3.275 \(\int \frac{a+\frac{b}{x^3}}{c+\frac{d}{x^3}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{(b c-a d) \log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{c} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} c^{4/3} d^{2/3}}+\frac{a x}{c} \]

[Out]

(a*x)/c - ((b*c - a*d)*ArcTan[(d^(1/3) - 2*c^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*c^(4/3)*d^(2/3)) + ((b*c -
a*d)*Log[d^(1/3) + c^(1/3)*x])/(3*c^(4/3)*d^(2/3)) - ((b*c - a*d)*Log[d^(2/3) - c^(1/3)*d^(1/3)*x + c^(2/3)*x^
2])/(6*c^(4/3)*d^(2/3))

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Rubi [A]  time = 0.105609, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {374, 388, 200, 31, 634, 617, 204, 628} \[ -\frac{(b c-a d) \log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{c} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} c^{4/3} d^{2/3}}+\frac{a x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^3)/(c + d/x^3),x]

[Out]

(a*x)/c - ((b*c - a*d)*ArcTan[(d^(1/3) - 2*c^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*c^(4/3)*d^(2/3)) + ((b*c -
a*d)*Log[d^(1/3) + c^(1/3)*x])/(3*c^(4/3)*d^(2/3)) - ((b*c - a*d)*Log[d^(2/3) - c^(1/3)*d^(1/3)*x + c^(2/3)*x^
2])/(6*c^(4/3)*d^(2/3))

Rule 374

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[x^(n*(p + q))*(b + a/x^n)^
p*(d + c/x^n)^q, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[p, q] && NegQ[n]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{a+\frac{b}{x^3}}{c+\frac{d}{x^3}} \, dx &=\int \frac{b+a x^3}{d+c x^3} \, dx\\ &=\frac{a x}{c}-\frac{(-b c+a d) \int \frac{1}{d+c x^3} \, dx}{c}\\ &=\frac{a x}{c}+\frac{(b c-a d) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{c} x} \, dx}{3 c d^{2/3}}+\frac{(b c-a d) \int \frac{2 \sqrt [3]{d}-\sqrt [3]{c} x}{d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2} \, dx}{3 c d^{2/3}}\\ &=\frac{a x}{c}+\frac{(b c-a d) \log \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 c^{2/3} x}{d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2} \, dx}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \int \frac{1}{d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2} \, dx}{2 c \sqrt [3]{d}}\\ &=\frac{a x}{c}+\frac{(b c-a d) \log \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2\right )}{6 c^{4/3} d^{2/3}}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{c^{4/3} d^{2/3}}\\ &=\frac{a x}{c}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{c} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} c^{4/3} d^{2/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{3 c^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (d^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3} x^2\right )}{6 c^{4/3} d^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0835389, size = 129, normalized size = 0.89 \[ \frac{-(b c-a d) \log \left (c^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3}\right )+2 (b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )-2 \sqrt{3} (b c-a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{c} x}{\sqrt [3]{d}}}{\sqrt{3}}\right )+6 a \sqrt [3]{c} d^{2/3} x}{6 c^{4/3} d^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^3)/(c + d/x^3),x]

[Out]

(6*a*c^(1/3)*d^(2/3)*x - 2*Sqrt[3]*(b*c - a*d)*ArcTan[(1 - (2*c^(1/3)*x)/d^(1/3))/Sqrt[3]] + 2*(b*c - a*d)*Log
[d^(1/3) + c^(1/3)*x] - (b*c - a*d)*Log[d^(2/3) - c^(1/3)*d^(1/3)*x + c^(2/3)*x^2])/(6*c^(4/3)*d^(2/3))

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Maple [A]  time = 0.004, size = 195, normalized size = 1.3 \begin{align*}{\frac{ax}{c}}-{\frac{ad}{3\,{c}^{2}}\ln \left ( x+\sqrt [3]{{\frac{d}{c}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{3\,c}\ln \left ( x+\sqrt [3]{{\frac{d}{c}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ad}{6\,{c}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{d}{c}}}x+ \left ({\frac{d}{c}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{6\,c}\ln \left ({x}^{2}-\sqrt [3]{{\frac{d}{c}}}x+ \left ({\frac{d}{c}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}ad}{3\,{c}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{c}}}}}}-1 \right ) } \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{3\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{c}}}}}}-1 \right ) } \right ) \left ({\frac{d}{c}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^3)/(c+d/x^3),x)

