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

Optimal. Leaf size=67 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};2,\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 c x^2 \sqrt{c+d x^3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.205784, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};2,\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 c x^2 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.4225, size = 58, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{3}{2},2,\frac{1}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a^{2} c^{2} x^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)*appellf1(-2/3, 3/2, 2, 1/3, -d*x**3/c, -b*x**3/a)/(2*a**2*c**2
*x**2*sqrt(1 + d*x**3/c))

_______________________________________________________________________________________

Mathematica [B]  time = 2.13514, size = 483, normalized size = 7.21 \[ \frac{\frac{7 a b c d x^6 \left (7 a^2 d^2-6 a b c d+5 b^2 c^2\right ) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{16 a c x^3 \left (7 a^3 d^3-6 a^2 b c d^2-33 a b^2 c^2 d+20 b^3 c^3\right ) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}-4 \left (a^3 d^2 \left (3 c+7 d x^3\right )+a^2 b d \left (-6 c^2-3 c d x^3+7 d^2 x^6\right )+3 a b^2 c \left (c^2-c d x^3-2 d^2 x^6\right )+5 b^3 c^2 x^3 \left (c+d x^3\right )\right )}{24 a^2 c^2 x^2 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*(5*b^3*c^2*x^3*(c + d*x^3) + a^3*d^2*(3*c + 7*d*x^3) + 3*a*b^2*c*(c^2 - c*d*
x^3 - 2*d^2*x^6) + a^2*b*d*(-6*c^2 - 3*c*d*x^3 + 7*d^2*x^6)) + (16*a*c*(20*b^3*c
^3 - 33*a*b^2*c^2*d - 6*a^2*b*c*d^2 + 7*a^3*d^3)*x^3*AppellF1[1/3, 1/2, 1, 4/3,
-((d*x^3)/c), -((b*x^3)/a)])/(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -(
(b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -((b*x^3)/a)
] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])) + (7*a*b*c*d*(5
*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x^6*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -
((b*x^3)/a)])/(-14*a*c*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)] +
3*x^3*(2*b*c*AppellF1[7/3, 1/2, 2, 10/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*Appel
lF1[7/3, 3/2, 1, 10/3, -((d*x^3)/c), -((b*x^3)/a)])))/(24*a^2*c^2*(b*c - a*d)^2*
x^2*(a + b*x^3)*Sqrt[c + d*x^3])

_______________________________________________________________________________________

Maple [C]  time = 0.017, size = 1919, normalized size = 28.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a^2*(-1/2/c^2*(d*x^3+c)^(1/2)/x^2-2/3*d/c^2*x/((x^3+c/d)*d)^(1/2)+7/18*I/c^2*3
^(1/2)*(-c*d^2)^(1/3)*(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)*((x-1/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)*(-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)/(d*x^3+c)^(1/2)*EllipticF(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),(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)))-b/a*(1/3*b^2/a/(a*d-b*c)^2*x*(d*x^3+c)^(1/2)/(b*
x^3+a)+2/3*d^2/c*x/(a*d-b*c)^2/((x^3+c/d)*d)^(1/2)-2/3*I*(1/6*b*d/(a*d-b*c)^2/a+
1/3*d^2/c/(a*d-b*c)^2)*3^(1/2)/d*(-c*d^2)^(1/3)*(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)*((x-1/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)*(-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)/(d*
x^3+c)^(1/2)*EllipticF(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),(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))+1/18*I/a/d^2*b*2^(1/2)*
sum((13*a*d-4*b*c)/(a*d-b*c)^3/_alpha^2*(-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*d/c*x/(a*d-b*c)/((x^3+c/d)*d)^(1/2)-2/9*I/c/(a*d-b*c)*3^(1/2)*(-c*d^2)^(
1/3)*(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)*((x-1/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)*(-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)/(d*x^3+c)^(1/2)*EllipticF(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),(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))+1/3*I/d^2*b*2^(1/2)*sum(1/(a*d-b*c)^2/_alpha^2*(-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=R
ootOf(_Z^3*b+a)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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