3.15.6 \(\int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [1406]

Optimal. Leaf size=172 \[ -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}-\frac {18 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^3 (c e+d e x)^{7/2}}-\frac {30 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^5 (c e+d e x)^{3/2}}+\frac {30 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{77 d e^{13/2}} \]

[Out]

30/77*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(13/2)-2/11*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(1
1/2)-18/77*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^3/(d*e*x+c*e)^(7/2)-30/77*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^5/(
d*e*x+c*e)^(3/2)

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Rubi [A]
time = 0.08, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {707, 703, 227} \begin {gather*} \frac {30 F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{77 d e^{13/2}}-\frac {30 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{77 d e^5 (c e+d e x)^{3/2}}-\frac {18 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{77 d e^3 (c e+d e x)^{7/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*e + d*e*x)^(13/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(11*d*e*(c*e + d*e*x)^(11/2)) - (18*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/
(77*d*e^3*(c*e + d*e*x)^(7/2)) - (30*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(77*d*e^5*(c*e + d*e*x)^(3/2)) + (30*E
llipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(77*d*e^(13/2))

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 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 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*} \int \frac {1}{(c e+d e x)^{13/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}+\frac {9 \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{11 e^2}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}-\frac {18 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^3 (c e+d e x)^{7/2}}+\frac {45 \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{77 e^4}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}-\frac {18 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^3 (c e+d e x)^{7/2}}-\frac {30 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^5 (c e+d e x)^{3/2}}+\frac {15 \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{77 e^6}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}-\frac {18 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^3 (c e+d e x)^{7/2}}-\frac {30 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^5 (c e+d e x)^{3/2}}+\frac {30 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{77 d e^7}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{11 d e (c e+d e x)^{11/2}}-\frac {18 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^3 (c e+d e x)^{7/2}}-\frac {30 \sqrt {1-c^2-2 c d x-d^2 x^2}}{77 d e^5 (c e+d e x)^{3/2}}+\frac {30 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{77 d e^{13/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 40, normalized size = 0.23 \begin {gather*} -\frac {2 (c+d x) \, _2F_1\left (-\frac {11}{4},\frac {1}{2};-\frac {7}{4};(c+d x)^2\right )}{11 d (e (c+d x))^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*e + d*e*x)^(13/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

(-2*(c + d*x)*Hypergeometric2F1[-11/4, 1/2, -7/4, (c + d*x)^2])/(11*d*(e*(c + d*x))^(13/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(514\) vs. \(2(146)=292\).
time = 0.77, size = 515, normalized size = 2.99

method	result	size
elliptic	\(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{11 d^{7} e^{7} \left (x +\frac {c}{d}\right )^{6}}-\frac {18 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{77 d^{5} e^{7} \left (x +\frac {c}{d}\right )^{4}}-\frac {30 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{77 e^{7} d^{3} \left (x +\frac {c}{d}\right )^{2}}+\frac {30 \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{77 e^{6} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(457\)
default	\(\frac {\left (15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d^{5} x^{5}+75 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c \,d^{4} x^{4}+150 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{2} d^{3} x^{3}-30 d^{6} x^{6}+150 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{3} d^{2} x^{2}-180 c \,d^{5} x^{5}+75 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{4} d x -450 c^{2} d^{4} x^{4}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{5}-600 c^{3} d^{3} x^{3}-450 c^{4} d^{2} x^{2}+12 x^{4} d^{4}-180 c^{5} d x +48 c \,x^{3} d^{3}-30 c^{6}+72 c^{2} d^{2} x^{2}+48 c^{3} d x +12 c^{4}+4 d^{2} x^{2}+8 c d x +4 c^{2}+14\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {e \left (d x +c \right )}}{77 e^{7} \left (d x +c \right )^{6} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d}\)	\(515\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)^(13/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/77*(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))*d^
5*x^5+75*(-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))*c*
d^4*x^4+150*(-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))
*c^2*d^3*x^3-30*d^6*x^6+150*(-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))*c^3*d^2*x^2-180*c*d^5*x^5+75*(-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))*c^4*d*x-450*c^2*d^4*x^4+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))*c^5-600*c^3*d^3*x^3-450*c^4*d^2*x^2+12*x^4*d^4-180*c^5*d*x+
48*c*x^3*d^3-30*c^6+72*c^2*d^2*x^2+48*c^3*d*x+12*c^4+4*d^2*x^2+8*c*d*x+4*c^2+14)/e^7*(-d^2*x^2-2*c*d*x-c^2+1)^
(1/2)*(e*(d*x+c))^(1/2)/(d*x+c)^6/(d^2*x^2+2*c*d*x+c^2-1)/d

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(13/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*x*e + c*e)^(13/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.83, size = 256, normalized size = 1.49 \begin {gather*} -\frac {2 \, {\left ({\left (15 \, d^{6} x^{4} + 60 \, c d^{5} x^{3} + 9 \, {\left (10 \, c^{2} + 1\right )} d^{4} x^{2} + 6 \, {\left (10 \, c^{3} + 3 \, c\right )} d^{3} x + {\left (15 \, c^{4} + 9 \, c^{2} + 7\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {1}{2}} + 15 \, {\left (d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )} e^{\left (-7\right )}}{77 \, {\left (d^{9} x^{6} + 6 \, c d^{8} x^{5} + 15 \, c^{2} d^{7} x^{4} + 20 \, c^{3} d^{6} x^{3} + 15 \, c^{4} d^{5} x^{2} + 6 \, c^{5} d^{4} x + c^{6} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(13/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="fricas")

[Out]

-2/77*((15*d^6*x^4 + 60*c*d^5*x^3 + 9*(10*c^2 + 1)*d^4*x^2 + 6*(10*c^3 + 3*c)*d^3*x + (15*c^4 + 9*c^2 + 7)*d^2
)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*x + c)*e^(1/2) + 15*(d^6*x^6 + 6*c*d^5*x^5 + 15*c^2*d^4*x^4 + 20*c
^3*d^3*x^3 + 15*c^4*d^2*x^2 + 6*c^5*d*x + c^6)*sqrt(-d^3*e)*weierstrassPInverse(4/d^2, 0, (d*x + c)/d))*e^(-7)
/(d^9*x^6 + 6*c*d^8*x^5 + 15*c^2*d^7*x^4 + 20*c^3*d^6*x^3 + 15*c^4*d^5*x^2 + 6*c^5*d^4*x + c^6*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {13}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)**(13/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Integral(1/((e*(c + d*x))**(13/2)*sqrt(-(c + d*x - 1)*(c + d*x + 1))), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(13/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*x*e + c*e)^(13/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{13/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*e + d*e*x)^(13/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)),x)

[Out]

int(1/((c*e + d*e*x)^(13/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)), x)

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