3.4.39 \(\int \frac {1}{(8 c-d x^3) (c+d x^3)^{3/2}} \, dx\) [339]
Optimal. Leaf size=64 \[ \frac {x \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {1}{3};1,\frac {3}{2};\frac {4}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c^2 \sqrt {c+d x^3}} \]
[Out]
1/8*x*AppellF1(1/3,3/2,1,4/3,-d*x^3/c,1/8*d*x^3/c)*(1+d*x^3/c)^(1/2)/c^2/(d*x^3+c)^(1/2)
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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {441, 440}
\begin {gather*} \frac {x \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {1}{3};1,\frac {3}{2};\frac {4}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c^2 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
(x*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1, 3/2, 4/3, (d*x^3)/(8*c), -((d*x^3)/c)])/(8*c^2*Sqrt[c + d*x^3])
Rule 440
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Rule 441
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Rubi steps
\begin {align*} \int \frac {1}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {1}{\left (8 c-d x^3\right ) \left (1+\frac {d x^3}{c}\right )^{3/2}} \, dx}{c \sqrt {c+d x^3}}\\ &=\frac {x \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {1}{3};1,\frac {3}{2};\frac {4}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c^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. \(230\) vs. \(2(64)=128\).
time = 10.08, size = 230, normalized size = 3.59 \begin {gather*} \frac {x \left (-\frac {d x^3 \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}+64 \left (\frac {1}{c^2}+\frac {176 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 )\right )}{864 \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
(x*(-((d*x^3*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) + 64*(c^(-2) +
(176*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)]))))))/(864*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 = 0.30, size = 721, normalized size = 11.27 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
[Out]
2/27*x/c^2/((x^3+c/d)*d)^(1/2)-2/81*I/c^2*3^(1/2)/d*(-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))-1/243*I/c^2/d^3*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))*Ell
ipticPi(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)),_a
lpha=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/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="maxima")
[Out]
-integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2572 vs.
\(2 (50) = 100\).
time = 6.58, size = 2572, normalized size = 40.19 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="fricas")
[Out]
