3.4.22 \(\int \frac {1}{(8 c-d x^3) \sqrt {c+d x^3}} \, dx\) [322]

Optimal. Leaf size=64 \[ \frac {x \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c \sqrt {c+d x^3}} \]

[Out]

1/8*x*AppellF1(1/3,1/2,1,4/3,-d*x^3/c,1/8*d*x^3/c)*(1+d*x^3/c)^(1/2)/c/(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 {1}{2};\frac {4}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]

[Out]

(x*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1, 1/2, 4/3, (d*x^3)/(8*c), -((d*x^3)/c)])/(8*c*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 ) \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {c+d x^3}}\\ &=\frac {x \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{8 c \sqrt {c+d x^3}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(166\) vs. \(2(64)=128\).
time = 10.10, size = 166, normalized size = 2.59 \begin {gather*} \frac {32 c x 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 ) \sqrt {c+d x^3} \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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]

[Out]

(32*c*x*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*Sqrt[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)])))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 6.
time = 0.30, size = 416, normalized size = 6.50

method	result	size
default	\(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{27 d^{3} c}\)	\(416\)
elliptic	\(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{27 d^{3} c}\)	\(416\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/27*I/d^3/c*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))*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),-1/18/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)),_alpha=Roo
tOf(_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)^(1/2),x, algorithm="maxima")

[Out]

-integrate(1/(sqrt(d*x^3 + c)*(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. 2448 vs. \(2 (50) = 100\).
time = 6.36, size = 2448, normalized size = 38.25 \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)^(1/2),x, algorithm="fricas")

[Out]

