3.15.11 \(\int \frac {1}{(c e+d e x)^{7/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [1411]

Optimal. Leaf size=159 \[ -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e (c e+d e x)^{5/2}}-\frac {6 \sqrt {1-c^2-2 c d x-d^2 x^2}}{5 d e^3 \sqrt {c e+d e x}}-\frac {6 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}+\frac {6 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}} \]

[Out]

-6/5*EllipticE((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(7/2)+6/5*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(7/2)-2/5
*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(5/2)-6/5*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^3/(d*e*x+c*e)^(1/
2)

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Rubi [A]
time = 0.08, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {707, 704, 313, 227, 1213, 435} \begin {gather*} \frac {6 F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac {6 E\left (\left .\text {ArcSin}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac {6 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt {c e+d e x}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*e + d*e*x)^(7/2)*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])/(5*d*e*(c*e + d*e*x)^(5/2)) - (6*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(5*
d*e^3*Sqrt[c*e + d*e*x]) - (6*EllipticE[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(5*d*e^(7/2)) + (6*EllipticF[A
rcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(5*d*e^(7/2))

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 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 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])

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 40, normalized size = 0.25 \begin {gather*} -\frac {2 (c+d x) \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};(c+d x)^2\right )}{5 d (e (c+d x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*e + d*e*x)^(7/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

(-2*(c + d*x)*Hypergeometric2F1[-5/4, 1/2, -1/4, (c + d*x)^2])/(5*d*(e*(c + d*x))^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(281\) vs. \(2(131)=262\).
time = 0.76, size = 282, normalized size = 1.77

method	result	size
default	\(-\frac {\left (3 \EllipticE \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d^{2} x^{2} \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}+6 x^{4} d^{4}+6 \EllipticE \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c d x \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}+24 c \,x^{3} d^{3}+3 \EllipticE \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{2} \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}+36 c^{2} d^{2} x^{2}+24 c^{3} d x +6 c^{4}-4 d^{2} x^{2}-8 c d x -4 c^{2}-2\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {e \left (d x +c \right )}}{5 e^{4} \left (d x +c \right )^{3} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d}\)	\(282\)
elliptic	\(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{5 d^{4} e^{4} \left (x +\frac {c}{d}\right )^{3}}-\frac {6 \left (-d^{3} e \,x^{2}-2 c \,d^{2} e x -c^{2} d e +d e \right )}{5 d^{2} e^{4} \sqrt {\left (x +\frac {c}{d}\right ) \left (-d^{3} e \,x^{2}-2 c \,d^{2} e x -c^{2} d e +d e \right )}}-\frac {6 c \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{5 e^{3} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}-\frac {6 d \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c \EllipticF \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{5 e^{3} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(717\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*(3*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*d^2*x^2*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(
1/2)+6*x^4*d^4+6*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c*d*x*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2
*c+2)^(1/2)+24*c*x^3*d^3+3*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c^2*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*
(2*d*x+2*c+2)^(1/2)+36*c^2*d^2*x^2+24*c^3*d*x+6*c^4-4*d^2*x^2-8*c*d*x-4*c^2-2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*
(e*(d*x+c))^(1/2)/e^4/(d*x+c)^3/(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)^(7/2)/(-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)^(7/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.38, size = 158, normalized size = 0.99 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, d^{3} x^{2} + 6 \, c d^{2} x + {\left (3 \, c^{2} + 1\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {1}{2}} + 3 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )\right )} e^{\left (-4\right )}}{5 \, {\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="fricas")

[Out]

-2/5*((3*d^3*x^2 + 6*c*d^2*x + (3*c^2 + 1)*d)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*x + c)*e^(1/2) + 3*(d^
3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*sqrt(-d^3*e)*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d
*x + c)/d)))*e^(-4)/(d^5*x^3 + 3*c*d^4*x^2 + 3*c^2*d^3*x + c^3*d^2)

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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 {7}{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)**(7/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Integral(1/((e*(c + d*x))**(7/2)*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)^(7/2)/(-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)^(7/2)), 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 )}^{7/2}\,\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)^(7/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)),x)

[Out]

int(1/((c*e + d*e*x)^(7/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)), x)

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