1/3888*(4*sqrt(3)*(c^2*d^2*x^3 + c^3*d)*(1/(c^13*d^2))^(1/6)*arctan(1/9*((9*sqrt(3)*c^11*d^3*x^5*(1/(c^13*d^2)
)^(5/6) + 3*sqrt(3)*(5*c^7*d^2*x^4 + 8*c^8*d*x)*sqrt(1/(c^13*d^2)) - sqrt(3)*(c^2*d^2*x^6 - 40*c^3*d*x^3 - 32*
c^4)*(1/(c^13*d^2))^(1/6))*sqrt(d*x^3 + c) - (12*sqrt(3)*(c^9*d^3*x^6 - c^10*d^2*x^3 - 2*c^11*d)*(1/(c^13*d^2)
)^(2/3) + 18*sqrt(3)*(c^5*d^2*x^5 + c^6*d*x^2)*(1/(c^13*d^2))^(1/3) + 3*sqrt(3)*(d^2*x^7 + 5*c*d*x^4 + 4*c^2*x
) - sqrt(d*x^3 + c)*(9*sqrt(3)*(c^11*d^3*x^5 + 2*c^12*d^2*x^2)*(1/(c^13*d^2))^(5/6) + 3*sqrt(3)*(7*c^7*d^2*x^4
+ 4*c^8*d*x)*sqrt(1/(c^13*d^2)) + sqrt(3)*(c^2*d^2*x^6 + 32*c^3*d*x^3 + 40*c^4)*(1/(c^13*d^2))^(1/6)))*sqrt((
d^3*x^9 - 276*c*d^2*x^6 - 1608*c^2*d*x^3 - 1088*c^3 + 18*(c^9*d^4*x^8 + 20*c^10*d^3*x^5 - 8*c^11*d^2*x^2)*(1/(
c^13*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c^11*d^4*x^7 - 28*c^12*d^3*x^4 - 272*c^13*d^2*x)*(1/(c^13*d^2))^(5/6) +
4*(c^7*d^3*x^6 + 41*c^8*d^2*x^3 + 40*c^9*d)*sqrt(1/(c^13*d^2)) - 24*(c^3*d^2*x^5 + c^4*d*x^2)*(1/(c^13*d^2))^
(1/6)) - 18*(c^5*d^3*x^7 - 52*c^6*d^2*x^4 - 80*c^7*d*x)*(1/(c^13*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^
2*d*x^3 - 512*c^3)))/(d^2*x^7 - 7*c*d*x^4 - 8*c^2*x)) + 4*sqrt(3)*(c^2*d^2*x^3 + c^3*d)*(1/(c^13*d^2))^(1/6)*a
rctan(1/9*((9*sqrt(3)*c^11*d^3*x^5*(1/(c^13*d^2))^(5/6) + 3*sqrt(3)*(5*c^7*d^2*x^4 + 8*c^8*d*x)*sqrt(1/(c^13*d
^2)) - sqrt(3)*(c^2*d^2*x^6 - 40*c^3*d*x^3 - 32*c^4)*(1/(c^13*d^2))^(1/6))*sqrt(d*x^3 + c) + (12*sqrt(3)*(c^9*
d^3*x^6 - c^10*d^2*x^3 - 2*c^11*d)*(1/(c^13*d^2))^(2/3) + 18*sqrt(3)*(c^5*d^2*x^5 + c^6*d*x^2)*(1/(c^13*d^2))^
(1/3) + 3*sqrt(3)*(d^2*x^7 + 5*c*d*x^4 + 4*c^2*x) + sqrt(d*x^3 + c)*(9*sqrt(3)*(c^11*d^3*x^5 + 2*c^12*d^2*x^2)
*(1/(c^13*d^2))^(5/6) + 3*sqrt(3)*(7*c^7*d^2*x^4 + 4*c^8*d*x)*sqrt(1/(c^13*d^2)) + sqrt(3)*(c^2*d^2*x^6 + 32*c
^3*d*x^3 + 40*c^4)*(1/(c^13*d^2))^(1/6)))*sqrt((d^3*x^9 - 276*c*d^2*x^6 - 1608*c^2*d*x^3 - 1088*c^3 + 18*(c^9*
d^4*x^8 + 20*c^10*d^3*x^5 - 8*c^11*d^2*x^2)*(1/(c^13*d^2))^(2/3) - 6*sqrt(d*x^3 + c)*((c^11*d^4*x^7 - 28*c^12*
d^3*x^4 - 272*c^13*d^2*x)*(1/(c^13*d^2))^(5/6) + 4*(c^7*d^3*x^6 + 41*c^8*d^2*x^3 + 40*c^9*d)*sqrt(1/(c^13*d^2)
) - 24*(c^3*d^2*x^5 + c^4*d*x^2)*(1/(c^13*d^2))^(1/6)) - 18*(c^5*d^3*x^7 - 52*c^6*d^2*x^4 - 80*c^7*d*x)*(1/(c^
13*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^2*x^7 - 7*c*d*x^4 - 8*c^2*x)) + 288*sq
rt(d*x^3 + c)*d*x + 360*(d*x^3 + c)*sqrt(d)*weierstrassPInverse(0, -4*c/d, x) + 2*(c^2*d^2*x^3 + c^3*d)*(1/(c^
13*d^2))^(1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3 + 640*c^3 + 18*(c^9*d^4*x^8 + 38*c^10*d^3*x^5 + 6
4*c^11*d^2*x^2)*(1/(c^13*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c^11*d^4*x^7 + 80*c^12*d^3*x^4 + 160*c^13*d^2*x)*(1
