3.493 \(\int \frac{1}{x \left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\)
Optimal. Leaf size=172 \[ \frac{b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 (b c-a d)^{5/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 c^{3/2}}+\frac{b}{3 a \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{d (2 a d+b c)}{3 a c \sqrt{c+d x^3} (b c-a d)^2} \]
[Out]
(d*(b*c + 2*a*d))/(3*a*c*(b*c - a*d)^2*Sqrt[c + d*x^3]) + b/(3*a*(b*c - a*d)*(a
+ b*x^3)*Sqrt[c + d*x^3]) - (2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2*c^(3/2))
+ (b^(3/2)*(2*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/
(3*a^2*(b*c - a*d)^(5/2))
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Rubi [A] time = 0.732981, antiderivative size = 172, 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
\[ \frac{b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 (b c-a d)^{5/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 c^{3/2}}+\frac{b}{3 a \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{d (2 a d+b c)}{3 a c \sqrt{c+d x^3} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
(d*(b*c + 2*a*d))/(3*a*c*(b*c - a*d)^2*Sqrt[c + d*x^3]) + b/(3*a*(b*c - a*d)*(a
+ b*x^3)*Sqrt[c + d*x^3]) - (2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^2*c^(3/2))
+ (b^(3/2)*(2*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/
(3*a^2*(b*c - a*d)^(5/2))
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Rubi in Sympy [A] time = 76.2381, size = 148, normalized size = 0.86 \[ - \frac{b}{3 a \left (a + b x^{3}\right ) \sqrt{c + d x^{3}} \left (a d - b c\right )} + \frac{d \left (2 a d + b c\right )}{3 a c \sqrt{c + d x^{3}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} \left (5 a d - 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \left (a d - b c\right )^{\frac{5}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x**3+a)**2/(d*x**3+c)**(3/2),x)
[Out]
-b/(3*a*(a + b*x**3)*sqrt(c + d*x**3)*(a*d - b*c)) + d*(2*a*d + b*c)/(3*a*c*sqrt
(c + d*x**3)*(a*d - b*c)**2) + b**(3/2)*(5*a*d - 2*b*c)*atan(sqrt(b)*sqrt(c + d*
x**3)/sqrt(a*d - b*c))/(3*a**2*(a*d - b*c)**(5/2)) - 2*atanh(sqrt(c + d*x**3)/sq
rt(c))/(3*a**2*c**(3/2))
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Mathematica [C] time = 1.11039, size = 453, normalized size = 2.63 \[ \frac{\frac{3 \left (2 a^2 d^2+2 a b d^2 x^3+b^2 c \left (c+d x^3\right )\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )-5 b d x^3 \left (4 a^2 d^2+2 a b d \left (2 c+3 d x^3\right )+b^2 c \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}{a c \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}-\frac{6 b d x^3 (2 a d+b c) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}}{9 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
((-6*b*d*(b*c + 2*a*d)*x^3*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*(4*a^2*d^2 + b^2*c*(c + 3*d*x^3) + 2*a*b*d*(2
*c + 3*d*x^3))*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), -(a/(b*x^3))] + 3*(2*a^2
*d^2 + 2*a*b*d^2*x^3 + b^2*c*(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*c*(-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*(b*c - a*d)^2*(a + b*x^3)*Sqrt[
c + d*x^3])
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Maple [C] time = 0.017, size = 1002, normalized size = 5.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x^3+a)^2/(d*x^3+c)^(3/2),x)
[Out]
1/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))-b
/a*(-1/3*b/(a*d-b*c)^2*(d*x^3+c)^(1/2)/(b*x^3+a)-2/3*d/(a*d-b*c)^2/((x^3+c/d)*d)
