3.4.24 \(\int \frac {1}{x^6 (8 c-d x^3) \sqrt {c+d x^3}} \, dx\) [324]

Optimal. Leaf size=66 \[ -\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {5}{3};1,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c x^5 \sqrt {c+d x^3}} \]

[Out]

-1/40*AppellF1(-5/3,1/2,1,-2/3,-d*x^3/c,1/8*d*x^3/c)*(1+d*x^3/c)^(1/2)/c/x^5/(d*x^3+c)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {525, 524} \begin {gather*} -\frac {\sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {5}{3};1,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c x^5 \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]

[Out]

-1/40*(Sqrt[1 + (d*x^3)/c]*AppellF1[-5/3, 1, 1/2, -2/3, (d*x^3)/(8*c), -((d*x^3)/c)])/(c*x^5*Sqrt[c + d*x^3])

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {1}{x^6 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {c+d x^3}}\\ &=-\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {5}{3};1,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c x^5 \sqrt {c+d x^3}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(66)=132\).
time = 20.14, size = 261, normalized size = 3.95 \begin {gather*} \frac {-23 d^3 x^9 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+64 c \left (-16 c^2+7 c d x^3+23 d^2 x^6+\frac {3264 c^2 d^2 x^6 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (32 c F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+3 d x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-4 F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )\right )}\right )}{40960 c^4 x^5 \sqrt {c+d x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^6*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]

[Out]

(-23*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 64*c*(-16*c^2 + 7*c
*d*x^3 + 23*d^2*x^6 + (3264*c^2*d^2*x^6*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3
)*(32*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3
)/c), (d*x^3)/(8*c)] - 4*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])))))/(40960*c^4*x^5*Sqrt[c +
d*x^3])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 6.
time = 0.36, size = 1047, normalized size = 15.86 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/8/c*(-1/5/c*(d*x^3+c)^(1/2)/x^5+7/20*d/c^2*(d*x^3+c)^(1/2)/x^2-7/60*I/c^2*d*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2
/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/
2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)
^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I
*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+
1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+1/64*d/c^2*(-1/2/c*(d*x^3+c)^(1/2)/x^2+1/6*I/c*3^(1/2)*(-c*d^2)^(1/3)
*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(
1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/
d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(
1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d
^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))-1/1728*I/c^3/d*2^(1/2)*sum(1/_alpha^2*(-c*d^2)^(1/3)*(1/2*I
*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-
c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-
c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3)+2*_alpha^2*d^2
-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c
*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d-I*(-c*d^2)^(2/3)*3
^(1/2)*_alpha+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3
)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c))

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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/x^6/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="maxima")

[Out]

-integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^6), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2603 vs. \(2 (52) = 104\).
time = 9.51, size = 2603, normalized size = 39.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")

[Out]

