3.276 \(\int \frac{x}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx\)
Optimal. Leaf size=206 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3\ 2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3\ 2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{9\ 2^{2/3} c^{5/6} d^{2/3}} \]
[Out]
-ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]]/(3*2^(2
/3)*Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])]/(3*2^(2
/3)*Sqrt[3]*c^(5/6)*d^(2/3)) - ArcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*x))/S
qrt[c + d*x^3]]/(3*2^(2/3)*c^(5/6)*d^(2/3)) + ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(
9*2^(2/3)*c^(5/6)*d^(2/3))
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Rubi [A] time = 0.146796, antiderivative size = 206, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042
\[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3\ 2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3\ 2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{9\ 2^{2/3} c^{5/6} d^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[x/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
-ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x))/Sqrt[c + d*x^3]]/(3*2^(2
/3)*Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])]/(3*2^(2
/3)*Sqrt[3]*c^(5/6)*d^(2/3)) - ArcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*x))/S
qrt[c + d*x^3]]/(3*2^(2/3)*c^(5/6)*d^(2/3)) + ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]]/(
9*2^(2/3)*c^(5/6)*d^(2/3))
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Rubi in Sympy [A] time = 11.6196, size = 277, normalized size = 1.34 \[ \frac{\sqrt [3]{2} \log{\left (1 - \frac{\sqrt{c + d x^{3}}}{\sqrt{c}} - \frac{\sqrt [3]{2} \sqrt [3]{d} x}{\sqrt [3]{c}} \right )}}{12 c^{\frac{5}{6}} d^{\frac{2}{3}}} - \frac{\sqrt [3]{2} \log{\left (1 + \frac{\sqrt{c + d x^{3}}}{\sqrt{c}} - \frac{\sqrt [3]{2} \sqrt [3]{d} x}{\sqrt [3]{c}} \right )}}{12 c^{\frac{5}{6}} d^{\frac{2}{3}}} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \left (\sqrt{c} - \sqrt{c + d x^{3}}\right )}{3 \sqrt [6]{c} \sqrt [3]{d} x} \right )}}{18 c^{\frac{5}{6}} d^{\frac{2}{3}}} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \left (\sqrt{c} + \sqrt{c + d x^{3}}\right )}{3 \sqrt [6]{c} \sqrt [3]{d} x} \right )}}{18 c^{\frac{5}{6}} d^{\frac{2}{3}}} + \frac{\sqrt [3]{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{18 c^{\frac{5}{6}} d^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
2**(1/3)*log(1 - sqrt(c + d*x**3)/sqrt(c) - 2**(1/3)*d**(1/3)*x/c**(1/3))/(12*c*
*(5/6)*d**(2/3)) - 2**(1/3)*log(1 + sqrt(c + d*x**3)/sqrt(c) - 2**(1/3)*d**(1/3)
*x/c**(1/3))/(12*c**(5/6)*d**(2/3)) - 2**(1/3)*sqrt(3)*atan(sqrt(3)/3 + 2**(2/3)
*sqrt(3)*(sqrt(c) - sqrt(c + d*x**3))/(3*c**(1/6)*d**(1/3)*x))/(18*c**(5/6)*d**(
2/3)) + 2**(1/3)*sqrt(3)*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*(sqrt(c) + sqrt(c + d
*x**3))/(3*c**(1/6)*d**(1/3)*x))/(18*c**(5/6)*d**(2/3)) + 2**(1/3)*atanh(sqrt(c
+ d*x**3)/sqrt(c))/(18*c**(5/6)*d**(2/3))
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Mathematica [C] time = 0.078473, size = 167, normalized size = 0.81 \[ \frac{10 c x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\sqrt{c+d x^3} \left (4 c+d x^3\right ) \left (20 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
(10*c*x^2*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/(Sqrt[c + d*
