3.3.80 \(\int \frac {1}{x^3 \sqrt {c+d x^3} (4 c+d x^3)} \, dx\) [280]
Optimal. Leaf size=66 \[ -\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {2}{3};1,\frac {1}{2};\frac {1}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{8 c x^2 \sqrt {c+d x^3}} \]
[Out]
-1/8*AppellF1(-2/3,1/2,1,1/3,-d*x^3/c,-1/4*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 = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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}{4 c},-\frac {d x^3}{c}\right )}{8 c x^2 \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^3*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
-1/8*(Sqrt[1 + (d*x^3)/c]*AppellF1[-2/3, 1, 1/2, 1/3, -1/4*(d*x^3)/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 \sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {1}{x^3 \left (4 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}{4 c},-\frac {d x^3}{c}\right )}{8 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. \(243\) vs. \(2(66)=132\).
time = 20.13, size = 243, normalized size = 3.68 \begin {gather*} \frac {-\frac {32 \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}{4 c}\right )}{c^3}+\frac {2048 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}{4 c}\right )}{\left (4 c+d x^3\right ) \left (-16 c F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 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}{4 c}\right )+2 F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )\right )\right )}}{256 x^2 \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(x^3*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
((-32*(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), -1/4*(d*x^3)/c]
)/c^3 + (2048*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c])/((4*c + d*x^3)*(-16*c*AppellF1[1
/3, 1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d*x^3)/c] + 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d*x^3
)/c] + 2*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c]))))/(256*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 = 0.41, 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+4*c)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/4/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)))+1/36*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))*Ellip
ticPi(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
/6/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)*_al
pha-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+4*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+4*c)/(d*x^3+c)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^3), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2392 vs.
\(2 (52) = 104\).
time = 6.00, size = 2392, normalized size = 36.24 \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+4*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")
[Out]
1/288*(4*sqrt(3)*(1/108)^(1/6)*c^2*x^2*(-d^4/c^13)^(1/6)*arctan(1/3*((108*sqrt(3)*(1/108)^(5/6)*c^11*d^3*x^2*(
-d^4/c^13)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^7*d^4*x*sqrt(-d^4/c^13) + sqrt(3)*(1/108)^(1/6)*(c^2*d^6*x^3 + 4*c^3*
d^5)*(-d^4/c^13)^(1/6))*sqrt(d*x^3 + c) - (4*sqrt(3)*(1/4)^(2/3)*(c^9*d*x^3 + c^10)*(-d^4/c^13)^(2/3) - sqrt(3
)*(d^4*x^4 + c*d^3*x) - (108*sqrt(3)*(1/108)^(5/6)*c^11*x^2*(-d^4/c^13)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^7*d*x*sq
rt(-d^4/c^13) - sqrt(3)*(1/108)^(1/6)*(c^2*d^3*x^3 - 2*c^3*d^2)*(-d^4/c^13)^(1/6))*sqrt(d*x^3 + c))*sqrt((d^9*
x^9 + 60*c*d^8*x^6 - 32*c^3*d^6 - 24*(1/4)^(2/3)*(c^9*d^6*x^8 - 7*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(-d^4/c^13)^(
2/3) + 24*(1/4)^(1/3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x)*(-d^4/c^13)^(1/3) + 12*(9*(1/108)^(1/6)*c^3*
d^7*x^5*(-d^4/c^13)^(1/6) - 18*(1/108)^(5/6)*(c^11*d^5*x^7 + 2*c^12*d^4*x^4 - 8*c^13*d^3*x)*(-d^4/c^13)^(5/6)
- sqrt(1/3)*(c^7*d^6*x^6 - 16*c^8*d^5*x^3 - 8*c^9*d^4)*sqrt(-d^4/c^13))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x