1/138240*(20*sqrt(3)*c^3*x^5*(d^10/c^19)^(1/6)*arctan(1/9*((9*sqrt(3)*c^16*d^9*x^5*(d^10/c^19)^(5/6) + 3*sqrt(
3)*(5*c^10*d^12*x^4 + 8*c^11*d^11*x)*sqrt(d^10/c^19) - sqrt(3)*(c^3*d^16*x^6 - 40*c^4*d^15*x^3 - 32*c^5*d^14)*
(d^10/c^19)^(1/6))*sqrt(d*x^3 + c) - (12*sqrt(3)*(c^13*d^3*x^6 - c^14*d^2*x^3 - 2*c^15*d)*(d^10/c^19)^(2/3) +
18*sqrt(3)*(c^7*d^6*x^5 + c^8*d^5*x^2)*(d^10/c^19)^(1/3) + 3*sqrt(3)*(d^10*x^7 + 5*c*d^9*x^4 + 4*c^2*d^8*x) -
sqrt(d*x^3 + c)*(9*sqrt(3)*(c^16*d*x^5 + 2*c^17*x^2)*(d^10/c^19)^(5/6) + 3*sqrt(3)*(7*c^10*d^4*x^4 + 4*c^11*d^
3*x)*sqrt(d^10/c^19) + sqrt(3)*(c^3*d^8*x^6 + 32*c^4*d^7*x^3 + 40*c^5*d^6)*(d^10/c^19)^(1/6)))*sqrt((d^19*x^9
- 276*c*d^18*x^6 - 1608*c^2*d^17*x^3 - 1088*c^3*d^16 + 18*(c^13*d^12*x^8 + 20*c^14*d^11*x^5 - 8*c^15*d^10*x^2)
*(d^10/c^19)^(2/3) + 6*sqrt(d*x^3 + c)*((c^16*d^10*x^7 - 28*c^17*d^9*x^4 - 272*c^18*d^8*x)*(d^10/c^19)^(5/6) +
4*(c^10*d^13*x^6 + 41*c^11*d^12*x^3 + 40*c^12*d^11)*sqrt(d^10/c^19) - 24*(c^4*d^16*x^5 + c^5*d^15*x^2)*(d^10/
c^19)^(1/6)) - 18*(c^7*d^15*x^7 - 52*c^8*d^14*x^4 - 80*c^9*d^13*x)*(d^10/c^19)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6
+ 192*c^2*d*x^3 - 512*c^3)))/(d^18*x^7 - 7*c*d^17*x^4 - 8*c^2*d^16*x)) + 20*sqrt(3)*c^3*x^5*(d^10/c^19)^(1/6)*
arctan(1/9*((9*sqrt(3)*c^16*d^9*x^5*(d^10/c^19)^(5/6) + 3*sqrt(3)*(5*c^10*d^12*x^4 + 8*c^11*d^11*x)*sqrt(d^10/
c^19) - sqrt(3)*(c^3*d^16*x^6 - 40*c^4*d^15*x^3 - 32*c^5*d^14)*(d^10/c^19)^(1/6))*sqrt(d*x^3 + c) + (12*sqrt(3
)*(c^13*d^3*x^6 - c^14*d^2*x^3 - 2*c^15*d)*(d^10/c^19)^(2/3) + 18*sqrt(3)*(c^7*d^6*x^5 + c^8*d^5*x^2)*(d^10/c^
19)^(1/3) + 3*sqrt(3)*(d^10*x^7 + 5*c*d^9*x^4 + 4*c^2*d^8*x) + sqrt(d*x^3 + c)*(9*sqrt(3)*(c^16*d*x^5 + 2*c^17
*x^2)*(d^10/c^19)^(5/6) + 3*sqrt(3)*(7*c^10*d^4*x^4 + 4*c^11*d^3*x)*sqrt(d^10/c^19) + sqrt(3)*(c^3*d^8*x^6 + 3
2*c^4*d^7*x^3 + 40*c^5*d^6)*(d^10/c^19)^(1/6)))*sqrt((d^19*x^9 - 276*c*d^18*x^6 - 1608*c^2*d^17*x^3 - 1088*c^3
*d^16 + 18*(c^13*d^12*x^8 + 20*c^14*d^11*x^5 - 8*c^15*d^10*x^2)*(d^10/c^19)^(2/3) - 6*sqrt(d*x^3 + c)*((c^16*d
^10*x^7 - 28*c^17*d^9*x^4 - 272*c^18*d^8*x)*(d^10/c^19)^(5/6) + 4*(c^10*d^13*x^6 + 41*c^11*d^12*x^3 + 40*c^12*
d^11)*sqrt(d^10/c^19) - 24*(c^4*d^16*x^5 + c^5*d^15*x^2)*(d^10/c^19)^(1/6)) - 18*(c^7*d^15*x^7 - 52*c^8*d^14*x
^4 - 80*c^9*d^13*x)*(d^10/c^19)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^18*x^7 - 7*c*d^
17*x^4 - 8*c^2*d^16*x)) + 5*c^3*x^5*(d^10/c^19)^(1/6)*log((d^19*x^9 - 276*c*d^18*x^6 - 1608*c^2*d^17*x^3 - 108
8*c^3*d^16 + 18*(c^13*d^12*x^8 + 20*c^14*d^11*x^5 - 8*c^15*d^10*x^2)*(d^10/c^19)^(2/3) + 6*sqrt(d*x^3 + c)*((c