x^3]*(4*c + d*x^3)*(20*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -(d*x^3)/(4*c)
] - 3*d*x^3*(AppellF1[5/3, 1/2, 2, 8/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2*Appell
F1[5/3, 3/2, 1, 8/3, -((d*x^3)/c), -(d*x^3)/(4*c)])))
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Maple [C] time = 0.008, size = 416, normalized size = 2. \[{\frac{-{\frac{i}{9}}\sqrt{2}}{{d}^{3}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d+4\,c \right ) }{\frac{1}{{\it \_alpha}}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)
[Out]
-1/9*I/d^3/c*2^(1/2)*sum(1/_alpha*(-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+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-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*_alpha^2*
(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*
(-c*d^2)^(2/3)-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., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="maxima")
[Out]
integrate(x/((d*x^3 + 4*c)*sqrt(d*x^3 + c)), x)
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Fricas [A] time = 0.805948, size = 3645, normalized size = 17.69 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="fricas")
[Out]
1/9*sqrt(3)*(1/432)^(1/6)*(-1/(c^5*d^4))^(1/6)*arctan(6*(4*sqrt(3)*(1/2)^(2/3)*(
c^4*d^5*x^7 - c^5*d^4*x^4 - 2*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) - sqrt(3)*(1/2)^(1
/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) + (648*sq
rt(3)*(1/432)^(5/6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/6) - sqrt(3)*(1/432)^(1/6)*(c*
d^3*x^7 - 16*c^2*d^2*x^4 - 8*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt(d*x^3 + c))/(d^
3*x^9 - 66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 - 24*(1/2)^(2/3)*(c^4*d^5*x^7 - c^5
*d^4*x^4 - 2*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) - 6*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^
3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) + 6*(648*(1/432)^(5/6)*c^5*d^5*x
^5*(-1/(c^5*d^4))^(5/6) - sqrt(1/3)*(5*c^3*d^4*x^6 - 20*c^4*d^3*x^3 - 16*c^5*d^2
)*sqrt(-1/(c^5*d^4)) + (1/432)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x^4 - 8*c^3*d*x)*(-
1/(c^5*d^4))^(1/6))*sqrt(d*x^3 + c) + (d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 6
4*c^3)*sqrt((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 - 24*(1/2)^(2/3)*(c^4*d^5*x^7 + 5*c
^5*d^4*x^4 + 4*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) + 12*(1/2)^(1/3)*(c^2*d^4*x^8 - 7
*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) - 12*(648*(1/432)^(5/6)*c^5*d
^5*x^5*(-1/(c^5*d^4))^(5/6) - sqrt(1/3)*(c^3*d^4*x^6 - 16*c^4*d^3*x^3 - 8*c^5*d^
2)*sqrt(-1/(c^5*d^4)) - (1/432)^(1/6)*(c*d^3*x^7 + 2*c^2*d^2*x^4 - 8*c^3*d*x)*(-
1/(c^5*d^4))^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64
*c^3)))) + 1/9*sqrt(3)*(1/432)^(1/6)*(-1/(c^5*d^4))^(1/6)*arctan(-6*(4*sqrt(3)*(
1/2)^(2/3)*(c^4*d^5*x^7 - c^5*d^4*x^4 - 2*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) - sqrt
(3)*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/
3) - (648*sqrt(3)*(1/432)^(5/6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/6) - sqrt(3)*(1/43
2)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x^4 - 8*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt(d*x
^3 + c))/(d^3*x^9 - 66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 - 24*(1/2)^(2/3)*(c^4*d
^5*x^7 - c^5*d^4*x^4 - 2*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) - 6*(1/2)^(1/3)*(c^2*d^
