3.259 \(\int \frac{x^8 \sqrt{c+d x^3}}{4 c+d x^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{32 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d^3}+\frac{32 c^2 \sqrt{c+d x^3}}{3 d^3}-\frac{10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]

[Out]

(32*c^2*Sqrt[c + d*x^3])/(3*d^3) - (10*c*(c + d*x^3)^(3/2))/(9*d^3) + (2*(c + d*
x^3)^(5/2))/(15*d^3) - (32*c^(5/2)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(S
qrt[3]*d^3)

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Rubi [A]  time = 0.282701, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{32 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{\sqrt{3} d^3}+\frac{32 c^2 \sqrt{c+d x^3}}{3 d^3}-\frac{10 c \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*Sqrt[c + d*x^3])/(4*c + d*x^3),x]

[Out]

(32*c^2*Sqrt[c + d*x^3])/(3*d^3) - (10*c*(c + d*x^3)^(3/2))/(9*d^3) + (2*(c + d*
x^3)^(5/2))/(15*d^3) - (32*c^(5/2)*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(S
qrt[3]*d^3)

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Rubi in Sympy [A]  time = 27.3981, size = 95, normalized size = 0.98 \[ - \frac{32 \sqrt{3} c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{3 d^{3}} + \frac{32 c^{2} \sqrt{c + d x^{3}}}{3 d^{3}} - \frac{10 c \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 d^{3}} + \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(d*x**3+c)**(1/2)/(d*x**3+4*c),x)

[Out]

-32*sqrt(3)*c**(5/2)*atan(sqrt(3)*sqrt(c + d*x**3)/(3*sqrt(c)))/(3*d**3) + 32*c*
*2*sqrt(c + d*x**3)/(3*d**3) - 10*c*(c + d*x**3)**(3/2)/(9*d**3) + 2*(c + d*x**3
)**(5/2)/(15*d**3)

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Mathematica [A]  time = 0.101277, size = 77, normalized size = 0.79 \[ \frac{2 \sqrt{c+d x^3} \left (218 c^2-19 c d x^3+3 d^2 x^6\right )-480 \sqrt{3} c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{3} \sqrt{c}}\right )}{45 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*Sqrt[c + d*x^3])/(4*c + d*x^3),x]

[Out]

(2*Sqrt[c + d*x^3]*(218*c^2 - 19*c*d*x^3 + 3*d^2*x^6) - 480*Sqrt[3]*c^(5/2)*ArcT
an[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(45*d^3)

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Maple [C]  time = 0.216, size = 506, normalized size = 5.2 \[{\frac{1}{{d}^{2}} \left ( d \left ({\frac{2\,{x}^{6}}{15}\sqrt{d{x}^{3}+c}}+{\frac{2\,c{x}^{3}}{45\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,{c}^{2}}{45\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) -{\frac{8\,c}{9\,d} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}} \right ) }+16\,{\frac{{c}^{2}}{{d}^{2}} \left ( 2/3\,{\frac{\sqrt{d{x}^{3}+c}}{d}}+{\frac{i/3\sqrt{2}}{{d}^{3}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}+4\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{id\sqrt{3}}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},1/6\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(d*x^3+c)^(1/2)/(d*x^3+4*c),x)

[Out]

1/d^2*(d*(2/15*x^6*(d*x^3+c)^(1/2)+2/45*c/d*x^3*(d*x^3+c)^(1/2)-4/45*c^2*(d*x^3+
c)^(1/2)/d^2)-8/9*c/d*(d*x^3+c)^(3/2))+16*c^2/d^2*(2/3*(d*x^3+c)^(1/2)/d+1/3*I/d
^3*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(sqrt(d*x^3 + c)*x^8/(d*x^3 + 4*c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284489, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{3}{\left (360 \, \sqrt{-c} c^{2} \log \left (\frac{\sqrt{3}{\left (d x^{3} - 2 \, c\right )} - 6 \, \sqrt{d x^{3} + c} \sqrt{-c}}{d x^{3} + 4 \, c}\right ) + \sqrt{3}{\left (3 \, d^{2} x^{6} - 19 \, c d x^{3} + 218 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{135 \, d^{3}}, -\frac{2 \, \sqrt{3}{\left (720 \, c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right ) - \sqrt{3}{\left (3 \, d^{2} x^{6} - 19 \, c d x^{3} + 218 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{135 \, d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)*x^8/(d*x^3 + 4*c),x, algorithm="fricas")

[Out]

[2/135*sqrt(3)*(360*sqrt(-c)*c^2*log((sqrt(3)*(d*x^3 - 2*c) - 6*sqrt(d*x^3 + c)*
sqrt(-c))/(d*x^3 + 4*c)) + sqrt(3)*(3*d^2*x^6 - 19*c*d*x^3 + 218*c^2)*sqrt(d*x^3
+ c))/d^3, -2/135*sqrt(3)*(720*c^(5/2)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(
c)) - sqrt(3)*(3*d^2*x^6 - 19*c*d*x^3 + 218*c^2)*sqrt(d*x^3 + c))/d^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8} \sqrt{c + d x^{3}}}{4 c + d x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(d*x**3+c)**(1/2)/(d*x**3+4*c),x)

[Out]

Integral(x**8*sqrt(c + d*x**3)/(4*c + d*x**3), x)

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GIAC/XCAS [A]  time = 0.214804, size = 111, normalized size = 1.14 \[ -\frac{32 \, \sqrt{3} c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{3} \sqrt{d x^{3} + c}}{3 \, \sqrt{c}}\right )}{3 \, d^{3}} + \frac{2 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} d^{12} - 25 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c d^{12} + 240 \, \sqrt{d x^{3} + c} c^{2} d^{12}\right )}}{45 \, d^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)*x^8/(d*x^3 + 4*c),x, algorithm="giac")

[Out]

-32/3*sqrt(3)*c^(5/2)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c))/d^3 + 2/45*(3*
(d*x^3 + c)^(5/2)*d^12 - 25*(d*x^3 + c)^(3/2)*c*d^12 + 240*sqrt(d*x^3 + c)*c^2*d
^12)/d^15