3.4.41 \(\int \frac {1}{x^6 (8 c-d x^3) (c+d x^3)^{3/2}} \, dx\) [341]

Optimal. Leaf size=66 \[ -\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {5}{3};1,\frac {3}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c^2 x^5 \sqrt {c+d x^3}} \]

[Out]

-1/40*AppellF1(-5/3,3/2,1,-2/3,-d*x^3/c,1/8*d*x^3/c)*(1+d*x^3/c)^(1/2)/c^2/x^5/(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 = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {525, 524} \begin {gather*} -\frac {\sqrt {\frac {d x^3}{c}+1} F_1\left (-\frac {5}{3};1,\frac {3}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c^2 x^5 \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]

[Out]

-1/40*(Sqrt[1 + (d*x^3)/c]*AppellF1[-5/3, 1, 3/2, -2/3, (d*x^3)/(8*c), -((d*x^3)/c)])/(c^2*x^5*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^6 \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}{x^6 \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 {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {5}{3};1,\frac {3}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{40 c^2 x^5 \sqrt {c+d x^3}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(66)=132\).
time = 20.15, size = 261, normalized size = 3.95 \begin {gather*} \frac {-2981 d^3 x^9 \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 )+64 c \left (-432 c^2+1269 c d x^3+2981 d^2 x^6+\frac {382528 c^2 d^2 x^6 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 )}{1105920 c^5 x^5 \sqrt {c+d x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^6*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]

[Out]

(-2981*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 64*c*(-432*c^2 +
1269*c*d*x^3 + 2981*d^2*x^6 + (382528*c^2*d^2*x^6*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)])))))/(1105920*c^5*x
^5*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 = 1402, normalized size = 21.24

method	result	size
elliptic	\(-\frac {\sqrt {d \,x^{3}+c}}{40 c^{3} x^{5}}+\frac {63 d \sqrt {d \,x^{3}+c}}{640 c^{4} x^{2}}+\frac {2 d^{2} x}{27 c^{4} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {2981 i d \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{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}}}}\, \sqrt {\frac {x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}}{-\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}}}\, \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}}}}\, \EllipticF \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}, \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 )}{51840 c^{4} \sqrt {d \,x^{3}+c}}-\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 )}{15552 d \,c^{4}}\)	\(757\)
risch	\(\text {Expression too large to display}\)	\(1040\)
default	\(\text {Expression too large to display}\)	\(1402\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8/c*(-1/5/c^2*(d*x^3+c)^(1/2)/x^5+17/20/c^3*d*(d*x^3+c)^(1/2)/x^2+2/3*d^2/c^3*x/((x^3+c/d)*d)^(1/2)-91/180*I
/c^3*d*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)))+1/64*d/c^2*(-1/2/c^2*(d*x^3+c)
^(1/2)/x^2-2/3*d*x/c^2/((x^3+c/d)*d)^(1/2)+7/18*I/c^2*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)))-1/64/c^2*d^2*(-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))*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)*_alph
a+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-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/x^6/(-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^6), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2697 vs. \(2 (52) = 104\).
time = 16.56, size = 2697, normalized size = 40.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="fricas")

[Out]

