3.4.23 \(\int \frac {1}{x^3 (8 c-d x^3) \sqrt {c+d x^3}} \, dx\) [323]
Optimal. Leaf size=66 \[ -\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {2}{3};1,\frac {1}{2};\frac {1}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{16 c x^2 \sqrt {c+d x^3}} \]
[Out]
-1/16*AppellF1(-2/3,1/2,1,1/3,-d*x^3/c,1/8*d*x^3/c)*(1+d*x^3/c)^(1/2)/c/x^2/(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 {2}{3};1,\frac {1}{2};\frac {1}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{16 c x^2 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
[Out]
-1/16*(Sqrt[1 + (d*x^3)/c]*AppellF1[-2/3, 1, 1/2, 1/3, (d*x^3)/(8*c), -((d*x^3)/c)])/(c*x^2*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^3 \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^3 \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 {2}{3};1,\frac {1}{2};\frac {1}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{16 c x^2 \sqrt {c+d x^3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(66)=132\).
time = 20.12, size = 242, normalized size = 3.67 \begin {gather*} \frac {-\frac {64 \left (c+d x^3\right )}{c^2}+\frac {d^2 x^6 \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 )}{c^3}+\frac {4096 d x^3 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 )}}{1024 x^2 \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(x^3*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
[Out]
((-64*(c + d*x^3))/c^2 + (d^2*x^6*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])
/c^3 + (4096*d*x^3*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)]))))/(1024*x^2*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 = 0.40, size = 722, normalized size = 10.94 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/8/c*(-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)))-1/216*I/d^2/c^2*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))*Elli
pticPi(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)),_al
pha=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^3/(-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^3), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2574 vs.
\(2 (52) = 104\).
time = 6.63, size = 2574, normalized size = 39.00 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")
[Out]
1/3456*(4*sqrt(3)*c^2*x^2*(d^4/c^13)^(1/6)*arctan(1/9*((9*sqrt(3)*c^11*d^4*x^5*(d^4/c^13)^(5/6) + 3*sqrt(3)*(5
*c^7*d^5*x^4 + 8*c^8*d^4*x)*sqrt(d^4/c^13) - sqrt(3)*(c^2*d^7*x^6 - 40*c^3*d^6*x^3 - 32*c^4*d^5)*(d^4/c^13)^(1
/6))*sqrt(d*x^3 + c) - (12*sqrt(3)*(c^9*d^2*x^6 - c^10*d*x^3 - 2*c^11)*(d^4/c^13)^(2/3) + 18*sqrt(3)*(c^5*d^3*
x^5 + c^6*d^2*x^2)*(d^4/c^13)^(1/3) + 3*sqrt(3)*(d^5*x^7 + 5*c*d^4*x^4 + 4*c^2*d^3*x) - sqrt(d*x^3 + c)*(9*sqr
t(3)*(c^11*d*x^5 + 2*c^12*x^2)*(d^4/c^13)^(5/6) + 3*sqrt(3)*(7*c^7*d^2*x^4 + 4*c^8*d*x)*sqrt(d^4/c^13) + sqrt(
3)*(c^2*d^4*x^6 + 32*c^3*d^3*x^3 + 40*c^4*d^2)*(d^4/c^13)^(1/6)))*sqrt((d^9*x^9 - 276*c*d^8*x^6 - 1608*c^2*d^7
*x^3 - 1088*c^3*d^6 + 18*(c^9*d^6*x^8 + 20*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(d^4/c^13)^(2/3) + 6*sqrt(d*x^3 + c)
*((c^11*d^5*x^7 - 28*c^12*d^4*x^4 - 272*c^13*d^3*x)*(d^4/c^13)^(5/6) + 4*(c^7*d^6*x^6 + 41*c^8*d^5*x^3 + 40*c^
9*d^4)*sqrt(d^4/c^13) - 24*(c^3*d^7*x^5 + c^4*d^6*x^2)*(d^4/c^13)^(1/6)) - 18*(c^5*d^7*x^7 - 52*c^6*d^6*x^4 -
80*c^7*d^5*x)*(d^4/c^13)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^8*x^7 - 7*c*d^7*x^4 -
8*c^2*d^6*x)) + 4*sqrt(3)*c^2*x^2*(d^4/c^13)^(1/6)*arctan(1/9*((9*sqrt(3)*c^11*d^4*x^5*(d^4/c^13)^(5/6) + 3*sq
rt(3)*(5*c^7*d^5*x^4 + 8*c^8*d^4*x)*sqrt(d^4/c^13) - sqrt(3)*(c^2*d^7*x^6 - 40*c^3*d^6*x^3 - 32*c^4*d^5)*(d^4/
