3.5.38 \(\int \frac {x^3}{(8 c-d x^3)^2 \sqrt {c+d x^3}} \, dx\) [438]

Optimal. Leaf size=66 \[ \frac {x^4 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};2,\frac {1}{2};\frac {7}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{256 c^2 \sqrt {c+d x^3}} \]

[Out]

1/256*x^4*AppellF1(4/3,1/2,2,7/3,-d*x^3/c,1/8*d*x^3/c)*(1+d*x^3/c)^(1/2)/c^2/(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 {x^4 \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {4}{3};2,\frac {1}{2};\frac {7}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{256 c^2 \sqrt {c+d x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 2, 1/2, 7/3, (d*x^3)/(8*c), -((d*x^3)/c)])/(256*c^2*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 {x^3}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {d x^3}{c}} \int \frac {x^3}{\left (8 c-d x^3\right )^2 \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {c+d x^3}}\\ &=\frac {x^4 \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {4}{3};2,\frac {1}{2};\frac {7}{3};\frac {d x^3}{8 c},-\frac {d x^3}{c}\right )}{256 c^2 \sqrt {c+d x^3}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(237\) vs. \(2(66)=132\).
time = 10.13, size = 237, normalized size = 3.59 \begin {gather*} \frac {x \left (x^3 \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 )-\frac {64 c \left (c+d x^3-\frac {32 c^2 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 )}{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 )}{d \left (-8 c+d x^3\right )}\right )}{1728 c^2 \sqrt {c+d x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(x^3*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*(c + d*x^3 - (32*c
^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(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)]))))/(d*(-8*c + d*x^3))))/(1728*c^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 = 0.38, size = 1151, normalized size = 17.44

