3.5.41 \(\int \frac {1}{x^6 (8 c-d x^3)^2 \sqrt {c+d x^3}} \, dx\) [441]
Optimal. Leaf size=66 \[ -\frac {\sqrt {1+\frac {d x^3}{c}} F_1\left (-\frac {5}{3};2,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{320 c^2 x^5 \sqrt {c+d x^3}} \]
[Out]
-1/320*AppellF1(-5/3,1/2,2,-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};2,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{320 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)^2*Sqrt[c + d*x^3]),x]
[Out]
-1/320*(Sqrt[1 + (d*x^3)/c]*AppellF1[-5/3, 2, 1/2, -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 )^2 \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {1}{x^6 \left (8 c-d x^3\right )^2 \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 {5}{3};2,\frac {1}{2};-\frac {2}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{320 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. \(279\) vs. \(2(66)=132\).
time = 10.15, size = 279, normalized size = 4.23 \begin {gather*} \frac {\frac {64 \left (c+d x^3\right ) \left (864 c^2-1080 c d x^3+119 d^2 x^6\right )}{c^4 x^5 \left (-8 c+d x^3\right )}-\frac {119 d^3 x^4 \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 )}{c^5}+\frac {1404928 d^2 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 )}{c^2 \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 )}}{2211840 \sqrt {c+d x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(x^6*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
[Out]
((64*(c + d*x^3)*(864*c^2 - 1080*c*d*x^3 + 119*d^2*x^6))/(c^4*x^5*(-8*c + d*x^3)) - (119*d^3*x^4*Sqrt[1 + (d*x
^3)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), (d*x^3)/(8*c)])/c^5 + (1404928*d^2*x*AppellF1[1/3, 1/2, 1, 4/3
, -((d*x^3)/c), (d*x^3)/(8*c)])/(c^2*(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)]))))/(2211840*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 = 0.44, size = 1783, normalized size = 27.02
| | |
| method |
result |
size |
| | |
| elliptic |
\(-\frac {\sqrt {d \,x^{3}+c}}{320 c^{3} x^{5}}+\frac {9 d \sqrt {d \,x^{3}+c}}{2560 c^{4} x^{2}}+\frac {d^{2} x \sqrt {d \,x^{3}+c}}{13824 c^{4} \left (-d \,x^{3}+8 c \right )}-\frac {119 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 )}{103680 c^{4} \sqrt {d \,x^{3}+c}}-\frac {7 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 )}{31104 d \,c^{4}}\) |
\(765\) |
| risch |
\(\text {Expression too large to display}\) |
\(1464\) |
| default |
\(\text {Expression too large to display}\) |
\(1783\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^6/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/64/c^2*(-1/5/c*(d*x^3+c)^(1/2)/x^5+7/20*d/c^2*(d*x^3+c)^(1/2)/x^2-7/60*I/c^2*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/c^2*d^2*(1/216*x/c^2*(d*x^3+c)^(1/2)/(-d*x^3+8*c)+1/648*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))-5/972*I/c^2/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),-1/18/d*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alp
ha^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-8*c)))+1/256/c^3*d*(-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)))-1/69
12*I/c^4/d*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=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)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*x^6), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2709 vs.
\(2 (52) = 104\).
time = 12.39, size = 2709, normalized size = 41.05 \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)^2/(d*x^3+c)^(1/2),x, algorithm="fricas")
[Out]
1/2488320*(140*sqrt(3)*(c^4*d*x^8 - 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^2
5)^(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)) + 140*sqrt(3)*(c^4
*d*x^8 - 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^1
3*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^2
5)^(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)*sqr
t(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)) + 11088*(d^2*x^8 - 8*c*d*x^5)*sqrt(d)*weierst
rassPInverse(0, -4*c/d, x) + 35*(c^4*d*x^8 - 8*c^5*x^5)*(d^10/c^25)^(1/6)*log(282475249*(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^1
3*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)) - 35*(c^4*d*x^8 - 8*c^5*x^5)*(d^10/c^25)^(1/6)*log(282475249*(d^19*x^9 - 276*c*d^18*x^6 - 16
08*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 + 4
1*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 - 5
12*c^3)) + 70*(c^4*d*x^8 - 8*c^5*x^5)*(d^10/c^25)^(1/6)*log(16807*(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)) - 70*(c^4*d*x^8 -
8*c^5*x^5)*(d^10/c^25)^(1/6)*log(16807*(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^1
0/c^25)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 72*(119*d^2*x^6 - 1080*c*d*x^3 + 864*c^2)
*sqrt(d*x^3 + c))/(c^4*d*x^8 - 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}{x^{6} \left (- 8 c + d x^{3}\right )^{2} \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)**2/(d*x**3+c)**(1/2),x)
[Out]
Integral(1/(x**6*(-8*c + d*x**3)**2*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)^2/(d*x^3+c)^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*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\,\sqrt {d\,x^3+c}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^6*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2),x)
[Out]
int(1/(x^6*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2), x)
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