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

Optimal. Leaf size=121 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b} \sqrt{b c-a d}}-\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}+\frac{\sqrt{c+d x^3}}{3 a \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.36369, antiderivative size = 121, 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 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b} \sqrt{b c-a d}}-\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}+\frac{\sqrt{c+d x^3}}{3 a \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.262, size = 105, normalized size = 0.87 \[ \frac{\sqrt{c + d x^{3}}}{3 a \left (a + b x^{3}\right )} - \frac{2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2}} + \frac{2 \left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \sqrt{b} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.368235, size = 306, normalized size = 2.53 \[ \frac{\frac{\frac{10 b c d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}{-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}+3 \left (c+d x^3\right )}{a}-\frac{6 c d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}}{9 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-6*c*d*x^3*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(-4*a*c*AppellF
1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)] + x^3*(2*b*c*AppellF1[2, 1/2, 2, 3,
-((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[2, 3/2, 1, 3, -((d*x^3)/c), -((b*x^3)
/a)])) + (3*(c + d*x^3) + (10*b*c*d*x^3*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)),
-(a/(b*x^3))])/(-5*b*d*x^3*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3)
)] + 2*a*d*AppellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^3)), -(a/(b*x^3))] + b*c*AppellF1
[5/2, 3/2, 1, 7/2, -(c/(d*x^3)), -(a/(b*x^3))]))/a)/(9*(a + b*x^3)*Sqrt[c + d*x^
3])

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Maple [C]  time = 0.018, size = 934, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a^2*(2/3*(d*x^3+c)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))*c^(1/2))-b/a*(-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))*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)),_alp
ha=RootOf(_Z^3*b+a)))-b/a^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*_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. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (b x^{3} + a\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x), x)

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Fricas [A]  time = 0.250116, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b x^{3} + a\right )} \sqrt{b^{2} c - a b d} \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} a -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{6 \,{\left (a^{2} b x^{3} + a^{3}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (b x^{3} + a\right )} \sqrt{-b^{2} c + a b d} \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} a +{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (a^{2} b x^{3} + a^{3}\right )} \sqrt{-b^{2} c + a b d}}, -\frac{4 \,{\left (b x^{3} + a\right )} \sqrt{b^{2} c - a b d} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) - 2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} a +{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{6 \,{\left (a^{2} b x^{3} + a^{3}\right )} \sqrt{b^{2} c - a b d}}, -\frac{2 \,{\left (b x^{3} + a\right )} \sqrt{-b^{2} c + a b d} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) - \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} a -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (a^{2} b x^{3} + a^{3}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(2*(b*x^3 + a)*sqrt(b^2*c - a*b*d)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 + c)*s
qrt(c) + 2*c)/x^3) + 2*sqrt(d*x^3 + c)*sqrt(b^2*c - a*b*d)*a - ((2*b^2*c - a*b*d
)*x^3 + 2*a*b*c - a^2*d)*log(((b*d*x^3 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) - 2*sq
rt(d*x^3 + c)*(b^2*c - a*b*d))/(b*x^3 + a)))/((a^2*b*x^3 + a^3)*sqrt(b^2*c - a*b
*d)), 1/3*((b*x^3 + a)*sqrt(-b^2*c + a*b*d)*sqrt(c)*log((d*x^3 - 2*sqrt(d*x^3 +
c)*sqrt(c) + 2*c)/x^3) + sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d)*a + ((2*b^2*c - a*
b*d)*x^3 + 2*a*b*c - a^2*d)*arctan(-(b*c - a*d)/(sqrt(d*x^3 + c)*sqrt(-b^2*c + a
*b*d))))/((a^2*b*x^3 + a^3)*sqrt(-b^2*c + a*b*d)), -1/6*(4*(b*x^3 + a)*sqrt(b^2*
c - a*b*d)*sqrt(-c)*arctan(sqrt(d*x^3 + c)/sqrt(-c)) - 2*sqrt(d*x^3 + c)*sqrt(b^
2*c - a*b*d)*a + ((2*b^2*c - a*b*d)*x^3 + 2*a*b*c - 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)))/(
(a^2*b*x^3 + a^3)*sqrt(b^2*c - a*b*d)), -1/3*(2*(b*x^3 + a)*sqrt(-b^2*c + a*b*d)
*sqrt(-c)*arctan(sqrt(d*x^3 + c)/sqrt(-c)) - sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d
)*a - ((2*b^2*c - a*b*d)*x^3 + 2*a*b*c - a^2*d)*arctan(-(b*c - a*d)/(sqrt(d*x^3
+ c)*sqrt(-b^2*c + a*b*d))))/((a^2*b*x^3 + a^3)*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((d*x**3+c)**(1/2)/x/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219857, size = 170, normalized size = 1.4 \[ \frac{1}{3} \, d^{2}{\left (\frac{\sqrt{d x^{3} + c}}{{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} a d} - \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} + \frac{2 \, c \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^2*(sqrt(d*x^3 + c)/(((d*x^3 + c)*b - b*c + a*d)*a*d) - (2*b*c - a*d)*arcta
n(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^2*d^2) + 2*c*a
rctan(sqrt(d*x^3 + c)/sqrt(-c))/(a^2*sqrt(-c)*d^2))