1/432*(4*sqrt(3)*c*d*(1/(c^7*d^2))^(1/6)*arctan(1/9*((9*sqrt(3)*c^6*d^3*x^5*(1/(c^7*d^2))^(5/6) + 3*sqrt(3)*(5
*c^4*d^2*x^4 + 8*c^5*d*x)*sqrt(1/(c^7*d^2)) - sqrt(3)*(c*d^2*x^6 - 40*c^2*d*x^3 - 32*c^3)*(1/(c^7*d^2))^(1/6))
*sqrt(d*x^3 + c) - (12*sqrt(3)*(c^5*d^3*x^6 - c^6*d^2*x^3 - 2*c^7*d)*(1/(c^7*d^2))^(2/3) + 18*sqrt(3)*(c^3*d^2
*x^5 + c^4*d*x^2)*(1/(c^7*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^6*d^3*x^5 + 2*c^7*d^2*x^2)*(1/(c^7*d^2))^(5/6) + 3*sqrt(3)*(7*c^4*d^2*x^4 + 4*c^5*d*x)*sqrt(1/(c^7*d^2)) +
sqrt(3)*(c*d^2*x^6 + 32*c^2*d*x^3 + 40*c^3)*(1/(c^7*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^5*d^4*x^8 + 20*c^6*d^3*x^5 - 8*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c
^6*d^4*x^7 - 28*c^7*d^3*x^4 - 272*c^8*d^2*x)*(1/(c^7*d^2))^(5/6) + 4*(c^4*d^3*x^6 + 41*c^5*d^2*x^3 + 40*c^6*d)
*sqrt(1/(c^7*d^2)) - 24*(c^2*d^2*x^5 + c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) - 18*(c^3*d^3*x^7 - 52*c^4*d^2*x^4 - 80
*c^5*d*x)*(1/(c^7*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*d*(1/(c^7*d^2))^(1/6)*arctan(1/9*((9*sqrt(3)*c^6*d^3*x^5*(1/(c^7*d^2))^(5/6) + 3*sqrt(3)*
(5*c^4*d^2*x^4 + 8*c^5*d*x)*sqrt(1/(c^7*d^2)) - sqrt(3)*(c*d^2*x^6 - 40*c^2*d*x^3 - 32*c^3)*(1/(c^7*d^2))^(1/6
))*sqrt(d*x^3 + c) + (12*sqrt(3)*(c^5*d^3*x^6 - c^6*d^2*x^3 - 2*c^7*d)*(1/(c^7*d^2))^(2/3) + 18*sqrt(3)*(c^3*d
^2*x^5 + c^4*d*x^2)*(1/(c^7*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^6*d^3*x^5 + 2*c^7*d^2*x^2)*(1/(c^7*d^2))^(5/6) + 3*sqrt(3)*(7*c^4*d^2*x^4 + 4*c^5*d*x)*sqrt(1/(c^7*d^2))
+ sqrt(3)*(c*d^2*x^6 + 32*c^2*d*x^3 + 40*c^3)*(1/(c^7*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^5*d^4*x^8 + 20*c^6*d^3*x^5 - 8*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) - 6*sqrt(d*x^3 + c)*(
(c^6*d^4*x^7 - 28*c^7*d^3*x^4 - 272*c^8*d^2*x)*(1/(c^7*d^2))^(5/6) + 4*(c^4*d^3*x^6 + 41*c^5*d^2*x^3 + 40*c^6*
d)*sqrt(1/(c^7*d^2)) - 24*(c^2*d^2*x^5 + c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) - 18*(c^3*d^3*x^7 - 52*c^4*d^2*x^4 -
80*c^5*d*x)*(1/(c^7*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)) + 2*c*d*(1/(c^7*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^5*d^4*x^8
+ 38*c^6*d^3*x^5 + 64*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^4*x^7 + 80*c^7*d^3*x^4 + 1
60*c^8*d^2*x)*(1/(c^7*d^2))^(5/6) + (7*c^4*d^3*x^6 + 152*c^5*d^2*x^3 + 64*c^6*d)*sqrt(1/(c^7*d^2)) + 6*(5*c^2*
d^2*x^5 + 32*c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) + 18*(5*c^3*d^3*x^7 + 64*c^4*d^2*x^4 + 32*c^5*d*x)*(1/(c^7*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*d*(1/(c^7*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^5*d^4*x^8 + 38*c^6*d^3*x^5 + 64*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) - 6*
sqrt(d*x^3 + c)*((c^6*d^4*x^7 + 80*c^7*d^3*x^4 + 160*c^8*d^2*x)*(1/(c^7*d^2))^(5/6) + (7*c^4*d^3*x^6 + 152*c^5
*d^2*x^3 + 64*c^6*d)*sqrt(1/(c^7*d^2)) + 6*(5*c^2*d^2*x^5 + 32*c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) + 18*(5*c^3*d^3
*x^7 + 64*c^4*d^2*x^4 + 32*c^5*d*x)*(1/(c^7*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) +
c*d*(1/(c^7*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^5*d^4*x^8 + 20*c^6*d
^3*x^5 - 8*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^4*x^7 - 28*c^7*d^3*x^4 - 272*c^8*d^2*x
)*(1/(c^7*d^2))^(5/6) + 4*(c^4*d^3*x^6 + 41*c^5*d^2*x^3 + 40*c^6*d)*sqrt(1/(c^7*d^2)) - 24*(c^2*d^2*x^5 + c^3*
d*x^2)*(1/(c^7*d^2))^(1/6)) - 18*(c^3*d^3*x^7 - 52*c^4*d^2*x^4 - 80*c^5*d*x)*(1/(c^7*d^2))^(1/3))/(d^3*x^9 - 2
4*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - c*d*(1/(c^7*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^5*d^4*x^8 + 20*c^6*d^3*x^5 - 8*c^7*d^2*x^2)*(1/(c^7*d^2))^(2/3) - 6*sqrt(d*x^3 + c)*((c^6
*d^4*x^7 - 28*c^7*d^3*x^4 - 272*c^8*d^2*x)*(1/(c^7*d^2))^(5/6) + 4*(c^4*d^3*x^6 + 41*c^5*d^2*x^3 + 40*c^6*d)*s
qrt(1/(c^7*d^2)) - 24*(c^2*d^2*x^5 + c^3*d*x^2)*(1/(c^7*d^2))^(1/6)) - 18*(c^3*d^3*x^7 - 52*c^4*d^2*x^4 - 80*c
^5*d*x)*(1/(c^7*d^2))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 72*sqrt(d)*weierstrassPInve
rse(0, -4*c/d, x))/(c*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- 8 c \sqrt {c + d x^{3}} + d x^{3} \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)**(1/2),x)

[Out]

-Integral(1/(-8*c*sqrt(c + d*x**3) + d*x**3*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)^(1/2),x, algorithm="giac")

[Out]

integrate(-1/(sqrt(d*x^3 + c)*(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}{\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/((c + d*x^3)^(1/2)*(8*c - d*x^3)),x)

[Out]

int(1/((c + d*x^3)^(1/2)*(8*c - d*x^3)), x)

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