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

Optimal. Leaf size=114 \[ \frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a (b c-a d)^{3/2}}-\frac{2 d}{3 c \sqrt{c+d x^3} (b c-a d)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.349047, antiderivative size = 114, 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 \[ \frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a (b c-a d)^{3/2}}-\frac{2 d}{3 c \sqrt{c+d x^3} (b c-a d)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.6775, size = 97, normalized size = 0.85 \[ \frac{2 d}{3 c \sqrt{c + d x^{3}} \left (a d - b c\right )} + \frac{2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a \left (a d - b c\right )^{\frac{3}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.804994, size = 396, normalized size = 3.47 \[ \frac{2 d \left (\frac{6 a b x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{5 b x^3 \left (2 a d+b \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (a+b x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{c \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}\right )}{9 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*d*((6*a*b*x^3*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), -((b*x^3)/a)])/(-4*a*c*App
ellF1[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*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*(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))
]))/(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)*(a + b*x^3)*Sqrt[c + d*
x^3])

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Maple [C]  time = 0.013, size = 512, normalized size = 4.5 \[{\frac{1}{a} \left ({\frac{2}{3\,c}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{2}{3}{\it Artanh} \left ({1\sqrt{d{x}^{3}+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \right ) }-{\frac{b}{a} \left ( -{\frac{2}{3\,ad-3\,bc}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{{\frac{i}{3}}b\sqrt{2}}{{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ \left ( -ad+bc \right ) \left ( ad-bc \right ) }\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(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
*(-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*_a
lpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*Elli
pticPi(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. \[ \int \frac{1}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.267099, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{d x^{3} + c} b c^{\frac{3}{2}} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{3} + a}\right ) + 2 \, a \sqrt{c} d - \sqrt{d x^{3} + c}{\left (b c - a d\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{3 \,{\left (a b c^{2} - a^{2} c d\right )} \sqrt{d x^{3} + c} \sqrt{c}}, \frac{2 \, \sqrt{d x^{3} + c} b c^{\frac{3}{2}} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{3} + c} b}\right ) - 2 \, a \sqrt{c} d + \sqrt{d x^{3} + c}{\left (b c - a d\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{3 \,{\left (a b c^{2} - a^{2} c d\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{\sqrt{d x^{3} + c} b \sqrt{-c} c \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{3} + a}\right ) + 2 \, a \sqrt{-c} d - 2 \, \sqrt{d x^{3} + c}{\left (b c - a d\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{3 \,{\left (a b c^{2} - a^{2} c d\right )} \sqrt{d x^{3} + c} \sqrt{-c}}, \frac{2 \,{\left (\sqrt{d x^{3} + c} b \sqrt{-c} c \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{3} + c} b}\right ) - a \sqrt{-c} d + \sqrt{d x^{3} + c}{\left (b c - a d\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )\right )}}{3 \,{\left (a b c^{2} - a^{2} c d\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217764, size = 158, normalized size = 1.39 \[ -\frac{2}{3} \,{\left (\frac{b^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a b c d - a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{1}{\sqrt{d x^{3} + c}{\left (b c^{2} - a c d\right )}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} c d}\right )} d \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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