3.509 \(\int \frac{1}{x^4 \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=91 \[ \frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right )}{3 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.280019, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right )}{3 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.1217, size = 78, normalized size = 0.86 \[ - \frac{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}{3 a c x^{3}} + \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{3}}}{\sqrt{a} \sqrt{c + d x^{3}}} \right )}}{3 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.338298, size = 192, normalized size = 2.11 \[ \frac{\frac{2 b d x^6 (a d+b c) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )}{4 b d x^3 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )-b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )-a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )}-\left (a+b x^3\right ) \left (c+d x^3\right )}{3 a c x^3 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-((a + b*x^3)*(c + d*x^3)) + (2*b*d*(b*c + a*d)*x^6*AppellF1[1, 1/2, 1/2, 2, -(
a/(b*x^3)), -(c/(d*x^3))])/(4*b*d*x^3*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^3)), -(c
/(d*x^3))] - b*c*AppellF1[2, 1/2, 3/2, 3, -(a/(b*x^3)), -(c/(d*x^3))] - a*d*Appe
llF1[2, 3/2, 1/2, 3, -(a/(b*x^3)), -(c/(d*x^3))]))/(3*a*c*x^3*Sqrt[a + b*x^3]*Sq
rt[c + d*x^3])

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291949, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} +{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{6} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt{a c}}{x^{6}}\right ) - 4 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} \sqrt{a c}}{12 \, \sqrt{a c} a c x^{3}}, \frac{{\left (b c + a d\right )} x^{3} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{3} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} a c}\right ) - 2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} \sqrt{-a c}}{6 \, \sqrt{-a c} a c x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*((b*c + a*d)*x^3*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x^3)*sqrt(b*x^3 +
a)*sqrt(d*x^3 + c) + ((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 8*a^2*c^2 + 8*(a*b*
c^2 + a^2*c*d)*x^3)*sqrt(a*c))/x^6) - 4*sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(a*c
))/(sqrt(a*c)*a*c*x^3), 1/6*((b*c + a*d)*x^3*arctan(1/2*((b*c + a*d)*x^3 + 2*a*c
)*sqrt(-a*c)/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*a*c)) - 2*sqrt(b*x^3 + a)*sqrt(d*x
^3 + c)*sqrt(-a*c))/(sqrt(-a*c)*a*c*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.242471, size = 558, normalized size = 6.13 \[ \frac{\sqrt{b d} b^{4} d{\left (\frac{{\left (b c + a d\right )} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} a b^{3} c d} - \frac{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} -{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} b c -{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} a d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{2} a b d +{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{4}\right )} a b^{2} c d}\right )}}{3 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(b*d)*b^4*d*((b*c + a*d)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*x^3 + a)*s
qrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-
a*b*c*d)*a*b^3*c*d) - 2*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2 - (sqrt(b*x^3 + a)*sq
rt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2*b*c - (sqrt(b*x^3 + a)*sqrt(b
*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^2*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^
2*b^2*d^2 - 2*(sqrt(b*x^3 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d)
)^2*b^2*c - 2*(sqrt(b*x^3 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d)
)^2*a*b*d + (sqrt(b*x^3 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d))^
4)*a*b^2*c*d))/abs(b)