3.3.68 \(\int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx\) [268]
Optimal. Leaf size=64 \[ \frac {x \sqrt {c+d x^3} F_1\left (\frac {1}{3};1,-\frac {1}{2};\frac {4}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \sqrt {1+\frac {d x^3}{c}}} \]
[Out]
1/4*x*AppellF1(1/3,-1/2,1,4/3,-d*x^3/c,-1/4*d*x^3/c)*(d*x^3+c)^(1/2)/c/(1+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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {441, 440}
\begin {gather*} \frac {x \sqrt {c+d x^3} F_1\left (\frac {1}{3};1,-\frac {1}{2};\frac {4}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \sqrt {\frac {d x^3}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[c + d*x^3]/(4*c + d*x^3),x]
[Out]
(x*Sqrt[c + d*x^3]*AppellF1[1/3, 1, -1/2, 4/3, -1/4*(d*x^3)/c, -((d*x^3)/c)])/(4*c*Sqrt[1 + (d*x^3)/c])
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 {\sqrt {c+d x^3}}{4 c+d x^3} \, dx &=\frac {\sqrt {c+d x^3} \int \frac {\sqrt {1+\frac {d x^3}{c}}}{4 c+d x^3} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=\frac {x \sqrt {c+d x^3} F_1\left (\frac {1}{3};1,-\frac {1}{2};\frac {4}{3};-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \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. \(165\) vs. \(2(64)=128\).
time = 8.83, size = 165, normalized size = 2.58 \begin {gather*} \frac {16 c x \sqrt {c+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 {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[c + d*x^3]/(4*c + d*x^3),x]
[Out]
(16*c*x*Sqrt[c + 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*Appell
F1[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, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
6.
time = 0.35, size = 696, normalized size = 10.88 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x^3+c)^(1/2)/(d*x^3+4*c),x,method=_RETURNVERBOSE)
[Out]
-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)*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))+1/3*I/d^3*2^(1/2)*sum(1/_alph
a^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)/(d*x^3+4*c),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2187 vs.
\(2 (50) = 100\).
time = 4.25, size = 2187, normalized size = 34.17 \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)/(d*x^3+4*c),x, algorithm="fricas")
[Out]
1/24*(4*sqrt(3)*(1/108)^(1/6)*d*(-1/(c*d^2))^(1/6)*arctan(1/3*((108*sqrt(3)*(1/108)^(5/6)*c*d^2*x^2*(-1/(c*d^2
))^(5/6) + 3*sqrt(3)*sqrt(1/3)*c*d*x*sqrt(-1/(c*d^2)) + sqrt(3)*(1/108)^(1/6)*(d*x^3 + 4*c)*(-1/(c*d^2))^(1/6)
)*sqrt(d*x^3 + c) - (4*sqrt(3)*(1/4)^(2/3)*(c*d^2*x^3 + c^2*d)*(-1/(c*d^2))^(2/3) - sqrt(3)*(d*x^4 + c*x) - (1
08*sqrt(3)*(1/108)^(5/6)*c*d^2*x^2*(-1/(c*d^2))^(5/6) + 3*sqrt(3)*sqrt(1/3)*c*d*x*sqrt(-1/(c*d^2)) - sqrt(3)*(
1/108)^(1/6)*(d*x^3 - 2*c)*(-1/(c*d^2))^(1/6))*sqrt(d*x^3 + c))*sqrt((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 - 24*(1/
4)^(2/3)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2)*(-1/(c*d^2))^(2/3) + 24*(1/4)^(1/3)*(c*d^3*x^7 + 5*c^2*d^
2*x^4 + 4*c^3*d*x)*(-1/(c*d^2))^(1/3) + 12*(9*(1/108)^(1/6)*c*d^2*x^5*(-1/(c*d^2))^(1/6) - 18*(1/108)^(5/6)*(c
*d^4*x^7 + 2*c^2*d^3*x^4 - 8*c^3*d^2*x)*(-1/(c*d^2))^(5/6) - sqrt(1/3)*(c*d^3*x^6 - 16*c^2*d^2*x^3 - 8*c^3*d)*
sqrt(-1/(c*d^2)))*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*x^4 + c*x)) + 4*sqrt(
3)*(1/108)^(1/6)*d*(-1/(c*d^2))^(1/6)*arctan(1/3*((108*sqrt(3)*(1/108)^(5/6)*c*d^2*x^2*(-1/(c*d^2))^(5/6) + 3*
sqrt(3)*sqrt(1/3)*c*d*x*sqrt(-1/(c*d^2)) + sqrt(3)*(1/108)^(1/6)*(d*x^3 + 4*c)*(-1/(c*d^2))^(1/6))*sqrt(d*x^3
