3.471 \(\int \frac{x^5 \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=163 \[ -\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2}}+\frac{\sqrt{c+d x^3} (2 b c-5 a d)}{3 b^3}+\frac{\left (c+d x^3\right )^{3/2} (2 b c-5 a d)}{9 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b \left (a+b x^3\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.388328, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2}}+\frac{\sqrt{c+d x^3} (2 b c-5 a d)}{3 b^3}+\frac{\left (c+d x^3\right )^{3/2} (2 b c-5 a d)}{9 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b \left (a+b x^3\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 38.7524, size = 144, normalized size = 0.88 \[ - \frac{a \left (c + d x^{3}\right )^{\frac{5}{2}}}{3 b \left (a + b x^{3}\right ) \left (a d - b c\right )} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}} \left (\frac{5 a d}{2} - b c\right )}{9 b^{2} \left (a d - b c\right )} - \frac{2 \sqrt{c + d x^{3}} \left (\frac{5 a d}{2} - b c\right )}{3 b^{3}} + \frac{2 \sqrt{a d - b c} \left (\frac{5 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(d*x**3+c)**(3/2)/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.299008, size = 115, normalized size = 0.71 \[ \frac{\sqrt{c+d x^3} \left (-\frac{3 a (a d-b c)}{a+b x^3}-12 a d+8 b c+2 b d x^3\right )}{9 b^3}-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]

[Out]

(Sqrt[c + d*x^3]*(8*b*c - 12*a*d + 2*b*d*x^3 - (3*a*(-(b*c) + a*d))/(a + b*x^3))
)/(9*b^3) - ((2*b*c - 5*a*d)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/S
qrt[b*c - a*d]])/(3*b^(7/2))

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Maple [C]  time = 0.016, size = 983, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(d*x^3+c)^(3/2)/(b*x^3+a)^2,x)

[Out]

1/b*(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)))-a/b*(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
)))

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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((d*x^3 + c)^(3/2)*x^5/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224496, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \,{\left (2 \, b^{2} d x^{6} + 2 \,{\left (4 \, b^{2} c - 5 \, a b d\right )} x^{3} + 11 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c}}{18 \,{\left (b^{4} x^{3} + a b^{3}\right )}}, -\frac{3 \,{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (2 \, b^{2} d x^{6} + 2 \,{\left (4 \, b^{2} c - 5 \, a b d\right )} x^{3} + 11 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c}}{9 \,{\left (b^{4} x^{3} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)*x^5/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

[-1/18*(3*((2*b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 5*a^2*d)*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*(2*b^2*d*x^6 + 2*(4*b^2*c - 5*a*b*d)*x^3 + 11*a*b*c - 15*a^2*d)*sqrt(d*x^3 +
c))/(b^4*x^3 + a*b^3), -1/9*(3*((2*b^2*c - 5*a*b*d)*x^3 + 2*a*b*c - 5*a^2*d)*sqr
t(-(b*c - a*d)/b)*arctan(sqrt(d*x^3 + c)/sqrt(-(b*c - a*d)/b)) - (2*b^2*d*x^6 +
2*(4*b^2*c - 5*a*b*d)*x^3 + 11*a*b*c - 15*a^2*d)*sqrt(d*x^3 + c))/(b^4*x^3 + a*b
^3)]

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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**5*(d*x**3+c)**(3/2)/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219562, size = 234, normalized size = 1.44 \[ \frac{{\left (2 \, b^{2} c^{2} - 7 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{3}} + \frac{\sqrt{d x^{3} + c} a b c d - \sqrt{d x^{3} + c} a^{2} d^{2}}{3 \,{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{4} + 3 \, \sqrt{d x^{3} + c} b^{4} c - 6 \, \sqrt{d x^{3} + c} a b^{3} d\right )}}{9 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)*x^5/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

1/3*(2*b^2*c^2 - 7*a*b*c*d + 5*a^2*d^2)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a
*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) + 1/3*(sqrt(d*x^3 + c)*a*b*c*d - sqrt(d*x^3 +
c)*a^2*d^2)/(((d*x^3 + c)*b - b*c + a*d)*b^3) + 2/9*((d*x^3 + c)^(3/2)*b^4 + 3*s
qrt(d*x^3 + c)*b^4*c - 6*sqrt(d*x^3 + c)*a*b^3*d)/b^6