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

Optimal. Leaf size=136 \[ -\frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^3} (2 b c-3 a d)}{3 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right ) (b c-a d)} \]

[Out]

((2*b*c - 3*a*d)*Sqrt[c + d*x^3])/(3*b^2*(b*c - a*d)) + (a*(c + d*x^3)^(3/2))/(3
*b*(b*c - a*d)*(a + b*x^3)) - ((2*b*c - 3*a*d)*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 = 0.306957, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^3} (2 b c-3 a d)}{3 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{3/2}}{3 b \left (a+b x^3\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*c - 3*a*d)*Sqrt[c + d*x^3])/(3*b^2*(b*c - a*d)) + (a*(c + d*x^3)^(3/2))/(3
*b*(b*c - a*d)*(a + b*x^3)) - ((2*b*c - 3*a*d)*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 = 34.437, size = 117, normalized size = 0.86 \[ - \frac{a \left (c + d x^{3}\right )^{\frac{3}{2}}}{3 b \left (a + b x^{3}\right ) \left (a d - b c\right )} + \frac{2 \sqrt{c + d x^{3}} \left (\frac{3 a d}{2} - b c\right )}{3 b^{2} \left (a d - b c\right )} - \frac{2 \left (\frac{3 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{5}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(d*x**3+c)**(1/2)/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.156974, size = 91, normalized size = 0.67 \[ \frac{1}{3} \left (\frac{\left (\frac{a}{a+b x^3}+2\right ) \sqrt{c+d x^3}}{b^2}-\frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{5/2} \sqrt{b c-a d}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.016, size = 897, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(d*x^3+c)^(1/2)/(b*x^3+a)^2,x)

[Out]

1/b*(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*_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*(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)*_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))*Elliptic
Pi(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*_alph
a*(-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(sqrt(d*x^3 + c)*x^5/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223743, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b x^{3} + 3 \, a\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} -{\left ({\left (2 \, b^{2} c - 3 \, a b d\right )} x^{3} + 2 \, a b c - 3 \, a^{2} d\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{6 \,{\left (b^{3} x^{3} + a b^{2}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \, b x^{3} + 3 \, a\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} -{\left ({\left (2 \, b^{2} c - 3 \, a b d\right )} x^{3} + 2 \, a b c - 3 \, a^{2} d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (b^{3} x^{3} + a b^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(2*(2*b*x^3 + 3*a)*sqrt(d*x^3 + c)*sqrt(b^2*c - a*b*d) - ((2*b^2*c - 3*a*b*
d)*x^3 + 2*a*b*c - 3*a^2*d)*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^3*x^3 + a*b^2)*sqrt(b^2*c -
a*b*d)), 1/3*((2*b*x^3 + 3*a)*sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d) - ((2*b^2*c -
3*a*b*d)*x^3 + 2*a*b*c - 3*a^2*d)*arctan(-(b*c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^
2*c + a*b*d))))/((b^3*x^3 + a*b^2)*sqrt(-b^2*c + a*b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \sqrt{c + d x^{3}}}{\left (a + b x^{3}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(d*x**3+c)**(1/2)/(b*x**3+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219888, size = 150, normalized size = 1.1 \[ \frac{\frac{\sqrt{d x^{3} + c} a d^{2}}{{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{2}} + \frac{2 \, \sqrt{d x^{3} + c} d}{b^{2}} + \frac{{\left (2 \, b c d - 3 \, a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(sqrt(d*x^3 + c)*a*d^2/(((d*x^3 + c)*b - b*c + a*d)*b^2) + 2*sqrt(d*x^3 + c)
*d/b^2 + (2*b*c*d - 3*a*d^2)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqr
t(-b^2*c + a*b*d)*b^2))/d