3.272 \(\int \frac{x^2}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3 \sqrt{3} \sqrt{c} d} \]

[Out]

(2*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(3*Sqrt[3]*Sqrt[c]*d)

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Rubi [A]  time = 0.129408, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3 \sqrt{3} \sqrt{c} d} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

(2*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(3*Sqrt[3]*Sqrt[c]*d)

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Rubi in Sympy [A]  time = 12.6562, size = 37, normalized size = 0.92 \[ \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 \sqrt{c} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

2*sqrt(3)*atan(sqrt(3)*sqrt(c + d*x**3)/(3*sqrt(c)))/(9*sqrt(c)*d)

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Mathematica [A]  time = 0.0308774, size = 40, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{3 \sqrt{3} \sqrt{c} d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[c + d*x^3]*(4*c + d*x^3)),x]

[Out]

(2*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(3*Sqrt[3]*Sqrt[c]*d)

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Maple [C]  time = 0.011, size = 413, normalized size = 10.3 \[{\frac{-{\frac{i}{9}}\sqrt{2}}{{d}^{3}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d+4\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{1}{6\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)

[Out]

-1/9*I/d^3/c*2^(1/2)*sum((-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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244076, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{3} \log \left (\frac{\sqrt{3}{\left (d x^{3} - 2 \, c\right )} \sqrt{-c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} + 4 \, c}\right )}{9 \, \sqrt{-c} d}, -\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{c}}{\sqrt{d x^{3} + c}}\right )}{9 \, \sqrt{c} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="fricas")

[Out]

[1/9*sqrt(3)*log((sqrt(3)*(d*x^3 - 2*c)*sqrt(-c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 +
4*c))/(sqrt(-c)*d), -2/9*sqrt(3)*arctan(sqrt(3)*sqrt(c)/sqrt(d*x^3 + c))/(sqrt(
c)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)

[Out]

Integral(x**2/(sqrt(c + d*x**3)*(4*c + d*x**3)), x)

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GIAC/XCAS [A]  time = 0.21402, size = 39, normalized size = 0.98 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{9 \, \sqrt{c} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((d*x^3 + 4*c)*sqrt(d*x^3 + c)),x, algorithm="giac")

[Out]

2/9*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c))/(sqrt(c)*d)