,c^{2}-40 \sqrt {-4 a c +b^{2}}\, \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 {EllipticF}\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^{3} c +5 \sqrt {-4 a c +b^{2}}\, \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 {EllipticF}\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^{5}+152 a \,c^{5} x^{4}+52 x^{4} b^{2} c^{4}+304 a b \,c^{4} x^{3}-16 b^{3} c^{3} x^{3}+72 a^{2} c^{4} x^{2}+192 a \,b^{2} c^{3} x^{2}-30 c^{2} x^{2} b^{4}+72 a^{2} b \,c^{3} x +40 x a \,b^{3} c^{2}-10 x c \,b^{5}-56 a^{3} c^{3}+60 a^{2} b^{2} c^{2}-10 a \,b^{4} c \right ) \sqrt {d \left (2 c x +b \right )}}{168 d^{3} \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right )^{2} c^{4}}\)	\(1000\)
elliptic	\(\text {Expression too large to display}\)	\(1430\)
risch	\(\text {Expression too large to display}\)	\(1694\)

Input:

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

Output:

1/168*(24*x^6*c^6+160*(-4*a*c+b^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*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)*EllipticF(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*c^3*x-80* 
(-4*a*c+b^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*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)*EllipticF(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^2*x+10*(-4*a*c+b^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*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)*EllipticF(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^4*c*x+72*x^5*b*c^5+80*(-4*a*c+b^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*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)*Ellipti 
cF(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*c^2-40*(-4*a*c+b^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*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)*EllipticF(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^3*c+5*(-4* 
a*c+b^2)^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{5/2}} \, dx=\frac {5 \, \sqrt {2} {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 4 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (12 \, c^{6} x^{4} + 24 \, b c^{5} x^{3} - 5 \, b^{4} c^{2} + 30 \, a b^{2} c^{3} - 28 \, a^{2} c^{4} + 2 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 32 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{168 \, {\left (4 \, c^{7} d^{3} x^{2} + 4 \, b c^{6} d^{3} x + b^{2} c^{5} d^{3}\right )}} \] Input:

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

Output:

1/168*(5*sqrt(2)*(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2 + 4*(b^4*c^2 - 8*a*b^2* 
c^3 + 16*a^2*c^4)*x^2 + 4*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)*sqrt(c^2 
*d)*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(12*c 
^6*x^4 + 24*b*c^5*x^3 - 5*b^4*c^2 + 30*a*b^2*c^3 - 28*a^2*c^4 + 2*(b^2*c^4 
 + 32*a*c^5)*x^2 - 2*(5*b^3*c^3 - 32*a*b*c^4)*x)*sqrt(2*c*d*x + b*d)*sqrt( 
c*x^2 + b*x + a))/(4*c^7*d^3*x^2 + 4*b*c^6*d^3*x + b^2*c^5*d^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*( - 168*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**3*c**2 + 128*sq 
rt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b**2*c + 384*sqrt(b + 2*c*x)*sqr 
t(a + b*x + c*x**2)*a**2*b*c**2*x + 384*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x 
**2)*a**2*c**3*x**2 - 20*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**4 - 1 
24*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**3*c*x - 52*sqrt(b + 2*c*x)* 
sqrt(a + b*x + c*x**2)*a*b**2*c**2*x**2 + 144*sqrt(b + 2*c*x)*sqrt(a + b*x 
 + c*x**2)*a*b*c**3*x**3 + 72*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*c** 
4*x**4 + 10*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 - 24*sqrt(b + 2*c*x)*sqrt(a + b*x + 
 c*x**2)*b**3*c**2*x**3 - 12*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**2*c 
**3*x**4 + 5760*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(6*a**2* 
b**3*c + 36*a**2*b**2*c**2*x + 72*a**2*b*c**3*x**2 + 48*a**2*c**4*x**3 - a 
*b**5 + 30*a*b**3*c**2*x**2 + 100*a*b**2*c**3*x**3 + 120*a*b*c**4*x**4 + 4 
8*a*c**5*x**5 - b**6*x - 7*b**5*c*x**2 - 18*b**4*c**2*x**3 - 20*b**3*c**3* 
x**4 - 8*b**2*c**4*x**5),x)*a**4*b**2*c**5 + 23040*int((sqrt(b + 2*c*x)*sq 
rt(a + b*x + c*x**2)*x**2)/(6*a**2*b**3*c + 36*a**2*b**2*c**2*x + 72*a**2* 
b*c**3*x**2 + 48*a**2*c**4*x**3 - a*b**5 + 30*a*b**3*c**2*x**2 + 100*a*b** 
2*c**3*x**3 + 120*a*b*c**4*x**4 + 48*a*c**5*x**5 - b**6*x - 7*b**5*c*x**2 
- 18*b**4*c**2*x**3 - 20*b**3*c**3*x**4 - 8*b**2*c**4*x**5),x)*a**4*b*c**6 
*x + 23040*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(6*a**2*b*...