+ c) + (4*sqrt(3)*(1/4)^(2/3)*(c*d^2*x^3 + c^2*d)*(-1/(c*d^2))^(2/3) - sqrt(3)*(d*x^4 + c*x) + (108*sqrt(3)*(1
/108)^(5/6)*c*d^2*x^2*(-1/(c*d^2))^(5/6) + 3*sqrt(3)*sqrt(1/3)*c*d*x*sqrt(-1/(c*d^2)) - sqrt(3)*(1/108)^(1/6)*
(d*x^3 - 2*c)*(-1/(c*d^2))^(1/6))*sqrt(d*x^3 + c))*sqrt((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 - 24*(1/4)^(2/3)*(c*d
^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2)*(-1/(c*d^2))^(2/3) + 24*(1/4)^(1/3)*(c*d^3*x^7 + 5*c^2*d^2*x^4 + 4*c^3
*d*x)*(-1/(c*d^2))^(1/3) - 12*(9*(1/108)^(1/6)*c*d^2*x^5*(-1/(c*d^2))^(1/6) - 18*(1/108)^(5/6)*(c*d^4*x^7 + 2*
c^2*d^3*x^4 - 8*c^3*d^2*x)*(-1/(c*d^2))^(5/6) - sqrt(1/3)*(c*d^3*x^6 - 16*c^2*d^2*x^3 - 8*c^3*d)*sqrt(-1/(c*d^
2)))*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*x^4 + c*x)) - (1/108)^(1/6)*d*(-1/
(c*d^2))^(1/6)*log((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 - 24*(1/4)^(2/3)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^
2)*(-1/(c*d^2))^(2/3) + 24*(1/4)^(1/3)*(c*d^3*x^7 + 5*c^2*d^2*x^4 + 4*c^3*d*x)*(-1/(c*d^2))^(1/3) + 12*(9*(1/1
08)^(1/6)*c*d^2*x^5*(-1/(c*d^2))^(1/6) - 18*(1/108)^(5/6)*(c*d^4*x^7 + 2*c^2*d^3*x^4 - 8*c^3*d^2*x)*(-1/(c*d^2
))^(5/6) - sqrt(1/3)*(c*d^3*x^6 - 16*c^2*d^2*x^3 - 8*c^3*d)*sqrt(-1/(c*d^2)))*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)*d*(-1/(c*d^2))^(1/6)*log((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 -
24*(1/4)^(2/3)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2)*(-1/(c*d^2))^(2/3) + 24*(1/4)^(1/3)*(c*d^3*x^7 + 5
*c^2*d^2*x^4 + 4*c^3*d*x)*(-1/(c*d^2))^(1/3) - 12*(9*(1/108)^(1/6)*c*d^2*x^5*(-1/(c*d^2))^(1/6) - 18*(1/108)^(
5/6)*(c*d^4*x^7 + 2*c^2*d^3*x^4 - 8*c^3*d^2*x)*(-1/(c*d^2))^(5/6) - sqrt(1/3)*(c*d^3*x^6 - 16*c^2*d^2*x^3 - 8*
c^3*d)*sqrt(-1/(c*d^2)))*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)*
d*(-1/(c*d^2))^(1/6)*log((d^3*x^9 - 66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 - 24*(1/4)^(2/3)*(c*d^4*x^8 - 7*c^2*d
^3*x^5 - 8*c^3*d^2*x^2)*(-1/(c*d^2))^(2/3) - 48*(1/4)^(1/3)*(c*d^3*x^7 - c^2*d^2*x^4 - 2*c^3*d*x)*(-1/(c*d^2))
^(1/3) + 6*(18*(1/108)^(1/6)*c*d^2*x^5*(-1/(c*d^2))^(1/6) + 36*(1/108)^(5/6)*(c*d^4*x^7 - 16*c^2*d^3*x^4 - 8*c
^3*d^2*x)*(-1/(c*d^2))^(5/6) + sqrt(1/3)*(5*c*d^3*x^6 - 20*c^2*d^2*x^3 - 16*c^3*d)*sqrt(-1/(c*d^2)))*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)*d*(-1/(c*d^2))^(1/6)*log((d^3*x^9
- 66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 - 24*(1/4)^(2/3)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2)*(-1/(c*d^2
))^(2/3) - 48*(1/4)^(1/3)*(c*d^3*x^7 - c^2*d^2*x^4 - 2*c^3*d*x)*(-1/(c*d^2))^(1/3) - 6*(18*(1/108)^(1/6)*c*d^2
*x^5*(-1/(c*d^2))^(1/6) + 36*(1/108)^(5/6)*(c*d^4*x^7 - 16*c^2*d^3*x^4 - 8*c^3*d^2*x)*(-1/(c*d^2))^(5/6) + sqr
t(1/3)*(5*c*d^3*x^6 - 20*c^2*d^2*x^3 - 16*c^3*d)*sqrt(-1/(c*d^2)))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 +
48*c^2*d*x^3 + 64*c^3)) + 24*sqrt(d)*weierstrassPInverse(0, -4*c/d, x))/d
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{3}}}{4 c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**3+c)**(1/2)/(d*x**3+4*c),x)
[Out]
Integral(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((d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="giac")
[Out]
integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c), 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}}{d\,x^3+4\,c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^3)^(1/2)/(4*c + d*x^3),x)
[Out]
int((c + d*x^3)^(1/2)/(4*c + d*x^3), x)
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