\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{21/2}} \, dx\) [1356]

Optimal result

Integrand size = 28, antiderivative size = 305 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac {5 \sqrt {a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {5 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{17556 c^4 \left (b^2-4 a c\right )^{7/4} d^{21/2} \sqrt {a+b x+c x^2}} \]

[Out]

-1/114*(c*x^2+b*x+a)^(3/2)/c^2/d^3/(2*c*d*x+b*d)^(15/2)-1/19*(c*x^2+b*x+a)^(5/2)/c/d/(2*c*d*x+b*d)^(19/2)-1/83
6*(c*x^2+b*x+a)^(1/2)/c^3/d^5/(2*c*d*x+b*d)^(11/2)+1/2926*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)/d^7/(2*c*d*x+b*
d)^(7/2)+5/8778*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)^2/d^9/(2*c*d*x+b*d)^(3/2)+5/17556*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/(-4*a*c+b^2)^(7/4)/d^(21/2)/(c*
x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {698, 707, 705, 703, 227} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=\frac {5 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{17556 c^4 d^{21/2} \left (b^2-4 a c\right )^{7/4} \sqrt {a+b x+c x^2}}+\frac {5 \sqrt {a+b x+c x^2}}{8778 c^3 d^9 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}} \]

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2),x]

[Out]

-1/836*Sqrt[a + b*x + c*x^2]/(c^3*d^5*(b*d + 2*c*d*x)^(11/2)) + Sqrt[a + b*x + c*x^2]/(2926*c^3*(b^2 - 4*a*c)*
d^7*(b*d + 2*c*d*x)^(7/2)) + (5*Sqrt[a + b*x + c*x^2])/(8778*c^3*(b^2 - 4*a*c)^2*d^9*(b*d + 2*c*d*x)^(3/2)) -
(a + b*x + c*x^2)^(3/2)/(114*c^2*d^3*(b*d + 2*c*d*x)^(15/2)) - (a + b*x + c*x^2)^(5/2)/(19*c*d*(b*d + 2*c*d*x)
^(19/2)) + (5*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])/(17556*c^4*(b^2 - 4*a*c)^(7/4)*d^(21/2)*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 698

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 + 1))), x] - Dist[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] && 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] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && 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]

Rule 707

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> 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] + Dist[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] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{17/2}} \, dx}{38 c d^2} \\ & = -\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx}{76 c^2 d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}} \, dx}{1672 c^3 d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {5 \int \frac {1}{(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{11704 c^3 \left (b^2-4 a c\right ) d^8} \\ & = -\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac {5 \sqrt {a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {5 \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{35112 c^3 \left (b^2-4 a c\right )^2 d^{10}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac {5 \sqrt {a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {\left (5 \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}{35112 c^3 \left (b^2-4 a c\right )^2 d^{10} \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac {5 \sqrt {a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {\left (5 \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 )}{17556 c^4 \left (b^2-4 a c\right )^2 d^{11} \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac {5 \sqrt {a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac {5 \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 )}{17556 c^4 \left (b^2-4 a c\right )^{7/4} d^{21/2} \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 = 5.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.36 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=-\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {19}{4},-\frac {5}{2},-\frac {15}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{608 c^3 d^9 (b+2 c x)^8 (d (b+2 c x))^{3/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2),x]

[Out]

-1/608*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-19/4, -5/2, -15/4, (b + 2*c*x)^2/(b^2 - 4*a*c
)])/(c^3*d^9*(b + 2*c*x)^8*(d*(b + 2*c*x))^(3/2)*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. \(749\) vs. \(2(259)=518\).

