\(\int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [1400]

Optimal result

Integrand size = 37, antiderivative size = 170 \[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {30 e^5 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {18 e^3 (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {2 e (c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac {30 e^{11/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{77 d} \]

[Out]

30/77*e^(11/2)*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d-18/77*e^3*(d*e*x+c*e)^(5/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(
1/2)/d-2/11*e*(d*e*x+c*e)^(9/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d-30/77*e^5*(d*e*x+c*e)^(1/2)*(-d^2*x^2-2*c*d*x
-c^2+1)^(1/2)/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {706, 703, 227} \[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {30 e^{11/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{77 d}-\frac {30 e^5 \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{77 d}-\frac {18 e^3 \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d} \]

[In]

Int[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-30*e^5*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(77*d) - (18*e^3*(c*e + d*e*x)^(5/2)*Sqrt[1 - c^
2 - 2*c*d*x - d^2*x^2])/(77*d) - (2*e*(c*e + d*e*x)^(9/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(11*d) + (30*e^(1
1/2)*EllipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(77*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 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 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])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 e (c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac {1}{11} \left (9 e^2\right ) \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {18 e^3 (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {2 e (c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac {1}{77} \left (45 e^4\right ) \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {30 e^5 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {18 e^3 (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {2 e (c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac {1}{77} \left (15 e^6\right ) \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \\ & = -\frac {30 e^5 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {18 e^3 (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {2 e (c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac {\left (30 e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{77 d} \\ & = -\frac {30 e^5 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {18 e^3 (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac {2 e (c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac {30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{77 d} \\ \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 = 125, normalized size of antiderivative = 0.74 \[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e^5 \sqrt {e (c+d x)} \left (\sqrt {1-c^2-2 c d x-d^2 x^2} \left (15+7 c^4+28 c^3 d x+9 d^2 x^2+7 d^4 x^4+2 c d x \left (9+14 d^2 x^2\right )+c^2 \left (9+42 d^2 x^2\right )\right )-15 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{77 d} \]

[In]

Integrate[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-2*e^5*Sqrt[e*(c + d*x)]*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(15 + 7*c^4 + 28*c^3*d*x + 9*d^2*x^2 + 7*d^4*x^4
+ 2*c*d*x*(9 + 14*d^2*x^2) + c^2*(9 + 42*d^2*x^2)) - 15*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(77*d)

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e^{5} \left (-14 d^{7} x^{7}-98 c \,d^{6} x^{6}-294 c^{2} d^{5} x^{5}-490 c^{3} d^{4} x^{4}-490 c^{4} d^{3} x^{3}-4 d^{5} x^{5}-294 c^{5} d^{2} x^{2}-20 c \,d^{4} x^{4}-98 c^{6} d x -40 c^{2} d^{3} x^{3}-14 c^{7}-40 c^{3} d^{2} x^{2}-20 c^{4} d x -12 d^{3} x^{3}-4 c^{5}-36 c \,d^{2} x^{2}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-36 c^{2} d x -12 c^{3}+30 d x +30 c \right )}{77 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\)	\(285\)
risch	\(\frac {2 \left (7 d^{4} x^{4}+28 c \,d^{3} x^{3}+42 c^{2} d^{2} x^{2}+28 c^{3} d x +7 c^{4}+9 d^{2} x^{2}+18 c d x +9 c^{2}+15\right ) \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{6}}{77 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {30 \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{6}}{77 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(455\)
elliptic	\(\text {Expression too large to display}\)	\(1826\)

[In]

int((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/77*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^5*(-14*d^7*x^7-98*c*d^6*x^6-294*c^2*d^5*x^5-490*c^3*d^
4*x^4-490*c^4*d^3*x^3-4*d^5*x^5-294*c^5*d^2*x^2-20*c*d^4*x^4-98*c^6*d*x-40*c^2*d^3*x^3-14*c^7-40*c^3*d^2*x^2-2
0*c^4*d*x-12*d^3*x^3-4*c^5-36*c*d^2*x^2+15*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/
2*(-2*d*x-2*c+2)^(1/2),2^(1/2))-36*c^2*d*x-12*c^3+30*d*x+30*c)/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 = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.86 \[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (15 \, \sqrt {-d^{3} e} e^{5} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (7 \, d^{6} e^{5} x^{4} + 28 \, c d^{5} e^{5} x^{3} + 3 \, {\left (14 \, c^{2} + 3\right )} d^{4} e^{5} x^{2} + 2 \, {\left (14 \, c^{3} + 9 \, c\right )} d^{3} e^{5} x + {\left (7 \, c^{4} + 9 \, c^{2} + 15\right )} d^{2} e^{5}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{77 \, d^{3}} \]

[In]

integrate((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="fricas")

[Out]

-2/77*(15*sqrt(-d^3*e)*e^5*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) + (7*d^6*e^5*x^4 + 28*c*d^5*e^5*x^3 + 3*
(14*c^2 + 3)*d^4*e^5*x^2 + 2*(14*c^3 + 9*c)*d^3*e^5*x + (7*c^4 + 9*c^2 + 15)*d^2*e^5)*sqrt(-d^2*x^2 - 2*c*d*x
- c^2 + 1)*sqrt(d*e*x + c*e))/d^3

Sympy [F]

\[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {\left (e \left (c + d x\right )\right )^{\frac {11}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

[In]

integrate((d*e*x+c*e)**(11/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Integral((e*(c + d*x))**(11/2)/sqrt(-(c + d*x - 1)*(c + d*x + 1)), x)

Maxima [F]

\[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {11}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]

[In]

integrate((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)

Giac [F]

\[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{\frac {11}{2}}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}} \,d x } \]

[In]

integrate((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c e+d e x)^{11/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^{11/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]

[In]

int((c*e + d*e*x)^(11/2)/(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2),x)

[Out]

int((c*e + d*e*x)^(11/2)/(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2), x)