\(\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx\) [1331]

Optimal result

Integrand size = 28, antiderivative size = 184 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {2 \sqrt {a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{21 c^2 \left (b^2-4 a c\right )^{3/4} d^{9/2} \sqrt {a+b x+c x^2}} \]

[Out]

-1/7*(c*x^2+b*x+a)^(1/2)/c/d/(2*c*d*x+b*d)^(7/2)+2/21*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)/d^3/(2*c*d*x+b*d)^(3/
2)+1/21*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^2/
(-4*a*c+b^2)^(3/4)/d^(9/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {698, 707, 705, 703, 227} \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx=\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{21 c^2 d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {a+b x+c x^2}}{21 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}} \]

[In]

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

[Out]

-1/7*Sqrt[a + b*x + c*x^2]/(c*d*(b*d + 2*c*d*x)^(7/2)) + (2*Sqrt[a + b*x + c*x^2])/(21*c*(b^2 - 4*a*c)*d^3*(b*
d + 2*c*d*x)^(3/2)) + (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])/(21*c^2*(b^2 - 4*a*c)^(3/4)*d^(9/2)*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 {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{14 c d^2} \\ & = -\frac {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {2 \sqrt {a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac {\int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{42 c \left (b^2-4 a c\right ) d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {2 \sqrt {a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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}{42 c \left (b^2-4 a c\right ) d^4 \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {2 \sqrt {a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{21 c^2 \left (b^2-4 a c\right ) d^5 \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {2 \sqrt {a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac {\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 )}{21 c^2 \left (b^2-4 a c\right )^{3/4} d^{9/2} \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 = 10.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx=-\frac {\sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{14 c d^5 (b+2 c x)^4 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]

[In]

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

[Out]

-1/14*(Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-7/4, -1/2, -3/4, (b + 2*c*x)^2/(b^2 - 4*a*
c)])/(c*d^5*(b + 2*c*x)^4*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. \(546\) vs. \(2(156)=312\).

Time = 2.37 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.97

method	result	size
elliptic	\(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{112 d^{5} c^{5} \left (x +\frac {b}{2 c}\right )^{4}}-\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{42 c^{3} \left (4 a c -b^{2}\right ) d^{5} \left (x +\frac {b}{2 c}\right )^{2}}-\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{21 c \left (4 a c -b^{2}\right ) d^{4} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(547\)
default	\(-\frac {\left (8 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) c^{3} x^{3}+12 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b \,c^{2} x^{2}+6 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2} c x +\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{3}+16 c^{4} x^{4}+32 b \,c^{3} x^{3}+40 x^{2} c^{3} a +14 b^{2} c^{2} x^{2}+40 a b \,c^{2} x -2 b^{3} c x +24 a^{2} c^{2}-2 a \,b^{2} c \right ) \sqrt {d \left (2 c x +b \right )}}{42 d^{5} \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right ) \left (2 c x +b \right )^{4} c^{2}}\)	\(659\)

[In]

int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(9/2),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)*(-1/112/d^5/c^5*(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)^4-1/42/c^3/(4*a*c-b^2)/d^5*(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)^2-1/21/c/(4*a*c-b^2)/d^4*(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 = 0.15 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx=\frac {\sqrt {2} {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (8 \, c^{4} x^{2} + 8 \, b c^{3} x - b^{2} c^{2} + 12 \, a c^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{42 \, {\left (16 \, {\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{5} x^{4} + 32 \, {\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{5} x^{3} + 24 \, {\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{5} x^{2} + 8 \, {\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{5} x + {\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{5}\right )}} \]

[In]

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

[Out]

1/42*(sqrt(2)*(16*c^4*x^4 + 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4)*sqrt(c^2*d)*weierstrassPInverse((
b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(8*c^4*x^2 + 8*b*c^3*x - b^2*c^2 + 12*a*c^3)*sqrt(2*c*d*x + b*d)*s
qrt(c*x^2 + b*x + a))/(16*(b^2*c^7 - 4*a*c^8)*d^5*x^4 + 32*(b^3*c^6 - 4*a*b*c^7)*d^5*x^3 + 24*(b^4*c^5 - 4*a*b
^2*c^6)*d^5*x^2 + 8*(b^5*c^4 - 4*a*b^3*c^5)*d^5*x + (b^6*c^3 - 4*a*b^4*c^4)*d^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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