3.401 \(\int \frac{x^2 \sqrt{c+d x^3}}{\left (8 c-d x^3\right )^2} \, dx\)

Optimal. Leaf size=64 \[ \frac{\sqrt{c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 \sqrt{c} d} \]

[Out]

Sqrt[c + d*x^3]/(3*d*(8*c - d*x^3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(9*Sq
rt[c]*d)

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Rubi [A]  time = 0.143758, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\sqrt{c+d x^3}}{3 d \left (8 c-d x^3\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 \sqrt{c} d} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^3]/(3*d*(8*c - d*x^3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(9*Sq
rt[c]*d)

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Rubi in Sympy [A]  time = 17.8675, size = 48, normalized size = 0.75 \[ \frac{\sqrt{c + d x^{3}}}{3 d \left (8 c - d x^{3}\right )} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{9 \sqrt{c} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

sqrt(c + d*x**3)/(3*d*(8*c - d*x**3)) - atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(9*s
qrt(c)*d)

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Mathematica [A]  time = 0.102514, size = 63, normalized size = 0.98 \[ -\frac{\sqrt{c+d x^3}}{3 d \left (d x^3-8 c\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{9 \sqrt{c} d} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^3]/(3*d*(-8*c + d*x^3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(9*
Sqrt[c]*d)

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Maple [C]  time = 0.012, size = 439, normalized size = 6.9 \[ -{\frac{1}{3\,d \left ( d{x}^{3}-8\,c \right ) }\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{54}}\sqrt{2}}{{d}^{3}c}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},-{\frac{1}{18\,cd} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/d*(d*x^3+c)^(1/2)/(d*x^3-8*c)+1/54*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),-1/18/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-8*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(sqrt(d*x^3 + c)*x^2/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221441, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} - 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 6 \, \sqrt{d x^{3} + c} \sqrt{c}}{18 \,{\left (d^{2} x^{3} - 8 \, c d\right )} \sqrt{c}}, \frac{{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 3 \, \sqrt{d x^{3} + c} \sqrt{-c}}{9 \,{\left (d^{2} x^{3} - 8 \, c d\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)*x^2/(d*x^3 - 8*c)^2,x, algorithm="fricas")

[Out]

[1/18*((d*x^3 - 8*c)*log(((d*x^3 + 10*c)*sqrt(c) - 6*sqrt(d*x^3 + c)*c)/(d*x^3 -
8*c)) - 6*sqrt(d*x^3 + c)*sqrt(c))/((d^2*x^3 - 8*c*d)*sqrt(c)), 1/9*((d*x^3 - 8
*c)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 3*sqrt(d*x^3 + c)*sqrt(-c))/((d^2*x
^3 - 8*c*d)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{c + d x^{3}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(d*x**3+c)**(1/2)/(-d*x**3+8*c)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215357, size = 72, normalized size = 1.12 \[ \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{9 \, \sqrt{-c} d} - \frac{\sqrt{d x^{3} + c}}{3 \,{\left (d x^{3} - 8 \, c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)*x^2/(d*x^3 - 8*c)^2,x, algorithm="giac")

[Out]

1/9*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d) - 1/3*sqrt(d*x^3 + c)/((d*
x^3 - 8*c)*d)