^(1/2)+1/2*I*b/d*2^(1/2)*sum(1/(a*d-b*c)^3*(-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/(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)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{2}{\left (d x^{3} + c\right )}^{\frac{3}{2}} x}\,{d x} \]
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),x, algorithm="maxima")
[Out]
integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x), x)
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Fricas [A] time = 0.419228, 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)^2*(d*x^3 + c)^(3/2)*x),x, algorithm="fricas")
[Out]
[-1/6*((2*a*b^2*c^2 - 5*a^2*b*c*d + (2*b^3*c^2 - 5*a*b^2*c*d)*x^3)*sqrt(d*x^3 +
c)*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*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d
^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^3)*sqrt(d*x^3 + c)*log(((d*x^3 + 2*c)
*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) - 2*(a*b^2*c^2 + 2*a^3*d^2 + (a*b^2*c*d + 2
*a^2*b*d^2)*x^3)*sqrt(c))/((a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*d^2 + (a^2*b^3*c
^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(c)), 1/3*((2*a*b^2
*c^2 - 5*a^2*b*c*d + (2*b^3*c^2 - 5*a*b^2*c*d)*x^3)*sqrt(d*x^3 + c)*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)) +
(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^3)*s
qrt(d*x^3 + c)*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) + (a*b^2*c
^2 + 2*a^3*d^2 + (a*b^2*c*d + 2*a^2*b*d^2)*x^3)*sqrt(c))/((a^3*b^2*c^3 - 2*a^4*b
*c^2*d + a^5*c*d^2 + (a^2*b^3*c^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^3)*sqrt(d*x
^3 + c)*sqrt(c)), -1/6*((2*a*b^2*c^2 - 5*a^2*b*c*d + (2*b^3*c^2 - 5*a*b^2*c*d)*x
^3)*sqrt(d*x^3 + c)*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)) - 4*(a*b^2*c^2 - 2
*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^3)*sqrt(d*x^3 + c)*
arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) - 2*(a*b^2*c^2 + 2*a^3*d^2 + (a*b^2*c*d + 2
*a^2*b*d^2)*x^3)*sqrt(-c))/((a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5*c*d^2 + (a^2*b^3*
c^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt(-c)), 1/3*((2*a*b
^2*c^2 - 5*a^2*b*c*d + (2*b^3*c^2 - 5*a*b^2*c*d)*x^3)*sqrt(d*x^3 + c)*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^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x
^3)*sqrt(d*x^3 + c)*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) + (a*b^2*c^2 + 2*a^3*d^
2 + (a*b^2*c*d + 2*a^2*b*d^2)*x^3)*sqrt(-c))/((a^3*b^2*c^3 - 2*a^4*b*c^2*d + a^5
*c*d^2 + (a^2*b^3*c^3 - 2*a^3*b^2*c^2*d + a^4*b*c*d^2)*x^3)*sqrt(d*x^3 + c)*sqrt
(-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(1/x/(b*x**3+a)**2/(d*x**3+c)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.223288, size = 319, normalized size = 1.85 \[ -\frac{1}{3} \, d^{2}{\left (\frac{{\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{{\left (d x^{3} + c\right )} b^{2} c + 2 \,{\left (d x^{3} + c\right )} a b d - 2 \, a b c d + 2 \, a^{2} d^{2}}{{\left (a b^{2} c^{3} d - 2 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )}{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{3} + c} b c + \sqrt{d x^{3} + c} a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}}\right )} \]
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),x, algorithm="giac")
[Out]
-1/3*d^2*((2*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(
(a^2*b^2*c^2*d^2 - 2*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b^2*c + a*b*d)) - ((d*x^3 + c)
*b^2*c + 2*(d*x^3 + c)*a*b*d - 2*a*b*c*d + 2*a^2*d^2)/((a*b^2*c^3*d - 2*a^2*b*c^
2*d^2 + a^3*c*d^3)*((d*x^3 + c)^(3/2)*b - sqrt(d*x^3 + c)*b*c + sqrt(d*x^3 + c)*
a*d)) - 2*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(a^2*sqrt(-c)*c*d^2))