Time = 6.34 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.46

method	result	size
elliptic	\(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{311296 c^{13} d^{11} \left (x +\frac {b}{2 c}\right )^{10}}-\frac {\left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{29184 c^{11} d^{11} \left (x +\frac {b}{2 c}\right )^{8}}-\frac {67 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{642048 c^{9} d^{11} \left (x +\frac {b}{2 c}\right )^{6}}-\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{46816 c^{7} \left (4 a c -b^{2}\right ) d^{11} \left (x +\frac {b}{2 c}\right )^{4}}+\frac {5 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{35112 c^{5} \left (4 a c -b^{2}\right )^{2} d^{11} \left (x +\frac {b}{2 c}\right )^{2}}+\frac {5 \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{17556 c^{3} \left (4 a c -b^{2}\right )^{2} d^{10} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(750\)
default	\(\text {Expression too large to display}\)	\(1843\)

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(21/2),x,method=_RETURNVERBOSE)

[Out]

(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)*(-1/311296*(16*a^2*c^2-8*a*b^2*c+b^4
)/c^13/d^11*(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)/(x+1/2/c*b)^10-1/29184/c^11*(4*a*c-b^2)/d^
11*(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)/(x+1/2/c*b)^8-67/642048/c^9/d^11*(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)/(x+1/2/c*b)^6-1/46816/c^7/(4*a*c-b^2)/d^11*(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)/(x+1/2/c*b)^4+5/35112/c^5/(4*a*c-b^2)^2/d^11*(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)/(x+1/2/c*b)^2+5/17556/c^3/(4*a*c-b^2)^2/d^10*(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)))

Fricas [C] (verification not implemented)

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

Time = 0.35 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.66 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=\frac {5 \, \sqrt {2} {\left (1024 \, c^{10} x^{10} + 5120 \, b c^{9} x^{9} + 11520 \, b^{2} c^{8} x^{8} + 15360 \, b^{3} c^{7} x^{7} + 13440 \, b^{4} c^{6} x^{6} + 8064 \, b^{5} c^{5} x^{5} + 3360 \, b^{6} c^{4} x^{4} + 960 \, b^{7} c^{3} x^{3} + 180 \, b^{8} c^{2} x^{2} + 20 \, b^{9} c x + b^{10}\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 (2560 \, c^{10} x^{8} + 10240 \, b c^{9} x^{7} - 5 \, b^{8} c^{2} - 10 \, a b^{6} c^{3} - 28 \, a^{2} b^{4} c^{4} + 4928 \, a^{3} b^{2} c^{5} - 14784 \, a^{4} c^{6} + 128 \, {\left (143 \, b^{2} c^{8} - 12 \, a c^{9}\right )} x^{6} + 128 \, {\left (149 \, b^{3} c^{7} - 36 \, a b c^{8}\right )} x^{5} + 4 \, {\left (2691 \, b^{4} c^{6} + 2312 \, a b^{2} c^{7} - 7504 \, a^{2} c^{8}\right )} x^{4} + 8 \, {\left (211 \, b^{5} c^{5} + 3272 \, a b^{3} c^{6} - 7504 \, a^{2} b c^{7}\right )} x^{3} - 2 \, {\left (359 \, b^{6} c^{4} - 6840 \, a b^{4} c^{5} + 7728 \, a^{2} b^{2} c^{6} + 19712 \, a^{3} c^{7}\right )} x^{2} - 2 \, {\left (45 \, b^{7} c^{3} + 88 \, a b^{5} c^{4} - 7280 \, a^{2} b^{3} c^{5} + 19712 \, a^{3} b c^{6}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{35112 \, {\left (1024 \, {\left (b^{4} c^{15} - 8 \, a b^{2} c^{16} + 16 \, a^{2} c^{17}\right )} d^{11} x^{10} + 5120 \, {\left (b^{5} c^{14} - 8 \, a b^{3} c^{15} + 16 \, a^{2} b c^{16}\right )} d^{11} x^{9} + 11520 \, {\left (b^{6} c^{13} - 8 \, a b^{4} c^{14} + 16 \, a^{2} b^{2} c^{15}\right )} d^{11} x^{8} + 15360 \, {\left (b^{7} c^{12} - 8 \, a b^{5} c^{13} + 16 \, a^{2} b^{3} c^{14}\right )} d^{11} x^{7} + 13440 \, {\left (b^{8} c^{11} - 8 \, a b^{6} c^{12} + 16 \, a^{2} b^{4} c^{13}\right )} d^{11} x^{6} + 8064 \, {\left (b^{9} c^{10} - 8 \, a b^{7} c^{11} + 16 \, a^{2} b^{5} c^{12}\right )} d^{11} x^{5} + 3360 \, {\left (b^{10} c^{9} - 8 \, a b^{8} c^{10} + 16 \, a^{2} b^{6} c^{11}\right )} d^{11} x^{4} + 960 \, {\left (b^{11} c^{8} - 8 \, a b^{9} c^{9} + 16 \, a^{2} b^{7} c^{10}\right )} d^{11} x^{3} + 180 \, {\left (b^{12} c^{7} - 8 \, a b^{10} c^{8} + 16 \, a^{2} b^{8} c^{9}\right )} d^{11} x^{2} + 20 \, {\left (b^{13} c^{6} - 8 \, a b^{11} c^{7} + 16 \, a^{2} b^{9} c^{8}\right )} d^{11} x + {\left (b^{14} c^{5} - 8 \, a b^{12} c^{6} + 16 \, a^{2} b^{10} c^{7}\right )} d^{11}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(21/2),x, algorithm="fricas")

