3.464 \(\int \frac{\sqrt{c+d x^3}}{x^4 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=161 \[ -\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3 \sqrt{c}}-\frac{2 b \sqrt{c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac{\sqrt{c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]

[Out]

(-2*b*Sqrt[c + d*x^3])/(3*a^2*(a + b*x^3)) - Sqrt[c + d*x^3]/(3*a*x^3*(a + b*x^3
)) + ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^3*Sqrt[c]) - (Sqrt[b]
*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*a^3*Sqrt
[b*c - a*d])

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Rubi [A]  time = 0.657199, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{b} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 \sqrt{b c-a d}}+\frac{(4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3 \sqrt{c}}-\frac{2 b \sqrt{c+d x^3}}{3 a^2 \left (a+b x^3\right )}-\frac{\sqrt{c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^3]/(x^4*(a + b*x^3)^2),x]

[Out]

(-2*b*Sqrt[c + d*x^3])/(3*a^2*(a + b*x^3)) - Sqrt[c + d*x^3]/(3*a*x^3*(a + b*x^3
)) + ((4*b*c - a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^3*Sqrt[c]) - (Sqrt[b]
*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*a^3*Sqrt
[b*c - a*d])

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Rubi in Sympy [A]  time = 60.8262, size = 136, normalized size = 0.84 \[ \frac{\sqrt{c + d x^{3}}}{3 a x^{3} \left (a + b x^{3}\right )} - \frac{2 \sqrt{c + d x^{3}}}{3 a^{2} x^{3}} - \frac{\sqrt{b} \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{3} \sqrt{a d - b c}} - \frac{\left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{3} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**(1/2)/x**4/(b*x**3+a)**2,x)

[Out]

sqrt(c + d*x**3)/(3*a*x**3*(a + b*x**3)) - 2*sqrt(c + d*x**3)/(3*a**2*x**3) - sq
rt(b)*(3*a*d - 4*b*c)*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3*a**3*sqr
t(a*d - b*c)) - (a*d - 4*b*c)*atanh(sqrt(c + d*x**3)/sqrt(c))/(3*a**3*sqrt(c))

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Mathematica [C]  time = 0.620375, size = 410, normalized size = 2.55 \[ \frac{\frac{12 a b c d x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{5 b d x^3 \left (3 a c+4 a d x^3+2 b c x^3+6 b d x^6\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (a+2 b x^3\right ) \left (c+d x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}}{9 a^2 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[c + d*x^3]/(x^4*(a + b*x^3)^2),x]

[Out]

((12*a*b*c*d*x^6*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(-4*a*c*App
ellF1[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)])) + (5*b*d*x^3*(3*a*c + 2*b*c*x^3 + 4*a*d*x^3 + 6*b*d*x^6)*AppellF1[3/2
, 1/2, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3))] - 3*(a + 2*b*x^3)*(c + d*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))]))/(-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))]))/(9*a^2*x^
3*(a + b*x^3)*Sqrt[c + d*x^3])

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Maple [C]  time = 0.019, size = 978, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^(1/2)/x^4/(b*x^3+a)^2,x)

[Out]

1/a^2*(-1/3*(d*x^3+c)^(1/2)/x^3-1/3*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))+
1/a^2*b^2*(-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)),_alpha=RootOf(_Z^3*b+a)))-2*b/a^3*(2/3*(d*x^3+c)^(1/2)-2/3*arctanh((d*x
^3+c)^(1/2)/c^(1/2))*c^(1/2))+2/a^3*b^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))*E
llipticPi(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. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (b x^{3} + a\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^4),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^4), x)

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Fricas [A]  time = 0.255736, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^4),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228008, size = 258, normalized size = 1.6 \[ -\frac{1}{3} \, d^{3}{\left (\frac{2 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} b - 2 \, \sqrt{d x^{3} + c} b c + \sqrt{d x^{3} + c} a d}{{\left ({\left (d x^{3} + c\right )}^{2} b - 2 \,{\left (d x^{3} + c\right )} b c + b c^{2} +{\left (d x^{3} + c\right )} a d - a c d\right )} a^{2} d^{2}} - \frac{{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} d^{3}} + \frac{{\left (4 \, b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)/((b*x^3 + a)^2*x^4),x, algorithm="giac")

[Out]

-1/3*d^3*((2*(d*x^3 + c)^(3/2)*b - 2*sqrt(d*x^3 + c)*b*c + sqrt(d*x^3 + c)*a*d)/
(((d*x^3 + c)^2*b - 2*(d*x^3 + c)*b*c + b*c^2 + (d*x^3 + c)*a*d - a*c*d)*a^2*d^2
) - (4*b^2*c - 3*a*b*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^
2*c + a*b*d)*a^3*d^3) + (4*b*c - a*d)*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(a^3*sqrt
(-c)*d^3))