1/1244160*(20*sqrt(3)*(c^4*d*x^8 + c^5*x^5)*(d^10/c^25)^(1/6)*arctan(1/9*((9*sqrt(3)*c^21*d^9*x^5*(d^10/c^25)^
(5/6) + 3*sqrt(3)*(5*c^13*d^12*x^4 + 8*c^14*d^11*x)*sqrt(d^10/c^25) - sqrt(3)*(c^4*d^16*x^6 - 40*c^5*d^15*x^3
- 32*c^6*d^14)*(d^10/c^25)^(1/6))*sqrt(d*x^3 + c) - (12*sqrt(3)*(c^17*d^3*x^6 - c^18*d^2*x^3 - 2*c^19*d)*(d^10
/c^25)^(2/3) + 18*sqrt(3)*(c^9*d^6*x^5 + c^10*d^5*x^2)*(d^10/c^25)^(1/3) + 3*sqrt(3)*(d^10*x^7 + 5*c*d^9*x^4 +
4*c^2*d^8*x) - sqrt(d*x^3 + c)*(9*sqrt(3)*(c^21*d*x^5 + 2*c^22*x^2)*(d^10/c^25)^(5/6) + 3*sqrt(3)*(7*c^13*d^4
*x^4 + 4*c^14*d^3*x)*sqrt(d^10/c^25) + sqrt(3)*(c^4*d^8*x^6 + 32*c^5*d^7*x^3 + 40*c^6*d^6)*(d^10/c^25)^(1/6)))
*sqrt((d^19*x^9 - 276*c*d^18*x^6 - 1608*c^2*d^17*x^3 - 1088*c^3*d^16 + 18*(c^17*d^12*x^8 + 20*c^18*d^11*x^5 -
8*c^19*d^10*x^2)*(d^10/c^25)^(2/3) + 6*sqrt(d*x^3 + c)*((c^21*d^10*x^7 - 28*c^22*d^9*x^4 - 272*c^23*d^8*x)*(d^
10/c^25)^(5/6) + 4*(c^13*d^13*x^6 + 41*c^14*d^12*x^3 + 40*c^15*d^11)*sqrt(d^10/c^25) - 24*(c^5*d^16*x^5 + c^6*
d^15*x^2)*(d^10/c^25)^(1/6)) - 18*(c^9*d^15*x^7 - 52*c^10*d^14*x^4 - 80*c^11*d^13*x)*(d^10/c^25)^(1/3))/(d^3*x
^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)))/(d^18*x^7 - 7*c*d^17*x^4 - 8*c^2*d^16*x)) + 20*sqrt(3)*(c^4*d*x
^8 + c^5*x^5)*(d^10/c^25)^(1/6)*arctan(1/9*((9*sqrt(3)*c^21*d^9*x^5*(d^10/c^25)^(5/6) + 3*sqrt(3)*(5*c^13*d^12
*x^4 + 8*c^14*d^11*x)*sqrt(d^10/c^25) - sqrt(3)*(c^4*d^16*x^6 - 40*c^5*d^15*x^3 - 32*c^6*d^14)*(d^10/c^25)^(1/
6))*sqrt(d*x^3 + c) + (12*sqrt(3)*(c^17*d^3*x^6 - c^18*d^2*x^3 - 2*c^19*d)*(d^10/c^25)^(2/3) + 18*sqrt(3)*(c^9
*d^6*x^5 + c^10*d^5*x^2)*(d^10/c^25)^(1/3) + 3*sqrt(3)*(d^10*x^7 + 5*c*d^9*x^4 + 4*c^2*d^8*x) + sqrt(d*x^3 + c
)*(9*sqrt(3)*(c^21*d*x^5 + 2*c^22*x^2)*(d^10/c^25)^(5/6) + 3*sqrt(3)*(7*c^13*d^4*x^4 + 4*c^14*d^3*x)*sqrt(d^10
/c^25) + sqrt(3)*(c^4*d^8*x^6 + 32*c^5*d^7*x^3 + 40*c^6*d^6)*(d^10/c^25)^(1/6)))*sqrt((d^19*x^9 - 276*c*d^18*x
^6 - 1608*c^2*d^17*x^3 - 1088*c^3*d^16 + 18*(c^17*d^12*x^8 + 20*c^18*d^11*x^5 - 8*c^19*d^10*x^2)*(d^10/c^25)^(
2/3) - 6*sqrt(d*x^3 + c)*((c^21*d^10*x^7 - 28*c^22*d^9*x^4 - 272*c^23*d^8*x)*(d^10/c^25)^(5/6) + 4*(c^13*d^13*
x^6 + 41*c^14*d^12*x^3 + 40*c^15*d^11)*sqrt(d^10/c^25) - 24*(c^5*d^16*x^5 + c^6*d^15*x^2)*(d^10/c^25)^(1/6)) -
18*(c^9*d^15*x^7 - 52*c^10*d^14*x^4 - 80*c^11*d^13*x)*(d^10/c^25)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*
x^3 - 512*c^3)))/(d^18*x^7 - 7*c*d^17*x^4 - 8*c^2*d^16*x)) + 214992*(d^2*x^8 + c*d*x^5)*sqrt(d)*weierstrassPIn
verse(0, -4*c/d, x) + 5*(c^4*d*x^8 + c^5*x^5)*(d^10/c^25)^(1/6)*log((d^19*x^9 - 276*c*d^18*x^6 - 1608*c^2*d^17
*x^3 - 1088*c^3*d^16 + 18*(c^17*d^12*x^8 + 20*c^18*d^11*x^5 - 8*c^19*d^10*x^2)*(d^10/c^25)^(2/3) + 6*sqrt(d*x^