[Out]

1/35112*(5*sqrt(2)*(1024*c^10*x^10 + 5120*b*c^9*x^9 + 11520*b^2*c^8*x^8 + 15360*b^3*c^7*x^7 + 13440*b^4*c^6*x^
6 + 8064*b^5*c^5*x^5 + 3360*b^6*c^4*x^4 + 960*b^7*c^3*x^3 + 180*b^8*c^2*x^2 + 20*b^9*c*x + b^10)*sqrt(c^2*d)*w
eierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(2560*c^10*x^8 + 10240*b*c^9*x^7 - 5*b^8*c^2 -
10*a*b^6*c^3 - 28*a^2*b^4*c^4 + 4928*a^3*b^2*c^5 - 14784*a^4*c^6 + 128*(143*b^2*c^8 - 12*a*c^9)*x^6 + 128*(14
9*b^3*c^7 - 36*a*b*c^8)*x^5 + 4*(2691*b^4*c^6 + 2312*a*b^2*c^7 - 7504*a^2*c^8)*x^4 + 8*(211*b^5*c^5 + 3272*a*b
^3*c^6 - 7504*a^2*b*c^7)*x^3 - 2*(359*b^6*c^4 - 6840*a*b^4*c^5 + 7728*a^2*b^2*c^6 + 19712*a^3*c^7)*x^2 - 2*(45
*b^7*c^3 + 88*a*b^5*c^4 - 7280*a^2*b^3*c^5 + 19712*a^3*b*c^6)*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(1
024*(b^4*c^15 - 8*a*b^2*c^16 + 16*a^2*c^17)*d^11*x^10 + 5120*(b^5*c^14 - 8*a*b^3*c^15 + 16*a^2*b*c^16)*d^11*x^
9 + 11520*(b^6*c^13 - 8*a*b^4*c^14 + 16*a^2*b^2*c^15)*d^11*x^8 + 15360*(b^7*c^12 - 8*a*b^5*c^13 + 16*a^2*b^3*c
^14)*d^11*x^7 + 13440*(b^8*c^11 - 8*a*b^6*c^12 + 16*a^2*b^4*c^13)*d^11*x^6 + 8064*(b^9*c^10 - 8*a*b^7*c^11 + 1
6*a^2*b^5*c^12)*d^11*x^5 + 3360*(b^10*c^9 - 8*a*b^8*c^10 + 16*a^2*b^6*c^11)*d^11*x^4 + 960*(b^11*c^8 - 8*a*b^9
*c^9 + 16*a^2*b^7*c^10)*d^11*x^3 + 180*(b^12*c^7 - 8*a*b^10*c^8 + 16*a^2*b^8*c^9)*d^11*x^2 + 20*(b^13*c^6 - 8*
a*b^11*c^7 + 16*a^2*b^9*c^8)*d^11*x + (b^14*c^5 - 8*a*b^12*c^6 + 16*a^2*b^10*c^7)*d^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(21/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {21}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(21/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(21/2), x)

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {21}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(21/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(21/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{21/2}} \,d x \]

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2),x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2), x)