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

Optimal. Leaf size=189 \[ -\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}-\frac{a \sqrt{c+d x^3} (4 b c-7 a d)}{3 b^4}-\frac{a \left (c+d x^3\right )^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d} \]

[Out]

-(a*(4*b*c - 7*a*d)*Sqrt[c + d*x^3])/(3*b^4) - (a*(4*b*c - 7*a*d)*(c + d*x^3)^(3
/2))/(9*b^3*(b*c - a*d)) + (2*(c + d*x^3)^(5/2))/(15*b^2*d) - (a^2*(c + d*x^3)^(
5/2))/(3*b^2*(b*c - a*d)*(a + b*x^3)) + (a*(4*b*c - 7*a*d)*Sqrt[b*c - a*d]*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.663962, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}-\frac{a \sqrt{c+d x^3} (4 b c-7 a d)}{3 b^4}-\frac{a \left (c+d x^3\right )^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(4*b*c - 7*a*d)*Sqrt[c + d*x^3])/(3*b^4) - (a*(4*b*c - 7*a*d)*(c + d*x^3)^(3
/2))/(9*b^3*(b*c - a*d)) + (2*(c + d*x^3)^(5/2))/(15*b^2*d) - (a^2*(c + d*x^3)^(
5/2))/(3*b^2*(b*c - a*d)*(a + b*x^3)) + (a*(4*b*c - 7*a*d)*Sqrt[b*c - a*d]*ArcTa
nh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.4904, size = 167, normalized size = 0.88 \[ \frac{a^{2} \left (c + d x^{3}\right )^{\frac{5}{2}}}{3 b^{2} \left (a + b x^{3}\right ) \left (a d - b c\right )} - \frac{a \left (c + d x^{3}\right )^{\frac{3}{2}} \left (7 a d - 4 b c\right )}{9 b^{3} \left (a d - b c\right )} + \frac{a \sqrt{c + d x^{3}} \left (7 a d - 4 b c\right )}{3 b^{4}} - \frac{a \sqrt{a d - b c} \left (7 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{9}{2}}} + \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}}}{15 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(c + d*x**3)**(5/2)/(3*b**2*(a + b*x**3)*(a*d - b*c)) - a*(c + d*x**3)**(3/
2)*(7*a*d - 4*b*c)/(9*b**3*(a*d - b*c)) + a*sqrt(c + d*x**3)*(7*a*d - 4*b*c)/(3*
b**4) - a*sqrt(a*d - b*c)*(7*a*d - 4*b*c)*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d
- b*c))/(3*b**(9/2)) + 2*(c + d*x**3)**(5/2)/(15*b**2*d)

_______________________________________________________________________________________

Mathematica [A]  time = 0.316265, size = 162, normalized size = 0.86 \[ \frac{\sqrt{c+d x^3} \left (105 a^3 d^2+5 a^2 b d \left (14 d x^3-19 c\right )+2 a b^2 \left (3 c^2-34 c d x^3-7 d^2 x^6\right )+6 b^3 x^3 \left (c+d x^3\right )^2\right )}{45 b^4 d \left (a+b x^3\right )}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^3]*(105*a^3*d^2 + 6*b^3*x^3*(c + d*x^3)^2 + 5*a^2*b*d*(-19*c + 14*
d*x^3) + 2*a*b^2*(3*c^2 - 34*c*d*x^3 - 7*d^2*x^6)))/(45*b^4*d*(a + b*x^3)) + (a*
(4*b*c - 7*a*d)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d
]])/(3*b^(9/2))

_______________________________________________________________________________________

Maple [C]  time = 0.06, size = 1003, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(d*x^3+c)^(5/2)/b^2/d+a^2/b^2*(1/3*(a*d-b*c)/b^2*(d*x^3+c)^(1/2)/(b*x^3+a)+
2/3*d*(d*x^3+c)^(1/2)/b^2+1/2*I/d/b^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*_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*(2/9/b*d*x^3*(d*x^3+c)^(1/2)+2/3*(-d*(a*d-2*b*c)/b^2-2/3/b*d*c)/d
*(d*x^3+c)^(1/2)+1/3*I/b^2/d^2*2^(1/2)*sum((-a^2*d^2+2*a*b*c*d-b^2*c^2)/(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*_alp
ha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*Ellipt
icPi(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*_al
pha*(-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)))