3 + c)*((c^21*d^10*x^7 - 28*c^22*d^9*x^4 - 272*c^23*d^8*x)*(d^10/c^25)^(5/6) + 4*(c^13*d^13*x^6 + 41*c^14*d^12
*x^3 + 40*c^15*d^11)*sqrt(d^10/c^25) - 24*(c^5*d^16*x^5 + c^6*d^15*x^2)*(d^10/c^25)^(1/6)) - 18*(c^9*d^15*x^7
- 52*c^10*d^14*x^4 - 80*c^11*d^13*x)*(d^10/c^25)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) -
5*(c^4*d*x^8 + c^5*x^5)*(d^10/c^25)^(1/6)*log((d^19*x^9 - 276*c*d^18*x^6 - 1608*c^2*d^17*x^3 - 1088*c^3*d^16 +
18*(c^17*d^12*x^8 + 20*c^18*d^11*x^5 - 8*c^19*d^10*x^2)*(d^10/c^25)^(2/3) - 6*sqrt(d*x^3 + c)*((c^21*d^10*x^7
- 28*c^22*d^9*x^4 - 272*c^23*d^8*x)*(d^10/c^25)^(5/6) + 4*(c^13*d^13*x^6 + 41*c^14*d^12*x^3 + 40*c^15*d^11)*s
qrt(d^10/c^25) - 24*(c^5*d^16*x^5 + c^6*d^15*x^2)*(d^10/c^25)^(1/6)) - 18*(c^9*d^15*x^7 - 52*c^10*d^14*x^4 - 8
0*c^11*d^13*x)*(d^10/c^25)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 10*(c^4*d*x^8 + c^5*x^
5)*(d^10/c^25)^(1/6)*log((d^11*x^9 + 318*c*d^10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8 + 18*(c^17*d^4*x^8 + 38*c
^18*d^3*x^5 + 64*c^19*d^2*x^2)*(d^10/c^25)^(2/3) + 6*sqrt(d*x^3 + c)*((c^21*d^2*x^7 + 80*c^22*d*x^4 + 160*c^23
*x)*(d^10/c^25)^(5/6) + (7*c^13*d^5*x^6 + 152*c^14*d^4*x^3 + 64*c^15*d^3)*sqrt(d^10/c^25) + 6*(5*c^5*d^8*x^5 +
32*c^6*d^7*x^2)*(d^10/c^25)^(1/6)) + 18*(5*c^9*d^7*x^7 + 64*c^10*d^6*x^4 + 32*c^11*d^5*x)*(d^10/c^25)^(1/3))/
(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 10*(c^4*d*x^8 + c^5*x^5)*(d^10/c^25)^(1/6)*log((d^11*x^9
+ 318*c*d^10*x^6 + 1200*c^2*d^9*x^3 + 640*c^3*d^8 + 18*(c^17*d^4*x^8 + 38*c^18*d^3*x^5 + 64*c^19*d^2*x^2)*(d^
10/c^25)^(2/3) - 6*sqrt(d*x^3 + c)*((c^21*d^2*x^7 + 80*c^22*d*x^4 + 160*c^23*x)*(d^10/c^25)^(5/6) + (7*c^13*d^
5*x^6 + 152*c^14*d^4*x^3 + 64*c^15*d^3)*sqrt(d^10/c^25) + 6*(5*c^5*d^8*x^5 + 32*c^6*d^7*x^2)*(d^10/c^25)^(1/6)
) + 18*(5*c^9*d^7*x^7 + 64*c^10*d^6*x^4 + 32*c^11*d^5*x)*(d^10/c^25)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*
d*x^3 - 512*c^3)) + 72*(2981*d^2*x^6 + 1269*c*d*x^3 - 432*c^2)*sqrt(d*x^3 + c))/(c^4*d*x^8 + c^5*x^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- 8 c^{2} x^{6} \sqrt {c + d x^{3}} - 7 c d x^{9} \sqrt {c + d x^{3}} + d^{2} x^{12} \sqrt {c + d x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)

[Out]

-Integral(1/(-8*c**2*x**6*sqrt(c + d*x**3) - 7*c*d*x**9*sqrt(c + d*x**3) + d**2*x**12*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/x^6/(-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^6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^6\,{\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/(x^6*(c + d*x^3)^(3/2)*(8*c - d*x^3)),x)

[Out]

int(1/(x^6*(c + d*x^3)^(3/2)*(8*c - d*x^3)), x)

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