\(\int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [1407]

Optimal result

Integrand size = 37, antiderivative size = 157 \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {14 e^{9/2} E\left (\left .\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^{9/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{15 d} \]

[Out]

14/15*e^(9/2)*EllipticE((d*e*x+c*e)^(1/2)/e^(1/2),I)/d-14/15*e^(9/2)*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d-
14/45*e^3*(d*e*x+c*e)^(3/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d-2/9*e*(d*e*x+c*e)^(7/2)*(-d^2*x^2-2*c*d*x-c^2+1)^
(1/2)/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {706, 704, 313, 227, 1213, 435} \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {14 e^{9/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{15 d}+\frac {14 e^{9/2} E\left (\left .\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^3 \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{45 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{7/2}}{9 d} \]

[In]

Int[(c*e + d*e*x)^(9/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-14*e^3*(c*e + d*e*x)^(3/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(45*d) - (2*e*(c*e + d*e*x)^(7/2)*Sqrt[1 - c^2
- 2*c*d*x - d^2*x^2])/(9*d) + (14*e^(9/2)*EllipticE[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(15*d) - (14*e^(9
/2)*EllipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(15*d)

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 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 704

Int[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[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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]

Rule 706

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

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {1}{9} \left (7 e^2\right ) \int \frac {(c e+d e x)^{5/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {1}{15} \left (7 e^4\right ) \int \frac {\sqrt {c e+d e x}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {\left (14 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{15 d} \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}-\frac {\left (14 e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{15 d}+\frac {\left (14 e^4\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{e}}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{15 d} \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}-\frac {14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}+\frac {\left (14 e^4\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{e}}}{\sqrt {1-\frac {x^2}{e}}} \, dx,x,\sqrt {c e+d e x}\right )}{15 d} \\ & = -\frac {14 e^3 (c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{45 d}-\frac {2 e (c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{9 d}+\frac {14 e^{9/2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d}-\frac {14 e^{9/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{15 d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.55 \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e^3 (e (c+d x))^{3/2} \left (\sqrt {1-c^2-2 c d x-d^2 x^2} \left (7+5 c^2+10 c d x+5 d^2 x^2\right )-7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},(c+d x)^2\right )\right )}{45 d} \]

[In]

Integrate[(c*e + d*e*x)^(9/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-2*e^3*(e*(c + d*x))^(3/2)*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(7 + 5*c^2 + 10*c*d*x + 5*d^2*x^2) - 7*Hypergeo
metric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(45*d)

Maple [A] (verified)

Time = 2.63 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e^{4} \left (-10 d^{6} x^{6}-60 c \,d^{5} x^{5}-150 c^{2} d^{4} x^{4}-200 c^{3} d^{3} x^{3}-150 c^{4} d^{2} x^{2}-4 d^{4} x^{4}-60 c^{5} d x -16 c \,d^{3} x^{3}-10 c^{6}-24 c^{2} d^{2} x^{2}-16 c^{3} d x +21 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, E\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-4 c^{4}+14 d^{2} x^{2}+28 c d x +14 c^{2}\right )}{45 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\)	\(245\)
risch	\(\frac {2 \left (5 d^{3} x^{3}+15 c \,d^{2} x^{2}+15 c^{2} d x +5 c^{3}+7 d x +7 c \right ) \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{5}}{45 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {\left (\frac {14 c \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right )}{15 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}+\frac {14 d \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{15 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{5}}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(737\)
elliptic	\(\text {Expression too large to display}\)	\(1255\)

[In]

int((d*e*x+c*e)^(9/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/45*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^4*(-10*d^6*x^6-60*c*d^5*x^5-150*c^2*d^4*x^4-200*c^3*d^
3*x^3-150*c^4*d^2*x^2-4*d^4*x^4-60*c^5*d*x-16*c*d^3*x^3-10*c^6-24*c^2*d^2*x^2-16*c^3*d*x+21*(-2*d*x-2*c+2)^(1/
2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))-4*c^4+14*d^2*x^2+28*c*d*x+14*
c^2)/d/(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3-d*x-c)

Fricas [C] (verification not implemented)

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

Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82 \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {2 \, {\left (21 \, \sqrt {-d^{3} e} e^{4} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - {\left (5 \, d^{4} e^{4} x^{3} + 15 \, c d^{3} e^{4} x^{2} + {\left (15 \, c^{2} + 7\right )} d^{2} e^{4} x + {\left (5 \, c^{3} + 7 \, c\right )} d e^{4}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{45 \, d^{2}} \]

[In]

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

[Out]

2/45*(21*sqrt(-d^3*e)*e^4*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d*x + c)/d)) - (5*d^4*e^4*x
^3 + 15*c*d^3*e^4*x^2 + (15*c^2 + 7)*d^2*e^4*x + (5*c^3 + 7*c)*d*e^4)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(
d*e*x + c*e))/d^2

Sympy [F]

\[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {\left (e \left (c + d x\right )\right )^{\frac {9}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

[In]

integrate((d*e*x+c*e)**(9/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Integral((e*(c + d*x))**(9/2)/sqrt(-(c + d*x - 1)*(c + d*x + 1)), x)

Maxima [F]

\[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {9}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate((d*e*x + c*e)^(9/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)

Giac [F]

\[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {9}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate((d*e*x + c*e)^(9/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^{9/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^{9/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]

[In]

int((c*e + d*e*x)^(9/2)/(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2),x)

[Out]

int((c*e + d*e*x)^(9/2)/(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2), x)