^6 + 48*c^2*d*x^3 + 64*c^3)))/(d^7*x^4 + c*d^6*x)) + 4*sqrt(3)*(1/108)^(1/6)*c^2*x^2*(-d^4/c^13)^(1/6)*arctan(
1/3*((108*sqrt(3)*(1/108)^(5/6)*c^11*d^3*x^2*(-d^4/c^13)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^7*d^4*x*sqrt(-d^4/c^13)
+ sqrt(3)*(1/108)^(1/6)*(c^2*d^6*x^3 + 4*c^3*d^5)*(-d^4/c^13)^(1/6))*sqrt(d*x^3 + c) + (4*sqrt(3)*(1/4)^(2/3)
*(c^9*d*x^3 + c^10)*(-d^4/c^13)^(2/3) - sqrt(3)*(d^4*x^4 + c*d^3*x) + (108*sqrt(3)*(1/108)^(5/6)*c^11*x^2*(-d^
4/c^13)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^7*d*x*sqrt(-d^4/c^13) - sqrt(3)*(1/108)^(1/6)*(c^2*d^3*x^3 - 2*c^3*d^2)*
(-d^4/c^13)^(1/6))*sqrt(d*x^3 + c))*sqrt((d^9*x^9 + 60*c*d^8*x^6 - 32*c^3*d^6 - 24*(1/4)^(2/3)*(c^9*d^6*x^8 -
7*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(-d^4/c^13)^(2/3) + 24*(1/4)^(1/3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x
)*(-d^4/c^13)^(1/3) - 12*(9*(1/108)^(1/6)*c^3*d^7*x^5*(-d^4/c^13)^(1/6) - 18*(1/108)^(5/6)*(c^11*d^5*x^7 + 2*c
^12*d^4*x^4 - 8*c^13*d^3*x)*(-d^4/c^13)^(5/6) - sqrt(1/3)*(c^7*d^6*x^6 - 16*c^8*d^5*x^3 - 8*c^9*d^4)*sqrt(-d^4
/c^13))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)))/(d^7*x^4 + c*d^6*x)) - (1/108)^(1/
6)*c^2*x^2*(-d^4/c^13)^(1/6)*log((d^9*x^9 + 60*c*d^8*x^6 - 32*c^3*d^6 - 24*(1/4)^(2/3)*(c^9*d^6*x^8 - 7*c^10*d
^5*x^5 - 8*c^11*d^4*x^2)*(-d^4/c^13)^(2/3) + 24*(1/4)^(1/3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x)*(-d^4/
c^13)^(1/3) + 12*(9*(1/108)^(1/6)*c^3*d^7*x^5*(-d^4/c^13)^(1/6) - 18*(1/108)^(5/6)*(c^11*d^5*x^7 + 2*c^12*d^4*
x^4 - 8*c^13*d^3*x)*(-d^4/c^13)^(5/6) - sqrt(1/3)*(c^7*d^6*x^6 - 16*c^8*d^5*x^3 - 8*c^9*d^4)*sqrt(-d^4/c^13))*
sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) + (1/108)^(1/6)*c^2*x^2*(-d^4/c^13)^(1/6)*l
og((d^9*x^9 + 60*c*d^8*x^6 - 32*c^3*d^6 - 24*(1/4)^(2/3)*(c^9*d^6*x^8 - 7*c^10*d^5*x^5 - 8*c^11*d^4*x^2)*(-d^4
/c^13)^(2/3) + 24*(1/4)^(1/3)*(c^5*d^7*x^7 + 5*c^6*d^6*x^4 + 4*c^7*d^5*x)*(-d^4/c^13)^(1/3) - 12*(9*(1/108)^(1
/6)*c^3*d^7*x^5*(-d^4/c^13)^(1/6) - 18*(1/108)^(5/6)*(c^11*d^5*x^7 + 2*c^12*d^4*x^4 - 8*c^13*d^3*x)*(-d^4/c^13
)^(5/6) - sqrt(1/3)*(c^7*d^6*x^6 - 16*c^8*d^5*x^3 - 8*c^9*d^4)*sqrt(-d^4/c^13))*sqrt(d*x^3 + c))/(d^3*x^9 + 12
*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) + 2*(1/108)^(1/6)*c^2*x^2*(-d^4/c^13)^(1/6)*log((d^6*x^9 - 66*c*d^5*x^6 -
72*c^2*d^4*x^3 - 32*c^3*d^3 - 24*(1/4)^(2/3)*(c^9*d^3*x^8 - 7*c^10*d^2*x^5 - 8*c^11*d*x^2)*(-d^4/c^13)^(2/3)
- 48*(1/4)^(1/3)*(c^5*d^4*x^7 - c^6*d^3*x^4 - 2*c^7*d^2*x)*(-d^4/c^13)^(1/3) + 6*(18*(1/108)^(1/6)*c^3*d^4*x^5
*(-d^4/c^13)^(1/6) + 36*(1/108)^(5/6)*(c^11*d^2*x^7 - 16*c^12*d*x^4 - 8*c^13*x)*(-d^4/c^13)^(5/6) + sqrt(1/3)*
(5*c^7*d^3*x^6 - 20*c^8*d^2*x^3 - 16*c^9*d)*sqrt(-d^4/c^13))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2
*d*x^3 + 64*c^3)) - 2*(1/108)^(1/6)*c^2*x^2*(-d^4/c^13)^(1/6)*log((d^6*x^9 - 66*c*d^5*x^6 - 72*c^2*d^4*x^3 - 3
2*c^3*d^3 - 24*(1/4)^(2/3)*(c^9*d^3*x^8 - 7*c^10*d^2*x^5 - 8*c^11*d*x^2)*(-d^4/c^13)^(2/3) - 48*(1/4)^(1/3)*(c
^5*d^4*x^7 - c^6*d^3*x^4 - 2*c^7*d^2*x)*(-d^4/c^13)^(1/3) - 6*(18*(1/108)^(1/6)*c^3*d^4*x^5*(-d^4/c^13)^(1/6)
+ 36*(1/108)^(5/6)*(c^11*d^2*x^7 - 16*c^12*d*x^4 - 8*c^13*x)*(-d^4/c^13)^(5/6) + sqrt(1/3)*(5*c^7*d^3*x^6 - 20
*c^8*d^2*x^3 - 16*c^9*d)*sqrt(-d^4/c^13))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) -
60*sqrt(d)*x^2*weierstrassPInverse(0, -4*c/d, x) - 36*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}{x^{3} \sqrt {c + d x^{3}} \cdot \left (4 c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
Integral(1/(x**3*sqrt(c + d*x**3)*(4*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+4*c)/(d*x^3+c)^(1/2),x, algorithm="giac")
[Out]
integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + 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 (d\,x^3+4\,c\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)*(4*c + d*x^3)),x)
[Out]
int(1/(x^3*(c + d*x^3)^(1/2)*(4*c + d*x^3)), x)
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