^16*d^10*x^7 - 28*c^17*d^9*x^4 - 272*c^18*d^8*x)*(d^10/c^19)^(5/6) + 4*(c^10*d^13*x^6 + 41*c^11*d^12*x^3 + 40*
c^12*d^11)*sqrt(d^10/c^19) - 24*(c^4*d^16*x^5 + c^5*d^15*x^2)*(d^10/c^19)^(1/6)) - 18*(c^7*d^15*x^7 - 52*c^8*d
^14*x^4 - 80*c^9*d^13*x)*(d^10/c^19)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 5*c^3*x^5*(d
^10/c^19)^(1/6)*log((d^19*x^9 - 276*c*d^18*x^6 - 1608*c^2*d^17*x^3 - 1088*c^3*d^16 + 18*(c^13*d^12*x^8 + 20*c^
14*d^11*x^5 - 8*c^15*d^10*x^2)*(d^10/c^19)^(2/3) - 6*sqrt(d*x^3 + c)*((c^16*d^10*x^7 - 28*c^17*d^9*x^4 - 272*c
^18*d^8*x)*(d^10/c^19)^(5/6) + 4*(c^10*d^13*x^6 + 41*c^11*d^12*x^3 + 40*c^12*d^11)*sqrt(d^10/c^19) - 24*(c^4*d
^16*x^5 + c^5*d^15*x^2)*(d^10/c^19)^(1/6)) - 18*(c^7*d^15*x^7 - 52*c^8*d^14*x^4 - 80*c^9*d^13*x)*(d^10/c^19)^(
1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 10*c^3*x^5*(d^10/c^19)^(1/6)*log((d^11*x^9 + 318*c
*d^10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8 + 18*(c^13*d^4*x^8 + 38*c^14*d^3*x^5 + 64*c^15*d^2*x^2)*(d^10/c^19)
^(2/3) + 6*sqrt(d*x^3 + c)*((c^16*d^2*x^7 + 80*c^17*d*x^4 + 160*c^18*x)*(d^10/c^19)^(5/6) + (7*c^10*d^5*x^6 +
152*c^11*d^4*x^3 + 64*c^12*d^3)*sqrt(d^10/c^19) + 6*(5*c^4*d^8*x^5 + 32*c^5*d^7*x^2)*(d^10/c^19)^(1/6)) + 18*(
5*c^7*d^7*x^7 + 64*c^8*d^6*x^4 + 32*c^9*d^5*x)*(d^10/c^19)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 51
2*c^3)) - 10*c^3*x^5*(d^10/c^19)^(1/6)*log((d^11*x^9 + 318*c*d^10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8 + 18*(c
^13*d^4*x^8 + 38*c^14*d^3*x^5 + 64*c^15*d^2*x^2)*(d^10/c^19)^(2/3) - 6*sqrt(d*x^3 + c)*((c^16*d^2*x^7 + 80*c^1
7*d*x^4 + 160*c^18*x)*(d^10/c^19)^(5/6) + (7*c^10*d^5*x^6 + 152*c^11*d^4*x^3 + 64*c^12*d^3)*sqrt(d^10/c^19) +
6*(5*c^4*d^8*x^5 + 32*c^5*d^7*x^2)*(d^10/c^19)^(1/6)) + 18*(5*c^7*d^7*x^7 + 64*c^8*d^6*x^4 + 32*c^9*d^5*x)*(d^
10/c^19)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 5328*d^(3/2)*x^5*weierstrassPInverse(0,
-4*c/d, x) + 216*(23*d*x^3 - 16*c)*sqrt(d*x^3 + c))/(c^3*x^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- 8 c x^{6} \sqrt {c + d x^{3}} + d x^{9} \sqrt {c + d x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)

[Out]

-Integral(1/(-8*c*x**6*sqrt(c + d*x**3) + d*x**9*sqrt(c + d*x**3)), 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/x^6/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="giac")

[Out]

integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^6\,\sqrt {d\,x^3+c}\,\left (8\,c-d\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(c + d*x^3)^(1/2)*(8*c - d*x^3)),x)

[Out]

int(1/(x^6*(c + d*x^3)^(1/2)*(8*c - d*x^3)), x)

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