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

Optimal. Leaf size=116 \[ -\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}-\frac{c \sqrt{c+d x^3}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.401954, antiderivative size = 116, 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 \[ -\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}-\frac{c \sqrt{c+d x^3}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.1943, size = 104, normalized size = 0.9 \[ - \frac{c \sqrt{c + d x^{3}}}{3 a x^{3}} - \frac{\sqrt{c} \left (3 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2}} + \frac{2 \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.62109, size = 414, normalized size = 3.57 \[ \frac{c \left (\frac{5 b d x^3 \left (3 a \left (c+2 d x^3\right )+b x^3 \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{a \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}+\frac{6 d x^6 (b c-2 a d) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}\right )}{9 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(c*((6*d*(b*c - 2*a*d)*x^6*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(
-4*a*c*AppellF1[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*d*x^3) + b*x^3*(c + 3*d*x^3))*Appe
llF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3))] - 3*(a + 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))]))/(a*(-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*x^3*(a + b*x^3)*Sqrt[c + d*x^3])

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Maple [C]  time = 0.013, size = 620, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(-1/3*c*(d*x^3+c)^(1/2)/x^3+2/3*d*(d*x^3+c)^(1/2)-c^(1/2)*d*arctanh((d*x^3+c
)^(1/2)/c^(1/2)))+1/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)))-b/a^2*(2/9*d*x^3*(d*x^3+c)^(
1/2)+8/9*c*(d*x^3+c)^(1/2)-2/3*c^(3/2)*arctanh((d*x^3+c)^(1/2)/c^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.285111, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (b c - a d\right )} x^{3} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) +{\left (2 \, b c - 3 \, a d\right )} \sqrt{c} x^{3} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a c}{6 \, a^{2} x^{3}}, -\frac{4 \,{\left (b c - a d\right )} x^{3} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, b c - 3 \, a d\right )} \sqrt{c} x^{3} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a c}{6 \, a^{2} x^{3}}, \frac{{\left (2 \, b c - 3 \, a d\right )} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) -{\left (b c - a d\right )} x^{3} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - \sqrt{d x^{3} + c} a c}{3 \, a^{2} x^{3}}, \frac{{\left (2 \, b c - 3 \, a d\right )} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) - 2 \,{\left (b c - a d\right )} x^{3} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) - \sqrt{d x^{3} + c} a c}{3 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(2*(b*c - a*d)*x^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*b*c - 3*a*d)*sqrt(c)*x^3*lo
g((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*sqrt(d*x^3 + c)*a*c)/(a^2*x
^3), -1/6*(4*(b*c - a*d)*x^3*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x^3 + c)/sqrt(-(
b*c - a*d)/b)) + (2*b*c - 3*a*d)*sqrt(c)*x^3*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt
(c) + 2*c)/x^3) + 2*sqrt(d*x^3 + c)*a*c)/(a^2*x^3), 1/3*((2*b*c - 3*a*d)*sqrt(-c
)*x^3*arctan(sqrt(d*x^3 + c)/sqrt(-c)) - (b*c - a*d)*x^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))
- sqrt(d*x^3 + c)*a*c)/(a^2*x^3), 1/3*((2*b*c - 3*a*d)*sqrt(-c)*x^3*arctan(sqrt(
d*x^3 + c)/sqrt(-c)) - 2*(b*c - a*d)*x^3*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x^3
+ c)/sqrt(-(b*c - a*d)/b)) - sqrt(d*x^3 + c)*a*c)/(a^2*x^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((d*x**3+c)**(3/2)/x**4/(b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219405, size = 182, normalized size = 1.57 \[ \frac{1}{3} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{3} + c} c}{a d^{2} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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