3.378 \(\int \frac{x^8}{\left (a+b x^3\right ) \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c+d x^3} (a d+b c)}{3 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2} \]

[Out]

(-2*(b*c + a*d)*Sqrt[c + d*x^3])/(3*b^2*d^2) + (2*(c + d*x^3)^(3/2))/(9*b*d^2) -
(2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2)*Sqrt[b*c
- a*d])

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Rubi [A]  time = 0.298242, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c+d x^3} (a d+b c)}{3 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c + a*d)*Sqrt[c + d*x^3])/(3*b^2*d^2) + (2*(c + d*x^3)^(3/2))/(9*b*d^2) -
(2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2)*Sqrt[b*c
- a*d])

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Rubi in Sympy [A]  time = 30.7678, size = 94, normalized size = 0.9 \[ \frac{2 a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{5}{2}} \sqrt{a d - b c}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 b d^{2}} - \frac{2 \sqrt{c + d x^{3}} \left (a d + b c\right )}{3 b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3*b**(5/2)*sqrt(a*d - b*c
)) + 2*(c + d*x**3)**(3/2)/(9*b*d**2) - 2*sqrt(c + d*x**3)*(a*d + b*c)/(3*b**2*d
**2)

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Mathematica [A]  time = 0.226284, size = 91, normalized size = 0.88 \[ \frac{2 \sqrt{c+d x^3} \left (-3 a d-2 b c+b d x^3\right )}{9 b^2 d^2}-\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x^3]*(-2*b*c - 3*a*d + b*d*x^3))/(9*b^2*d^2) - (2*a^2*ArcTanh[(Sqr
t[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2)*Sqrt[b*c - a*d])

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Maple [C]  time = 0.049, size = 488, normalized size = 4.7 \[{\frac{1}{{b}^{2}} \left ( b \left ({\frac{2\,{x}^{3}}{9\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,c}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) -{\frac{2\,a}{3\,d}\sqrt{d{x}^{3}+c}} \right ) }-{\frac{{\frac{i}{3}}{a}^{2}\sqrt{2}}{{b}^{2}{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ad-bc}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{id\sqrt{3} \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b^2*(b*(2/9/d*x^3*(d*x^3+c)^(1/2)-4/9*c*(d*x^3+c)^(1/2)/d^2)-2/3*a/d*(d*x^3+c)
^(1/2))-1/3*I*a^2/b^2/d^2*2^(1/2)*sum(1/(a*d-b*c)*(-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),1/2*
b/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)/(a*d-b*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*
b+a))

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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^8/((b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231595, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} d^{2} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right ) + 2 \,{\left (b d x^{3} - 2 \, b c - 3 \, a d\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}{9 \, \sqrt{b^{2} c - a b d} b^{2} d^{2}}, -\frac{2 \,{\left (3 \, a^{2} d^{2} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ) -{\left (b d x^{3} - 2 \, b c - 3 \, a d\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}\right )}}{9 \, \sqrt{-b^{2} c + a b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/9*(3*a^2*d^2*log(((b*d*x^3 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) - 2*sqrt(d*x^3
+ c)*(b^2*c - a*b*d))/(b*x^3 + a)) + 2*(b*d*x^3 - 2*b*c - 3*a*d)*sqrt(d*x^3 + c)
*sqrt(b^2*c - a*b*d))/(sqrt(b^2*c - a*b*d)*b^2*d^2), -2/9*(3*a^2*d^2*arctan(-(b*
c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d))) - (b*d*x^3 - 2*b*c - 3*a*d)*sqr
t(d*x^3 + c)*sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215293, size = 143, normalized size = 1.38 \[ \frac{2 \, a^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{2}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{2} d^{4} - 3 \, \sqrt{d x^{3} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{3} + c} a b d^{5}\right )}}{9 \, b^{3} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*a^2*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2
) + 2/9*((d*x^3 + c)^(3/2)*b^2*d^4 - 3*sqrt(d*x^3 + c)*b^2*c*d^4 - 3*sqrt(d*x^3
+ c)*a*b*d^5)/(b^3*d^6)