_______________________________________________________________________________________

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((d*x^3 + c)^(3/2)*x^8/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225285, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (4 \, a^{2} b c d - 7 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 7 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \,{\left (6 \, b^{3} d^{2} x^{9} + 2 \,{\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{6} + 6 \, a b^{2} c^{2} - 95 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (3 \, b^{3} c^{2} - 34 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{90 \,{\left (b^{5} d x^{3} + a b^{4} d\right )}}, \frac{15 \,{\left (4 \, a^{2} b c d - 7 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 7 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (6 \, b^{3} d^{2} x^{9} + 2 \,{\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{6} + 6 \, a b^{2} c^{2} - 95 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (3 \, b^{3} c^{2} - 34 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{45 \,{\left (b^{5} d x^{3} + a b^{4} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/90*(15*(4*a^2*b*c*d - 7*a^3*d^2 + (4*a*b^2*c*d - 7*a^2*b*d^2)*x^3)*sqrt((b*c
- a*d)/b)*log((b*d*x^3 + 2*b*c - a*d - 2*sqrt(d*x^3 + c)*b*sqrt((b*c - a*d)/b))
/(b*x^3 + a)) - 2*(6*b^3*d^2*x^9 + 2*(6*b^3*c*d - 7*a*b^2*d^2)*x^6 + 6*a*b^2*c^2
- 95*a^2*b*c*d + 105*a^3*d^2 + 2*(3*b^3*c^2 - 34*a*b^2*c*d + 35*a^2*b*d^2)*x^3)
*sqrt(d*x^3 + c))/(b^5*d*x^3 + a*b^4*d), 1/45*(15*(4*a^2*b*c*d - 7*a^3*d^2 + (4*
a*b^2*c*d - 7*a^2*b*d^2)*x^3)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x^3 + c)/sqrt(-
(b*c - a*d)/b)) + (6*b^3*d^2*x^9 + 2*(6*b^3*c*d - 7*a*b^2*d^2)*x^6 + 6*a*b^2*c^2
- 95*a^2*b*c*d + 105*a^3*d^2 + 2*(3*b^3*c^2 - 34*a*b^2*c*d + 35*a^2*b*d^2)*x^3)
*sqrt(d*x^3 + c))/(b^5*d*x^3 + a*b^4*d)]

_______________________________________________________________________________________

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)**(3/2)/(b*x**3+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223333, size = 285, normalized size = 1.51 \[ -\frac{{\left (4 \, a b^{2} c^{2} - 11 \, a^{2} b c d + 7 \, a^{3} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{4}} - \frac{\sqrt{d x^{3} + c} a^{2} b c d - \sqrt{d x^{3} + c} a^{3} d^{2}}{3 \,{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{4}} + \frac{2 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} b^{8} d^{4} - 10 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} a b^{7} d^{5} - 30 \, \sqrt{d x^{3} + c} a b^{7} c d^{5} + 45 \, \sqrt{d x^{3} + c} a^{2} b^{6} d^{6}\right )}}{45 \, b^{10} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(4*a*b^2*c^2 - 11*a^2*b*c*d + 7*a^3*d^2)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2
*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^4) - 1/3*(sqrt(d*x^3 + c)*a^2*b*c*d - sqrt(
d*x^3 + c)*a^3*d^2)/(((d*x^3 + c)*b - b*c + a*d)*b^4) + 2/45*(3*(d*x^3 + c)^(5/2
)*b^8*d^4 - 10*(d*x^3 + c)^(3/2)*a*b^7*d^5 - 30*sqrt(d*x^3 + c)*a*b^7*c*d^5 + 45
*sqrt(d*x^3 + c)*a^2*b^6*d^6)/(b^10*d^5)