3.3.69 \(\int \frac {\sqrt {c+d x^3}}{x^3 (4 c+d x^3)} \, dx\) [269]
Optimal. Leaf size=66 \[ -\frac {\sqrt {c+d x^3} F_1\left (-\frac {2}{3};1,-\frac {1}{2};\frac {1}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{8 c x^2 \sqrt {1+\frac {d x^3}{c}}} \]
[Out]
-1/8*AppellF1(-2/3,-1/2,1,1/3,-d*x^3/c,-1/4*d*x^3/c)*(d*x^3+c)^(1/2)/c/x^2/(1+d*x^3/c)^(1/2)
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Rubi [A]
time = 0.03, 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 {c+d x^3} 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 {\frac {d x^3}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]
[Out]
-1/8*(Sqrt[c + d*x^3]*AppellF1[-2/3, 1, -1/2, 1/3, -1/4*(d*x^3)/c, -((d*x^3)/c)])/(c*x^2*Sqrt[1 + (d*x^3)/c])
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 {\sqrt {c+d x^3}}{x^3 \left (4 c+d x^3\right )} \, dx &=\frac {\sqrt {c+d x^3} \int \frac {\sqrt {1+\frac {d x^3}{c}}}{x^3 \left (4 c+d x^3\right )} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=-\frac {\sqrt {c+d x^3} 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 {1+\frac {d x^3}{c}}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(244\) vs. \(2(66)=132\).
time = 20.10, size = 244, normalized size = 3.70 \begin {gather*} \frac {-32 c \left (c+d x^3\right )-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 )+\frac {2048 c^3 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 c^2 x^2 \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]
[Out]
(-32*c*(c + d*x^3) - 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] + (2
048*c^3*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*c^2*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 = 1002, normalized size = 15.18 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x,method=_RETURNVERBOSE)
[Out]
1/4/c*(-1/2*(d*x^3+c)^(1/2)/x^2-1/2*I*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)))
-1/4*d/c*(-2/3*I*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)*Ell
ipticF(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))+1/3*I/d^3*2^(1/2)*s
um(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),1/6/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)*_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+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((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2372 vs.
\(2 (52) = 104\).
time = 4.66, size = 2372, normalized size = 35.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x, algorithm="fricas")
[Out]
-1/96*(4*sqrt(3)*(1/108)^(1/6)*c*x^2*(-d^4/c^7)^(1/6)*arctan(1/3*((108*sqrt(3)*(1/108)^(5/6)*c^6*d^3*x^2*(-d^4
/c^7)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^4*d^4*x*sqrt(-d^4/c^7) + sqrt(3)*(1/108)^(1/6)*(c*d^6*x^3 + 4*c^2*d^5)*(-d
^4/c^7)^(1/6))*sqrt(d*x^3 + c) - (4*sqrt(3)*(1/4)^(2/3)*(c^5*d*x^3 + c^6)*(-d^4/c^7)^(2/3) - sqrt(3)*(d^4*x^4
+ c*d^3*x) - (108*sqrt(3)*(1/108)^(5/6)*c^6*x^2*(-d^4/c^7)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^4*d*x*sqrt(-d^4/c^7)
- sqrt(3)*(1/108)^(1/6)*(c*d^3*x^3 - 2*c^2*d^2)*(-d^4/c^7)^(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^5*d^6*x^8 - 7*c^6*d^5*x^5 - 8*c^7*d^4*x^2)*(-d^4/c^7)^(2/3) + 24*(1/4)^(1/3
)*(c^3*d^7*x^7 + 5*c^4*d^6*x^4 + 4*c^5*d^5*x)*(-d^4/c^7)^(1/3) + 12*(9*(1/108)^(1/6)*c^2*d^7*x^5*(-d^4/c^7)^(1
/6) - 18*(1/108)^(5/6)*(c^6*d^5*x^7 + 2*c^7*d^4*x^4 - 8*c^8*d^3*x)*(-d^4/c^7)^(5/6) - sqrt(1/3)*(c^4*d^6*x^6 -
