3.392 \(\int \frac{1}{x^4 \left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 c^{5/2}}-\frac{d (b c-3 a d)}{3 a c^2 \sqrt{c+d x^3} (b c-a d)}-\frac{1}{3 a c x^3 \sqrt{c+d x^3}} \]

[Out]

-(d*(b*c - 3*a*d))/(3*a*c^2*(b*c - a*d)*Sqrt[c + d*x^3]) - 1/(3*a*c*x^3*Sqrt[c +
d*x^3]) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2*c^(5/2)) -
(2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*a^2*(b*c - a*d
)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.640556, antiderivative size = 158, 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 \[ -\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 c^{5/2}}-\frac{d (b c-3 a d)}{3 a c^2 \sqrt{c+d x^3} (b c-a d)}-\frac{1}{3 a c x^3 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(b*c - 3*a*d))/(3*a*c^2*(b*c - a*d)*Sqrt[c + d*x^3]) - 1/(3*a*c*x^3*Sqrt[c +
d*x^3]) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2*c^(5/2)) -
(2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*a^2*(b*c - a*d
)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 72.9312, size = 139, normalized size = 0.88 \[ - \frac{1}{3 a c x^{3} \sqrt{c + d x^{3}}} - \frac{d \left (3 a d - b c\right )}{3 a c^{2} \sqrt{c + d x^{3}} \left (a d - b c\right )} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\left (3 a d + 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a*c*x**3*sqrt(c + d*x**3)) - d*(3*a*d - b*c)/(3*a*c**2*sqrt(c + d*x**3)*(a
*d - b*c)) - 2*b**(5/2)*atan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3*a**2*(
a*d - b*c)**(3/2)) + (3*a*d + 2*b*c)*atanh(sqrt(c + d*x**3)/sqrt(c))/(3*a**2*c**
(5/2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.23161, size = 501, normalized size = 3.17 \[ \frac{\frac{6 b c d x^6 (b c-3 a d) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{(b c-a d) \left (x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}-\frac{5 b d x^3 \left (-3 a^2 d \left (c+2 d x^3\right )+a b \left (3 c^2-c d x^3-9 d^2 x^6\right )+b^2 c x^3 \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+3 \left (a^2 d \left (c+3 d x^3\right )-a b \left (c^2-3 d^2 x^6\right )-b^2 c x^3 \left (c+d x^3\right )\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{a (a d-b c) \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}}{9 c^2 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.014, size = 575, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(-1/3*(d*x^3+c)^(1/2)/c^2/x^3-2/3*d/c^2/((x^3+c/d)*d)^(1/2)+d*arctanh((d*x^3
+c)^(1/2)/c^(1/2))/c^(5/2))+1/a^2*b^2*(-2/3/(a*d-b*c)/((x^3+c/d)*d)^(1/2)-1/3*I/
d^2*b*2^(1/2)*sum(1/(-a*d+b*c)/(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*(2/3/c/((x^3+c/d)*d)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)^(3/2)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)*(d*x^3 + c)^(3/2)*x^4), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.364694, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)^(3/2)*x^4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2194, size = 248, normalized size = 1.57 \[ \frac{1}{3} \,{\left (\frac{2 \, b^{3} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{{\left (d x^{3} + c\right )} b c - 3 \,{\left (d x^{3} + c\right )} a d + 2 \, a c d}{{\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )}{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - \sqrt{d x^{3} + c} c\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{2} d^{2}}\right )} d^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)^(3/2)*x^4),x, algorithm="giac")

[Out]

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