/(c^13*d^2))^(5/6) + (7*c^7*d^3*x^6 + 152*c^8*d^2*x^3 + 64*c^9*d)*sqrt(1/(c^13*d^2)) + 6*(5*c^3*d^2*x^5 + 32*c
^4*d*x^2)*(1/(c^13*d^2))^(1/6)) + 18*(5*c^5*d^3*x^7 + 64*c^6*d^2*x^4 + 32*c^7*d*x)*(1/(c^13*d^2))^(1/3))/(d^3*
x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 2*(c^2*d^2*x^3 + c^3*d)*(1/(c^13*d^2))^(1/6)*log((d^3*x^9 + 3
18*c*d^2*x^6 + 1200*c^2*d*x^3 + 640*c^3 + 18*(c^9*d^4*x^8 + 38*c^10*d^3*x^5 + 64*c^11*d^2*x^2)*(1/(c^13*d^2))^
(2/3) - 6*sqrt(d*x^3 + c)*((c^11*d^4*x^7 + 80*c^12*d^3*x^4 + 160*c^13*d^2*x)*(1/(c^13*d^2))^(5/6) + (7*c^7*d^3
*x^6 + 152*c^8*d^2*x^3 + 64*c^9*d)*sqrt(1/(c^13*d^2)) + 6*(5*c^3*d^2*x^5 + 32*c^4*d*x^2)*(1/(c^13*d^2))^(1/6))
+ 18*(5*c^5*d^3*x^7 + 64*c^6*d^2*x^4 + 32*c^7*d*x)*(1/(c^13*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*
x^3 - 512*c^3)) + (c^2*d^2*x^3 + c^3*d)*(1/(c^13*d^2))^(1/6)*log((d^3*x^9 - 276*c*d^2*x^6 - 1608*c^2*d*x^3 - 1
088*c^3 + 18*(c^9*d^4*x^8 + 20*c^10*d^3*x^5 - 8*c^11*d^2*x^2)*(1/(c^13*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c^11*
d^4*x^7 - 28*c^12*d^3*x^4 - 272*c^13*d^2*x)*(1/(c^13*d^2))^(5/6) + 4*(c^7*d^3*x^6 + 41*c^8*d^2*x^3 + 40*c^9*d)
*sqrt(1/(c^13*d^2)) - 24*(c^3*d^2*x^5 + c^4*d*x^2)*(1/(c^13*d^2))^(1/6)) - 18*(c^5*d^3*x^7 - 52*c^6*d^2*x^4 -
80*c^7*d*x)*(1/(c^13*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - (c^2*d^2*x^3 + c^3*d)*
(1/(c^13*d^2))^(1/6)*log((d^3*x^9 - 276*c*d^2*x^6 - 1608*c^2*d*x^3 - 1088*c^3 + 18*(c^9*d^4*x^8 + 20*c^10*d^3*
x^5 - 8*c^11*d^2*x^2)*(1/(c^13*d^2))^(2/3) - 6*sqrt(d*x^3 + c)*((c^11*d^4*x^7 - 28*c^12*d^3*x^4 - 272*c^13*d^2
*x)*(1/(c^13*d^2))^(5/6) + 4*(c^7*d^3*x^6 + 41*c^8*d^2*x^3 + 40*c^9*d)*sqrt(1/(c^13*d^2)) - 24*(c^3*d^2*x^5 +
c^4*d*x^2)*(1/(c^13*d^2))^(1/6)) - 18*(c^5*d^3*x^7 - 52*c^6*d^2*x^4 - 80*c^7*d*x)*(1/(c^13*d^2))^(1/3))/(d^3*x
^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(c^2*d^2*x^3 + c^3*d)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- 8 c^{2} \sqrt {c + d x^{3}} - 7 c d x^{3} \sqrt {c + d x^{3}} + d^{2} x^{6} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)
[Out]
-Integral(1/(-8*c**2*sqrt(c + d*x**3) - 7*c*d*x**3*sqrt(c + d*x**3) + d**2*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/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="giac")
[Out]
integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (d\,x^3+c\right )}^{3/2}\,\left (8\,c-d\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((c + d*x^3)^(3/2)*(8*c - d*x^3)),x)
[Out]
int(1/((c + d*x^3)^(3/2)*(8*c - d*x^3)), x)
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