16*c^5*d^5*x^3 - 8*c^6*d^4)*sqrt(-d^4/c^7))*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*x^2*(-d^4/c^7)^(1/6)*arctan(1/3*((108*sqrt(3)*(1/108)^(5/6
)*c^6*d^3*x^2*(-d^4/c^7)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^4*d^4*x*sqrt(-d^4/c^7) + sqrt(3)*(1/108)^(1/6)*(c*d^6*x
^3 + 4*c^2*d^5)*(-d^4/c^7)^(1/6))*sqrt(d*x^3 + c) + (4*sqrt(3)*(1/4)^(2/3)*(c^5*d*x^3 + c^6)*(-d^4/c^7)^(2/3)
- sqrt(3)*(d^4*x^4 + c*d^3*x) + (108*sqrt(3)*(1/108)^(5/6)*c^6*x^2*(-d^4/c^7)^(5/6) + 3*sqrt(3)*sqrt(1/3)*c^4*
d*x*sqrt(-d^4/c^7) - sqrt(3)*(1/108)^(1/6)*(c*d^3*x^3 - 2*c^2*d^2)*(-d^4/c^7)^(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^5*d^6*x^8 - 7*c^6*d^5*x^5 - 8*c^7*d^4*x^2)*(-d^4/c^7)^(2
/3) + 24*(1/4)^(1/3)*(c^3*d^7*x^7 + 5*c^4*d^6*x^4 + 4*c^5*d^5*x)*(-d^4/c^7)^(1/3) - 12*(9*(1/108)^(1/6)*c^2*d^
7*x^5*(-d^4/c^7)^(1/6) - 18*(1/108)^(5/6)*(c^6*d^5*x^7 + 2*c^7*d^4*x^4 - 8*c^8*d^3*x)*(-d^4/c^7)^(5/6) - sqrt(
1/3)*(c^4*d^6*x^6 - 16*c^5*d^5*x^3 - 8*c^6*d^4)*sqrt(-d^4/c^7))*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*x^2*(-d^4/c^7)^(1/6)*log((d^9*x^9 + 60*c*d^8*x^6
- 32*c^3*d^6 - 24*(1/4)^(2/3)*(c^5*d^6*x^8 - 7*c^6*d^5*x^5 - 8*c^7*d^4*x^2)*(-d^4/c^7)^(2/3) + 24*(1/4)^(1/3)*
(c^3*d^7*x^7 + 5*c^4*d^6*x^4 + 4*c^5*d^5*x)*(-d^4/c^7)^(1/3) + 12*(9*(1/108)^(1/6)*c^2*d^7*x^5*(-d^4/c^7)^(1/6
) - 18*(1/108)^(5/6)*(c^6*d^5*x^7 + 2*c^7*d^4*x^4 - 8*c^8*d^3*x)*(-d^4/c^7)^(5/6) - sqrt(1/3)*(c^4*d^6*x^6 - 1
6*c^5*d^5*x^3 - 8*c^6*d^4)*sqrt(-d^4/c^7))*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*x^2*(-d^4/c^7)^(1/6)*log((d^9*x^9 + 60*c*d^8*x^6 - 32*c^3*d^6 - 24*(1/4)^(2/3)*(c^5*d^6*x^8
- 7*c^6*d^5*x^5 - 8*c^7*d^4*x^2)*(-d^4/c^7)^(2/3) + 24*(1/4)^(1/3)*(c^3*d^7*x^7 + 5*c^4*d^6*x^4 + 4*c^5*d^5*x)
*(-d^4/c^7)^(1/3) - 12*(9*(1/108)^(1/6)*c^2*d^7*x^5*(-d^4/c^7)^(1/6) - 18*(1/108)^(5/6)*(c^6*d^5*x^7 + 2*c^7*d
^4*x^4 - 8*c^8*d^3*x)*(-d^4/c^7)^(5/6) - sqrt(1/3)*(c^4*d^6*x^6 - 16*c^5*d^5*x^3 - 8*c^6*d^4)*sqrt(-d^4/c^7))*
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*x^2*(-d^4/c^7)^(1/6)*lo
g((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^5*d^3*x^8 - 7*c^6*d^2*x^5 - 8*c^7*
d*x^2)*(-d^4/c^7)^(2/3) - 48*(1/4)^(1/3)*(c^3*d^4*x^7 - c^4*d^3*x^4 - 2*c^5*d^2*x)*(-d^4/c^7)^(1/3) + 6*(18*(1
/108)^(1/6)*c^2*d^4*x^5*(-d^4/c^7)^(1/6) + 36*(1/108)^(5/6)*(c^6*d^2*x^7 - 16*c^7*d*x^4 - 8*c^8*x)*(-d^4/c^7)^
(5/6) + sqrt(1/3)*(5*c^4*d^3*x^6 - 20*c^5*d^2*x^3 - 16*c^6*d)*sqrt(-d^4/c^7))*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*x^2*(-d^4/c^7)^(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^5*d^3*x^8 - 7*c^6*d^2*x^5 - 8*c^7*d*x^2)*(-d^4/c^7)^(2/3) - 48*(1/
4)^(1/3)*(c^3*d^4*x^7 - c^4*d^3*x^4 - 2*c^5*d^2*x)*(-d^4/c^7)^(1/3) - 6*(18*(1/108)^(1/6)*c^2*d^4*x^5*(-d^4/c^
7)^(1/6) + 36*(1/108)^(5/6)*(c^6*d^2*x^7 - 16*c^7*d*x^4 - 8*c^8*x)*(-d^4/c^7)^(5/6) + sqrt(1/3)*(5*c^4*d^3*x^6
- 20*c^5*d^2*x^3 - 16*c^6*d)*sqrt(-d^4/c^7))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3
)) - 12*sqrt(d)*x^2*weierstrassPInverse(0, -4*c/d, x) + 12*sqrt(d*x^3 + c))/(c*x^2)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{3}}}{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((d*x**3+c)**(1/2)/x**3/(d*x**3+4*c),x)
[Out]
Integral(sqrt(c + d*x**3)/(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((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {d\,x^3+c}}{x^3\,\left (d\,x^3+4\,c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^3)^(1/2)/(x^3*(4*c + d*x^3)),x)
[Out]
int((c + d*x^3)^(1/2)/(x^3*(4*c + d*x^3)), x)
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