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

Optimal. Leaf size=132 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^3}}{3 a \left (a+b x^3\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.441004, antiderivative size = 132, 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 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 (b c-a d)^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^3}}{3 a \left (a+b x^3\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.0701, size = 117, normalized size = 0.89 \[ - \frac{b \sqrt{c + d x^{3}}}{3 a \left (a + b x^{3}\right ) \left (a d - b c\right )} - \frac{2 \sqrt{b} \left (\frac{3 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(c + d*x**3)/(3*a*(a + b*x**3)*(a*d - b*c)) - 2*sqrt(b)*(3*a*d/2 - b*c)*a
tan(sqrt(b)*sqrt(c + d*x**3)/sqrt(a*d - b*c))/(3*a**2*(a*d - b*c)**(3/2)) - 2*at
anh(sqrt(c + d*x**3)/sqrt(c))/(3*a**2*sqrt(c))

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Mathematica [C]  time = 0.44019, size = 396, normalized size = 3. \[ \frac{b \left (\frac{6 c d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{5 d x^3 \left (2 a d+b \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (c+d x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{a \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}\right )}{9 \left (a+b x^3\right ) \sqrt{c+d x^3} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(b*((6*c*d*x^3*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(-4*a*c*Appel
lF1[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*d*x^3*(2*a*d + b*(c + 3*d*x^3))*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*
x^3)), -(a/(b*x^3))] - 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*(-(b*c) + a*d)*(a + b*x^3)*Sqrt[c + d
*x^3])

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Maple [C]  time = 0.018, size = 915, normalized size = 6.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/a^2/c^(1/2)-b/a*(1/3/(a*d-b*c)*(d*x^3+c)^(
1/2)/(b*x^3+a)-1/6*I/d*2^(1/2)*sum(1/(a*d-b*c)^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)))+1/3*I*b/a^2/d^2*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)
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269009, 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*sqrt(d*x^3 + c)*x),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21869, size = 207, normalized size = 1.57 \[ -\frac{1}{3} \, d^{2}{\left (\frac{{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x^{3} + c} b}{{\left (a b c d - a^{2} d^{2}\right )}{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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