method	result	size
elliptic	\(\frac {x \sqrt {d \,x^{3}+c}}{27 c d \left (-d \,x^{3}+8 c \right )}+\frac {i \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 )}{81 d^{2} c \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 )}{243 d^{4} c}\)	\(732\)
default	\(\text {Expression too large to display}\)	\(1151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*c/d*(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/_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)))+1/27*I/d^4/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)*_a
lpha+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(x^3/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2637 vs. \(2 (52) = 104\).
time = 7.54, size = 2637, normalized size = 39.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3888*(4*sqrt(3)*(c*d^3*x^3 - 8*c^2*d^2)*(1/(c^7*d^8))^(1/6)*arctan(1/9*((9*sqrt(3)*c^6*d^8*x^5*(1/(c^7*d^8))
^(5/6) + 3*sqrt(3)*(5*c^4*d^5*x^4 + 8*c^5*d^4*x)*sqrt(1/(c^7*d^8)) - sqrt(3)*(c*d^3*x^6 - 40*c^2*d^2*x^3 - 32*
c^3*d)*(1/(c^7*d^8))^(1/6))*sqrt(d*x^3 + c) - (12*sqrt(3)*(c^5*d^7*x^6 - c^6*d^6*x^3 - 2*c^7*d^5)*(1/(c^7*d^8)
)^(2/3) + 18*sqrt(3)*(c^3*d^4*x^5 + c^4*d^3*x^2)*(1/(c^7*d^8))^(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^8*x^5 + 2*c^7*d^7*x^2)*(1/(c^7*d^8))^(5/6) + 3*sqrt(3)*(7*c^4*d^5*x^4 +
4*c^5*d^4*x)*sqrt(1/(c^7*d^8)) + sqrt(3)*(c*d^3*x^6 + 32*c^2*d^2*x^3 + 40*c^3*d)*(1/(c^7*d^8))^(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^8*x^8 + 20*c^6*d^7*x^5 - 8*c^7*d^6*x^2)*(1/(c^
7*d^8))^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^9*x^7 - 28*c^7*d^8*x^4 - 272*c^8*d^7*x)*(1/(c^7*d^8))^(5/6) + 4*(c^4
*d^6*x^6 + 41*c^5*d^5*x^3 + 40*c^6*d^4)*sqrt(1/(c^7*d^8)) - 24*(c^2*d^3*x^5 + c^3*d^2*x^2)*(1/(c^7*d^8))^(1/6)
) - 18*(c^3*d^5*x^7 - 52*c^4*d^4*x^4 - 80*c^5*d^3*x)*(1/(c^7*d^8))^(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^3*x^3 - 8*c^2*d^2)*(1/(c^7*d^8))^(1/6)*arct
an(1/9*((9*sqrt(3)*c^6*d^8*x^5*(1/(c^7*d^8))^(5/6) + 3*sqrt(3)*(5*c^4*d^5*x^4 + 8*c^5*d^4*x)*sqrt(1/(c^7*d^8))
- sqrt(3)*(c*d^3*x^6 - 40*c^2*d^2*x^3 - 32*c^3*d)*(1/(c^7*d^8))^(1/6))*sqrt(d*x^3 + c) + (12*sqrt(3)*(c^5*d^7
*x^6 - c^6*d^6*x^3 - 2*c^7*d^5)*(1/(c^7*d^8))^(2/3) + 18*sqrt(3)*(c^3*d^4*x^5 + c^4*d^3*x^2)*(1/(c^7*d^8))^(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^8*x^5 + 2*c^7*d^7*x^2)*(1/(
c^7*d^8))^(5/6) + 3*sqrt(3)*(7*c^4*d^5*x^4 + 4*c^5*d^4*x)*sqrt(1/(c^7*d^8)) + sqrt(3)*(c*d^3*x^6 + 32*c^2*d^2*
x^3 + 40*c^3*d)*(1/(c^7*d^8))^(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^8*
x^8 + 20*c^6*d^7*x^5 - 8*c^7*d^6*x^2)*(1/(c^7*d^8))^(2/3) - 6*sqrt(d*x^3 + c)*((c^6*d^9*x^7 - 28*c^7*d^8*x^4 -
272*c^8*d^7*x)*(1/(c^7*d^8))^(5/6) + 4*(c^4*d^6*x^6 + 41*c^5*d^5*x^3 + 40*c^6*d^4)*sqrt(1/(c^7*d^8)) - 24*(c^
2*d^3*x^5 + c^3*d^2*x^2)*(1/(c^7*d^8))^(1/6)) - 18*(c^3*d^5*x^7 - 52*c^4*d^4*x^4 - 80*c^5*d^3*x)*(1/(c^7*d^8))
^(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)) - 144*sqrt(d*x^3
+ c)*d*x - 72*(d*x^3 - 8*c)*sqrt(d)*weierstrassPInverse(0, -4*c/d, x) + 2*(c*d^3*x^3 - 8*c^2*d^2)*(1/(c^7*d^8
))^(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^8*x^8 + 38*c^6*d^7*x^5 + 64*c^7*d
^6*x^2)*(1/(c^7*d^8))^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^9*x^7 + 80*c^7*d^8*x^4 + 160*c^8*d^7*x)*(1/(c^7*d^8))^
(5/6) + (7*c^4*d^6*x^6 + 152*c^5*d^5*x^3 + 64*c^6*d^4)*sqrt(1/(c^7*d^8)) + 6*(5*c^2*d^3*x^5 + 32*c^3*d^2*x^2)*
(1/(c^7*d^8))^(1/6)) + 18*(5*c^3*d^5*x^7 + 64*c^4*d^4*x^4 + 32*c^5*d^3*x)*(1/(c^7*d^8))^(1/3))/(d^3*x^9 - 24*c
*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 2*(c*d^3*x^3 - 8*c^2*d^2)*(1/(c^7*d^8))^(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^8*x^8 + 38*c^6*d^7*x^5 + 64*c^7*d^6*x^2)*(1/(c^7*d^8))^(2/3) - 6*sq
rt(d*x^3 + c)*((c^6*d^9*x^7 + 80*c^7*d^8*x^4 + 160*c^8*d^7*x)*(1/(c^7*d^8))^(5/6) + (7*c^4*d^6*x^6 + 152*c^5*d
^5*x^3 + 64*c^6*d^4)*sqrt(1/(c^7*d^8)) + 6*(5*c^2*d^3*x^5 + 32*c^3*d^2*x^2)*(1/(c^7*d^8))^(1/6)) + 18*(5*c^3*d
^5*x^7 + 64*c^4*d^4*x^4 + 32*c^5*d^3*x)*(1/(c^7*d^8))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3
)) + (c*d^3*x^3 - 8*c^2*d^2)*(1/(c^7*d^8))^(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^8*x^8 + 20*c^6*d^7*x^5 - 8*c^7*d^6*x^2)*(1/(c^7*d^8))^(2/3) + 6*sqrt(d*x^3 + c)*((c^6*d^9*x^7 - 28*c^7
*d^8*x^4 - 272*c^8*d^7*x)*(1/(c^7*d^8))^(5/6) + 4*(c^4*d^6*x^6 + 41*c^5*d^5*x^3 + 40*c^6*d^4)*sqrt(1/(c^7*d^8)
) - 24*(c^2*d^3*x^5 + c^3*d^2*x^2)*(1/(c^7*d^8))^(1/6)) - 18*(c^3*d^5*x^7 - 52*c^4*d^4*x^4 - 80*c^5*d^3*x)*(1/
(c^7*d^8))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - (c*d^3*x^3 - 8*c^2*d^2)*(1/(c^7*d^8))^
(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^8*x^8 + 20*c^6*d^7*x^5 - 8*c^7*d^6*
x^2)*(1/(c^7*d^8))^(2/3) - 6*sqrt(d*x^3 + c)*((c^6*d^9*x^7 - 28*c^7*d^8*x^4 - 272*c^8*d^7*x)*(1/(c^7*d^8))^(5/
6) + 4*(c^4*d^6*x^6 + 41*c^5*d^5*x^3 + 40*c^6*d^4)*sqrt(1/(c^7*d^8)) - 24*(c^2*d^3*x^5 + c^3*d^2*x^2)*(1/(c^7*
d^8))^(1/6)) - 18*(c^3*d^5*x^7 - 52*c^4*d^4*x^4 - 80*c^5*d^3*x)*(1/(c^7*d^8))^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 +
192*c^2*d*x^3 - 512*c^3)))/(c*d^3*x^3 - 8*c^2*d^2)

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

[Out]

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

[Out]

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

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

[Out]

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

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