4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) - 6*(648*(1/432)^(5/
6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/6) - sqrt(1/3)*(5*c^3*d^4*x^6 - 20*c^4*d^3*x^3
- 16*c^5*d^2)*sqrt(-1/(c^5*d^4)) + (1/432)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x^4 - 8
*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt(d*x^3 + c) + (d^3*x^9 + 12*c*d^2*x^6 + 48*c
^2*d*x^3 + 64*c^3)*sqrt((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 - 24*(1/2)^(2/3)*(c^4*d
^5*x^7 + 5*c^5*d^4*x^4 + 4*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) + 12*(1/2)^(1/3)*(c^2
*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) + 12*(648*(1/432)
^(5/6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/6) - sqrt(1/3)*(c^3*d^4*x^6 - 16*c^4*d^3*x^
3 - 8*c^5*d^2)*sqrt(-1/(c^5*d^4)) - (1/432)^(1/6)*(c*d^3*x^7 + 2*c^2*d^2*x^4 - 8
*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^
2*d*x^3 + 64*c^3)))) - 1/36*(1/432)^(1/6)*(-1/(c^5*d^4))^(1/6)*log((d^3*x^9 + 60
*c*d^2*x^6 - 32*c^3 - 24*(1/2)^(2/3)*(c^4*d^5*x^7 + 5*c^5*d^4*x^4 + 4*c^6*d^3*x)
*(-1/(c^5*d^4))^(2/3) + 12*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*
x^2)*(-1/(c^5*d^4))^(1/3) + 12*(648*(1/432)^(5/6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/
6) - sqrt(1/3)*(c^3*d^4*x^6 - 16*c^4*d^3*x^3 - 8*c^5*d^2)*sqrt(-1/(c^5*d^4)) - (
1/432)^(1/6)*(c*d^3*x^7 + 2*c^2*d^2*x^4 - 8*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt(
d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) + 1/36*(1/432)^(1/
6)*(-1/(c^5*d^4))^(1/6)*log((d^3*x^9 + 60*c*d^2*x^6 - 32*c^3 - 24*(1/2)^(2/3)*(c
^4*d^5*x^7 + 5*c^5*d^4*x^4 + 4*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) + 12*(1/2)^(1/3)*
(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) - 12*(648*(1/
432)^(5/6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/6) - sqrt(1/3)*(c^3*d^4*x^6 - 16*c^4*d^
3*x^3 - 8*c^5*d^2)*sqrt(-1/(c^5*d^4)) - (1/432)^(1/6)*(c*d^3*x^7 + 2*c^2*d^2*x^4
- 8*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 4
8*c^2*d*x^3 + 64*c^3)) + 1/18*(1/432)^(1/6)*(-1/(c^5*d^4))^(1/6)*log((d^3*x^9 -
66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 + 48*(1/2)^(2/3)*(c^4*d^5*x^7 - c^5*d^4*x^4
- 2*c^6*d^3*x)*(-1/(c^5*d^4))^(2/3) + 12*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x
^5 - 8*c^4*d^2*x^2)*(-1/(c^5*d^4))^(1/3) + 6*(1296*(1/432)^(5/6)*c^5*d^5*x^5*(-1
/(c^5*d^4))^(5/6) + sqrt(1/3)*(5*c^3*d^4*x^6 - 20*c^4*d^3*x^3 - 16*c^5*d^2)*sqrt
(-1/(c^5*d^4)) + 2*(1/432)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x^4 - 8*c^3*d*x)*(-1/(c
^5*d^4))^(1/6))*sqrt(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3
)) - 1/18*(1/432)^(1/6)*(-1/(c^5*d^4))^(1/6)*log((d^3*x^9 - 66*c*d^2*x^6 - 72*c^
2*d*x^3 - 32*c^3 + 48*(1/2)^(2/3)*(c^4*d^5*x^7 - c^5*d^4*x^4 - 2*c^6*d^3*x)*(-1/
(c^5*d^4))^(2/3) + 12*(1/2)^(1/3)*(c^2*d^4*x^8 - 7*c^3*d^3*x^5 - 8*c^4*d^2*x^2)*
(-1/(c^5*d^4))^(1/3) - 6*(1296*(1/432)^(5/6)*c^5*d^5*x^5*(-1/(c^5*d^4))^(5/6) +
sqrt(1/3)*(5*c^3*d^4*x^6 - 20*c^4*d^3*x^3 - 16*c^5*d^2)*sqrt(-1/(c^5*d^4)) + 2*(
1/432)^(1/6)*(c*d^3*x^7 - 16*c^2*d^2*x^4 - 8*c^3*d*x)*(-1/(c^5*d^4))^(1/6))*sqrt
(d*x^3 + c))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3))
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
Integral(x/(sqrt(c + d*x**3)*(4*c + d*x**3)), x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="giac")
[Out]
integrate(x/((d*x^3 + 4*c)*sqrt(d*x^3 + c)), x)