\(\int \frac {(a+b x+c x^2)^{5/2}}{\sqrt {b d+2 c d x}} \, dx\) [1351]

Optimal result

Integrand size = 28, antiderivative size = 229 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{308 c^4 \sqrt {d} \sqrt {a+b x+c x^2}} \]

[Out]

-5/154*(-4*a*c+b^2)*(c*x^2+b*x+a)^(3/2)*(2*c*d*x+b*d)^(1/2)/c^2/d+1/11*(c*x^2+b*x+a)^(5/2)*(2*c*d*x+b*d)^(1/2)
/c/d+5/308*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3/d-5/308*(-4*a*c+b^2)^(13/4)*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)/(-4*a*c+b^2))^(1/2)/c^4/d^(1/2)/(c*x^2+b*x+a
)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {699, 705, 703, 227} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=-\frac {5 \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{308 c^4 \sqrt {d} \sqrt {a+b x+c x^2}}+\frac {5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{154 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} \sqrt {b d+2 c d x}}{11 c d} \]

[In]

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

[Out]

(5*(b^2 - 4*a*c)^2*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(308*c^3*d) - (5*(b^2 - 4*a*c)*Sqrt[b*d + 2*c*d*
x]*(a + b*x + c*x^2)^(3/2))/(154*c^2*d) + (Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(5/2))/(11*c*d) - (5*(b^2 - 4
*a*c)^(13/4)*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])/(308*c^4*Sqrt[d]*Sqrt[a + b*x + c*x^2])

Rule 227

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

Rule 699

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && 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 703

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]

Rule 705

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && EqQ[m^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx}{22 c} \\ & = -\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx}{308 c^2} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{616 c^3} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{616 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{308 c^4 d \sqrt {a+b x+c x^2}} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{308 c^4 \sqrt {d} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.44 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\frac {\left (b^2-4 a c\right )^2 \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(195)=390\).

Time = 2.68 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.40

method	result	size
risch	\(\frac {\left (28 c^{4} x^{4}+56 b \,c^{3} x^{3}+96 x^{2} c^{3} a +18 b^{2} c^{2} x^{2}+96 a b \,c^{2} x -10 b^{3} c x +148 a^{2} c^{2}-50 a \,b^{2} c +5 b^{4}\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{308 c^{3} \sqrt {d \left (2 c x +b \right )}}+\frac {5 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{308 c^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(549\)
default	\(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (112 c^{7} x^{7}+392 b \,c^{6} x^{6}+496 a \,c^{6} x^{5}+464 b^{2} c^{5} x^{5}+320 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-240 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+60 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a \,b^{4} c -5 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, b^{6}+1240 a b \,c^{5} x^{4}+180 b^{3} c^{4} x^{4}+976 a^{2} c^{5} x^{3}+752 a \,b^{2} c^{4} x^{3}-4 b^{4} c^{3} x^{3}+1464 a^{2} b \,c^{4} x^{2}-112 a \,b^{3} c^{3} x^{2}+10 b^{5} c^{2} x^{2}+592 a^{3} c^{4} x +288 a^{2} b^{2} c^{3} x -100 c^{2} a \,b^{4} x +10 b^{6} c x +296 a^{3} c^{3} b -100 a^{2} c^{2} b^{3}+10 a \,b^{5} c \right )}{616 d \,c^{4} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\)	\(798\)
elliptic	\(\text {Expression too large to display}\)	\(1885\)

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/308/c^3*(28*c^4*x^4+56*b*c^3*x^3+96*a*c^3*x^2+18*b^2*c^2*x^2+96*a*b*c^2*x-10*b^3*c*x+148*a^2*c^2-50*a*b^2*c+
5*b^4)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(d*(2*c*x+b))^(1/2)+5/308/c^3*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*
(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*
a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x+1/2/c*b)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(1/
2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)/(2*
c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)*EllipticF(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-
4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^
(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2/c*b))^(1/2))*(d*(2*c*x+b)*(c*x^2+b*x+a))^(1/2)/(d*(2*c*x+b))^(1/2)/
(c*x^2+b*x+a)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=-\frac {5 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (28 \, c^{6} x^{4} + 56 \, b c^{5} x^{3} + 5 \, b^{4} c^{2} - 50 \, a b^{2} c^{3} + 148 \, a^{2} c^{4} + 6 \, {\left (3 \, b^{2} c^{4} + 16 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 48 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{616 \, c^{5} d} \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(1/2),x, algorithm="fricas")

[Out]

-1/616*(5*sqrt(2)*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c^2*d)*weierstrassPInverse((b^2 - 4*a*
c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2*(28*c^6*x^4 + 56*b*c^5*x^3 + 5*b^4*c^2 - 50*a*b^2*c^3 + 148*a^2*c^4 + 6*(3*b
^2*c^4 + 16*a*c^5)*x^2 - 2*(5*b^3*c^3 - 48*a*b*c^4)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(c^5*d)

Sympy [F]

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

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(1/2),x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(1/2), x)