[Out]

a*x/c-1/3/c^2/(d/c)^(2/3)*ln(x+(d/c)^(1/3))*a*d+1/3/c/(d/c)^(2/3)*ln(x+(d/c)^(1/3))*b+1/6/c^2/(d/c)^(2/3)*ln(x
^2-(d/c)^(1/3)*x+(d/c)^(2/3))*a*d-1/6/c/(d/c)^(2/3)*ln(x^2-(d/c)^(1/3)*x+(d/c)^(2/3))*b-1/3/c^2/(d/c)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(d/c)^(1/3)*x-1))*a*d+1/3/c/(d/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/c)^(1/3)*
x-1))*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)/(c+d/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34319, size = 926, normalized size = 6.39 \begin{align*} \left [\frac{6 \, a c d^{2} x - 3 \, \sqrt{\frac{1}{3}}{\left (b c^{2} d - a c d^{2}\right )} \sqrt{\frac{\left (-c d^{2}\right )^{\frac{1}{3}}}{c}} \log \left (\frac{2 \, c d x^{3} + 3 \, \left (-c d^{2}\right )^{\frac{1}{3}} d x - d^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, c d x^{2} + \left (-c d^{2}\right )^{\frac{2}{3}} x + \left (-c d^{2}\right )^{\frac{1}{3}} d\right )} \sqrt{\frac{\left (-c d^{2}\right )^{\frac{1}{3}}}{c}}}{c x^{3} + d}\right ) - \left (-c d^{2}\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (c d x^{2} - \left (-c d^{2}\right )^{\frac{2}{3}} x - \left (-c d^{2}\right )^{\frac{1}{3}} d\right ) + 2 \, \left (-c d^{2}\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (c d x + \left (-c d^{2}\right )^{\frac{2}{3}}\right )}{6 \, c^{2} d^{2}}, \frac{6 \, a c d^{2} x + 6 \, \sqrt{\frac{1}{3}}{\left (b c^{2} d - a c d^{2}\right )} \sqrt{-\frac{\left (-c d^{2}\right )^{\frac{1}{3}}}{c}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (-c d^{2}\right )^{\frac{2}{3}} x + \left (-c d^{2}\right )^{\frac{1}{3}} d\right )} \sqrt{-\frac{\left (-c d^{2}\right )^{\frac{1}{3}}}{c}}}{d^{2}}\right ) - \left (-c d^{2}\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (c d x^{2} - \left (-c d^{2}\right )^{\frac{2}{3}} x - \left (-c d^{2}\right )^{\frac{1}{3}} d\right ) + 2 \, \left (-c d^{2}\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (c d x + \left (-c d^{2}\right )^{\frac{2}{3}}\right )}{6 \, c^{2} d^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)/(c+d/x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.51837, size = 71, normalized size = 0.49 \begin{align*} \frac{a x}{c} + \operatorname{RootSum}{\left (27 t^{3} c^{4} d^{2} + a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t c d}{a d - b c} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/c + RootSum(27*_t**3*c**4*d**2 + a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3, Lambda(_t, _t*
log(-3*_t*c*d/(a*d - b*c) + x)))

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Giac [A]  time = 1.13226, size = 217, normalized size = 1.5 \begin{align*} \frac{a x}{c} - \frac{{\left (b c - a d\right )} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{d}{c}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d} + \frac{\sqrt{3}{\left (\left (-c^{2} d\right )^{\frac{1}{3}} b c - \left (-c^{2} d\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{d}{c}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{d}{c}\right )^{\frac{1}{3}}}\right )}{3 \, c^{2} d} + \frac{{\left (\left (-c^{2} d\right )^{\frac{1}{3}} b c - \left (-c^{2} d\right )^{\frac{1}{3}} a d\right )} \log \left (x^{2} + x \left (-\frac{d}{c}\right )^{\frac{1}{3}} + \left (-\frac{d}{c}\right )^{\frac{2}{3}}\right )}{6 \, c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)/(c+d/x^3),x, algorithm="giac")

[Out]

a*x/c - 1/3*(b*c - a*d)*(-d/c)^(1/3)*log(abs(x - (-d/c)^(1/3)))/(c*d) + 1/3*sqrt(3)*((-c^2*d)^(1/3)*b*c - (-c^
2*d)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-d/c)^(1/3))/(-d/c)^(1/3))/(c^2*d) + 1/6*((-c^2*d)^(1/3)*b*c - (-c^
2*d)^(1/3)*a*d)*log(x^2 + x*(-d/c)^(1/3) + (-d/c)^(2/3))/(c^2*d)