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

Optimal. Leaf size=154 \[ -\frac{2 a^2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}+\frac{2 a^2 \sqrt{c+d x^3} (b c-a d)}{3 b^4}+\frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}-\frac{2 \left (c+d x^3\right )^{5/2} (a d+b c)}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2} \]

[Out]

(2*a^2*(b*c - a*d)*Sqrt[c + d*x^3])/(3*b^4) + (2*a^2*(c + d*x^3)^(3/2))/(9*b^3)
- (2*(b*c + a*d)*(c + d*x^3)^(5/2))/(15*b^2*d^2) + (2*(c + d*x^3)^(7/2))/(21*b*d
^2) - (2*a^2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]
])/(3*b^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.41926, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2 a^2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}+\frac{2 a^2 \sqrt{c+d x^3} (b c-a d)}{3 b^4}+\frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}-\frac{2 \left (c+d x^3\right )^{5/2} (a d+b c)}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*(b*c - a*d)*Sqrt[c + d*x^3])/(3*b^4) + (2*a^2*(c + d*x^3)^(3/2))/(9*b^3)
- (2*(b*c + a*d)*(c + d*x^3)^(5/2))/(15*b^2*d^2) + (2*(c + d*x^3)^(7/2))/(21*b*d
^2) - (2*a^2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]
])/(3*b^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 45.8148, size = 141, normalized size = 0.92 \[ \frac{2 a^{2} \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 b^{3}} - \frac{2 a^{2} \sqrt{c + d x^{3}} \left (a d - b c\right )}{3 b^{4}} + \frac{2 a^{2} \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{9}{2}}} + \frac{2 \left (c + d x^{3}\right )^{\frac{7}{2}}}{21 b d^{2}} - \frac{2 \left (c + d x^{3}\right )^{\frac{5}{2}} \left (a d + b c\right )}{15 b^{2} d^{2}} \]

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),x)

[Out]

2*a**2*(c + d*x**3)**(3/2)/(9*b**3) - 2*a**2*sqrt(c + d*x**3)*(a*d - b*c)/(3*b**
4) + 2*a**2*(a*d - b*c)**(3/2)*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3
*b**(9/2)) + 2*(c + d*x**3)**(7/2)/(21*b*d**2) - 2*(c + d*x**3)**(5/2)*(a*d + b*
c)/(15*b**2*d**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.376429, size = 143, normalized size = 0.93 \[ \frac{2 \sqrt{c+d x^3} \left (-105 a^3 d^3+35 a^2 b d^2 \left (4 c+d x^3\right )-21 a b^2 d \left (c+d x^3\right )^2-3 b^3 \left (2 c-5 d x^3\right ) \left (c+d x^3\right )^2\right )}{315 b^4 d^2}-\frac{2 a^2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x^3]*(-105*a^3*d^3 - 21*a*b^2*d*(c + d*x^3)^2 - 3*b^3*(2*c - 5*d*x
^3)*(c + d*x^3)^2 + 35*a^2*b*d^2*(4*c + d*x^3)))/(315*b^4*d^2) - (2*a^2*(b*c - a
*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(9/2))

_______________________________________________________________________________________

Maple [C]  time = 0.052, size = 605, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b^2*(b*(2/21*d*x^9*(d*x^3+c)^(1/2)+16/105*c*x^6*(d*x^3+c)^(1/2)+2/105*c^2/d*x^
3*(d*x^3+c)^(1/2)-4/105*c^3/d^2*(d*x^3+c)^(1/2))-2/15*a/d*(d*x^3+c)^(5/2))+a^2/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*_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)))

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240937, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \,{\left (15 \, b^{3} d^{3} x^{9} + 3 \,{\left (8 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{6} - 6 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 140 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} +{\left (3 \, b^{3} c^{2} d - 42 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{315 \, b^{4} d^{2}}, -\frac{2 \,{\left (105 \,{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (15 \, b^{3} d^{3} x^{9} + 3 \,{\left (8 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{6} - 6 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 140 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} +{\left (3 \, b^{3} c^{2} d - 42 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c}\right )}}{315 \, b^{4} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/315*(105*(a^2*b*c*d^2 - a^3*d^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*(15*b^3*d^3*x^9
+ 3*(8*b^3*c*d^2 - 7*a*b^2*d^3)*x^6 - 6*b^3*c^3 - 21*a*b^2*c^2*d + 140*a^2*b*c*d
^2 - 105*a^3*d^3 + (3*b^3*c^2*d - 42*a*b^2*c*d^2 + 35*a^2*b*d^3)*x^3)*sqrt(d*x^3
+ c))/(b^4*d^2), -2/315*(105*(a^2*b*c*d^2 - a^3*d^3)*sqrt(-(b*c - a*d)/b)*arcta
n(sqrt(d*x^3 + c)/sqrt(-(b*c - a*d)/b)) - (15*b^3*d^3*x^9 + 3*(8*b^3*c*d^2 - 7*a
*b^2*d^3)*x^6 - 6*b^3*c^3 - 21*a*b^2*c^2*d + 140*a^2*b*c*d^2 - 105*a^3*d^3 + (3*
b^3*c^2*d - 42*a*b^2*c*d^2 + 35*a^2*b*d^3)*x^3)*sqrt(d*x^3 + c))/(b^4*d^2)]

_______________________________________________________________________________________

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),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220906, size = 261, normalized size = 1.69 \[ \frac{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{4}} + \frac{2 \,{\left (15 \,{\left (d x^{3} + c\right )}^{\frac{7}{2}} b^{6} d^{12} - 21 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} b^{6} c d^{12} - 21 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} a b^{5} d^{13} + 35 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} a^{2} b^{4} d^{14} + 105 \, \sqrt{d x^{3} + c} a^{2} b^{4} c d^{14} - 105 \, \sqrt{d x^{3} + c} a^{3} b^{3} d^{15}\right )}}{315 \, b^{7} d^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*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) + 2/315*(15*(d*x^3 + c)^(7/2)*b^6*d^12 - 21*
(d*x^3 + c)^(5/2)*b^6*c*d^12 - 21*(d*x^3 + c)^(5/2)*a*b^5*d^13 + 35*(d*x^3 + c)^
(3/2)*a^2*b^4*d^14 + 105*sqrt(d*x^3 + c)*a^2*b^4*c*d^14 - 105*sqrt(d*x^3 + c)*a^
3*b^3*d^15)/(b^7*d^14)