c^13)^(1/6))*sqrt(d*x^3 + c) + (12*sqrt(3)*(c^9*d^2*x^6 - c^10*d*x^3 - 2*c^11)*(d^4/c^13)^(2/3) + 18*sqrt(3)*(
c^5*d^3*x^5 + c^6*d^2*x^2)*(d^4/c^13)^(1/3) + 3*sqrt(3)*(d^5*x^7 + 5*c*d^4*x^4 + 4*c^2*d^3*x) + sqrt(d*x^3 + c
)*(9*sqrt(3)*(c^11*d*x^5 + 2*c^12*x^2)*(d^4/c^13)^(5/6) + 3*sqrt(3)*(7*c^7*d^2*x^4 + 4*c^8*d*x)*sqrt(d^4/c^13)
+ sqrt(3)*(c^2*d^4*x^6 + 32*c^3*d^3*x^3 + 40*c^4*d^2)*(d^4/c^13)^(1/6)))*sqrt((d^9*x^9 - 276*c*d^8*x^6 - 1608
*c^2*d^7*x^3 - 1088*c^3*d^6 + 18*(c^9*d^6*x^8 + 20*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(d^4/c^13)^(2/3) - 6*sqrt(d*
x^3 + c)*((c^11*d^5*x^7 - 28*c^12*d^4*x^4 - 272*c^13*d^3*x)*(d^4/c^13)^(5/6) + 4*(c^7*d^6*x^6 + 41*c^8*d^5*x^3
+ 40*c^9*d^4)*sqrt(d^4/c^13) - 24*(c^3*d^7*x^5 + c^4*d^6*x^2)*(d^4/c^13)^(1/6)) - 18*(c^5*d^7*x^7 - 52*c^6*d^
6*x^4 - 80*c^7*d^5*x)*(d^4/c^13)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^8*x^7 - 7*c*d^
7*x^4 - 8*c^2*d^6*x)) + c^2*x^2*(d^4/c^13)^(1/6)*log((d^9*x^9 - 276*c*d^8*x^6 - 1608*c^2*d^7*x^3 - 1088*c^3*d^
6 + 18*(c^9*d^6*x^8 + 20*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(d^4/c^13)^(2/3) + 6*sqrt(d*x^3 + c)*((c^11*d^5*x^7 -
28*c^12*d^4*x^4 - 272*c^13*d^3*x)*(d^4/c^13)^(5/6) + 4*(c^7*d^6*x^6 + 41*c^8*d^5*x^3 + 40*c^9*d^4)*sqrt(d^4/c^
13) - 24*(c^3*d^7*x^5 + c^4*d^6*x^2)*(d^4/c^13)^(1/6)) - 18*(c^5*d^7*x^7 - 52*c^6*d^6*x^4 - 80*c^7*d^5*x)*(d^4
/c^13)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - c^2*x^2*(d^4/c^13)^(1/6)*log((d^9*x^9 - 27
6*c*d^8*x^6 - 1608*c^2*d^7*x^3 - 1088*c^3*d^6 + 18*(c^9*d^6*x^8 + 20*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(d^4/c^13)
^(2/3) - 6*sqrt(d*x^3 + c)*((c^11*d^5*x^7 - 28*c^12*d^4*x^4 - 272*c^13*d^3*x)*(d^4/c^13)^(5/6) + 4*(c^7*d^6*x^
6 + 41*c^8*d^5*x^3 + 40*c^9*d^4)*sqrt(d^4/c^13) - 24*(c^3*d^7*x^5 + c^4*d^6*x^2)*(d^4/c^13)^(1/6)) - 18*(c^5*d
^7*x^7 - 52*c^6*d^6*x^4 - 80*c^7*d^5*x)*(d^4/c^13)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3))
+ 2*c^2*x^2*(d^4/c^13)^(1/6)*log((d^6*x^9 + 318*c*d^5*x^6 + 1200*c^2*d^4*x^3 + 640*c^3*d^3 + 18*(c^9*d^3*x^8 +
38*c^10*d^2*x^5 + 64*c^11*d*x^2)*(d^4/c^13)^(2/3) + 6*sqrt(d*x^3 + c)*((c^11*d^2*x^7 + 80*c^12*d*x^4 + 160*c^
13*x)*(d^4/c^13)^(5/6) + (7*c^7*d^3*x^6 + 152*c^8*d^2*x^3 + 64*c^9*d)*sqrt(d^4/c^13) + 6*(5*c^3*d^4*x^5 + 32*c
^4*d^3*x^2)*(d^4/c^13)^(1/6)) + 18*(5*c^5*d^4*x^7 + 64*c^6*d^3*x^4 + 32*c^7*d^2*x)*(d^4/c^13)^(1/3))/(d^3*x^9
- 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 2*c^2*x^2*(d^4/c^13)^(1/6)*log((d^6*x^9 + 318*c*d^5*x^6 + 1200*c^
2*d^4*x^3 + 640*c^3*d^3 + 18*(c^9*d^3*x^8 + 38*c^10*d^2*x^5 + 64*c^11*d*x^2)*(d^4/c^13)^(2/3) - 6*sqrt(d*x^3 +
c)*((c^11*d^2*x^7 + 80*c^12*d*x^4 + 160*c^13*x)*(d^4/c^13)^(5/6) + (7*c^7*d^3*x^6 + 152*c^8*d^2*x^3 + 64*c^9*
d)*sqrt(d^4/c^13) + 6*(5*c^3*d^4*x^5 + 32*c^4*d^3*x^2)*(d^4/c^13)^(1/6)) + 18*(5*c^5*d^4*x^7 + 64*c^6*d^3*x^4
+ 32*c^7*d^2*x)*(d^4/c^13)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 144*sqrt(d)*x^2*weiers
trassPInverse(0, -4*c/d, x) - 216*sqrt(d*x^3 + c))/(c^2*x^2)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- 8 c x^{3} \sqrt {c + d x^{3}} + d x^{6} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
[Out]
-Integral(1/(-8*c*x**3*sqrt(c + d*x**3) + d*x**6*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^3/(-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^3), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^3\,\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^3*(c + d*x^3)^(1/2)*(8*c - d*x^3)),x)
[Out]
int(1/(x^3*(c + d*x^3)^(1/2)*(8*c - d*x^3)), x)
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