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

Optimal. Leaf size=161 \[ -\frac{a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2} \sqrt{b c-a d}}-\frac{a \sqrt{c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b^2 d} \]

[Out]

-(a*(4*b*c - 5*a*d)*Sqrt[c + d*x^3])/(3*b^3*(b*c - a*d)) + (2*(c + d*x^3)^(3/2))
/(9*b^2*d) - (a^2*(c + d*x^3)^(3/2))/(3*b^2*(b*c - a*d)*(a + b*x^3)) + (a*(4*b*c
- 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(7/2)*Sqrt[b*
c - a*d])

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Rubi [A]  time = 0.532934, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2} \sqrt{b c-a d}}-\frac{a \sqrt{c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(4*b*c - 5*a*d)*Sqrt[c + d*x^3])/(3*b^3*(b*c - a*d)) + (2*(c + d*x^3)^(3/2))
/(9*b^2*d) - (a^2*(c + d*x^3)^(3/2))/(3*b^2*(b*c - a*d)*(a + b*x^3)) + (a*(4*b*c
- 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(7/2)*Sqrt[b*
c - a*d])

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Rubi in Sympy [A]  time = 46.6628, size = 139, normalized size = 0.86 \[ \frac{a^{2} \left (c + d x^{3}\right )^{\frac{3}{2}}}{3 b^{2} \left (a + b x^{3}\right ) \left (a d - b c\right )} - \frac{a \sqrt{c + d x^{3}} \left (5 a d - 4 b c\right )}{3 b^{3} \left (a d - b c\right )} + \frac{a \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{7}{2}} \sqrt{a d - b c}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(c + d*x**3)**(3/2)/(3*b**2*(a + b*x**3)*(a*d - b*c)) - a*sqrt(c + d*x**3)*
(5*a*d - 4*b*c)/(3*b**3*(a*d - b*c)) + a*(5*a*d - 4*b*c)*atan(sqrt(b)*sqrt(c + d
*x**3)/sqrt(a*d - b*c))/(3*b**(7/2)*sqrt(a*d - b*c)) + 2*(c + d*x**3)**(3/2)/(9*
b**2*d)

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Mathematica [A]  time = 0.293466, size = 111, normalized size = 0.69 \[ \frac{\sqrt{c+d x^3} \left (-\frac{3 a^2}{a+b x^3}-12 a+\frac{2 b \left (c+d x^3\right )}{d}\right )}{9 b^3}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^3]*(-12*a - (3*a^2)/(a + b*x^3) + (2*b*(c + d*x^3))/d))/(9*b^3) +
(a*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(7/2
)*Sqrt[b*c - a*d])

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Maple [C]  time = 0.057, size = 917, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(d*x^3+c)^(3/2)/b^2/d+a^2/b^2*(-1/3*(d*x^3+c)^(1/2)/b/(b*x^3+a)-1/6*I/b/d*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)*_al
pha*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)))-2*a/b^2*(2/3*(d*x^3+c
)^(1/2)/b+1/3*I/b/d^2*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),1/2*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*_alph
a*(-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(sqrt(d*x^3 + c)*x^8/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226139, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b^{2} d x^{6} + 2 \,{\left (b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} - 3 \,{\left (4 \, a^{2} b c d - 5 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3}\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{18 \,{\left (b^{4} d x^{3} + a b^{3} d\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \, b^{2} d x^{6} + 2 \,{\left (b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} + 3 \,{\left (4 \, a^{2} b c d - 5 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{9 \,{\left (b^{4} d x^{3} + a b^{3} d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/18*(2*(2*b^2*d*x^6 + 2*(b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 15*a^2*d)*sqrt(d*x^3
+ c)*sqrt(b^2*c - a*b*d) - 3*(4*a^2*b*c*d - 5*a^3*d^2 + (4*a*b^2*c*d - 5*a^2*b*
d^2)*x^3)*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)))/((b^4*d*x^3 + a*b^3*d)*sqrt(b^2*c - a*b*d)), 1/9*(
(2*b^2*d*x^6 + 2*(b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 15*a^2*d)*sqrt(d*x^3 + c)*sqr
t(-b^2*c + a*b*d) + 3*(4*a^2*b*c*d - 5*a^3*d^2 + (4*a*b^2*c*d - 5*a^2*b*d^2)*x^3
)*arctan(-(b*c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d))))/((b^4*d*x^3 + a*b
^3*d)*sqrt(-b^2*c + a*b*d))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218818, size = 184, normalized size = 1.14 \[ -\frac{\sqrt{d x^{3} + c} a^{2} d}{3 \,{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{3}} - \frac{{\left (4 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{4} d^{2} - 6 \, \sqrt{d x^{3} + c} a b^{3} d^{3}\right )}}{9 \, b^{6} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(d*x^3 + c)*a^2*d/(((d*x^3 + c)*b - b*c + a*d)*b^3) - 1/3*(4*a*b*c - 5*
a^2*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3)
+ 2/9*((d*x^3 + c)^(3/2)*b^4*d^2 - 6*sqrt(d*x^3 + c)*a*b^3*d^3)/(b^6*d^3)