∫ 3 √ ____ a+bx3 _x(ad−bdx3 ) dx

3.571 \(\int \frac {\sqrt [3]{a+b x^3}}{x (a d-b d x^3)} \, dx\)

Optimal. Leaf size=214 \[ \frac {\log \left (a-b x^3\right )}{3\ 2^{2/3} a^{2/3} d}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{2/3} d}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}-\frac {\log (x)}{2 a^{2/3} d} \]

[Out]

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

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

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

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Rubi [A] time = 0.17, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {446, 83, 57, 617, 204, 31} \[ \frac {\log \left (a-b x^3\right )}{3\ 2^{2/3} a^{2/3} d}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{2/3} d}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}-\frac {\log (x)}{2 a^{2/3} d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

2^(2/3)*a^(2/3)*d) + Log[a^(1/3) - (a + b*x^3)^(1/3)]/(2*a^(2/3)*d) - Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)]

/(2^(2/3)*a^(2/3)*d)

Rule 31

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L

og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x

)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]

&& PosQ[(b*c - a*d)/b]

Rule 83

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[(b*e - a*f)/(b*c

- a*d), Int[(e + f*x)^(p - 1)/(a + b*x), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[(e + f*x)^(p - 1)/(c + d*

x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int

[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&

NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps

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

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Mathematica [A] time = 0.11, size = 233, normalized size = 1.09 \[ -\frac {\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )-2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{6 a^{2/3} d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3) - (a + b*x^3)^(1/3)] + 2*2^(1/3)*Log[2^(1/3)*a^(1/3) - (a

+ b*x^3)^(1/3)] + Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 2^(1/3)*Log[2^(2/3)*a^(2/3) +

2^(1/3)*a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(a^(2/3)*d)

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fricas [B] time = 0.74, size = 1046, normalized size = 4.89 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

-2/3*sqrt(3)*(1/2)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*arctan(1/12*(1/2)^(2/3)*(sqrt(3)

*a^3*d^5*sqrt(1/(a^4*d^6)) - 3*sqrt(3)*a*d^2)*sqrt(-4*(1/2)^(1/3)*(b*x^3 + a)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(

1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 4*(1/2)^(2/3)*a^2*d^2*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) +

1)/(a^2*d^3))^(2/3) + 4*(b*x^3 + a)^(2/3))*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(2/3) - 1/6*(1/2)^(2

/3)*(sqrt(3)*a^3*d^5*sqrt(1/(a^4*d^6)) - 3*sqrt(3)*a*d^2)*(b*x^3 + a)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1

)/(a^2*d^3))^(2/3) + 1/3*sqrt(3)) + 2/3*sqrt(3)*(1/2)^(1/3)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3

)*arctan(1/6*(1/2)^(2/3)*(sqrt(3)*a^3*d^5*sqrt(1/(a^4*d^6)) + 3*sqrt(3)*a*d^2)*sqrt((1/2)^(1/3)*(b*x^3 + a)^(1

/3)*a^3*d^4*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + (1/2)^(2/3)*a^2*d^2*((3*a^

2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(2/3) + (b*x^3 + a)^(2/3))*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3

))^(2/3) - 1/6*(1/2)^(2/3)*(sqrt(3)*a^3*d^5*sqrt(1/(a^4*d^6)) + 3*sqrt(3)*a*d^2)*(b*x^3 + a)^(1/3)*((3*a^2*d^3

*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(2/3) - 1/3*sqrt(3)) - 1/6*(1/2)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/

(a^2*d^3))^(1/3)*log(-4*(1/2)^(1/3)*(b*x^3 + a)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(

1/3)*sqrt(1/(a^4*d^6)) + 4*(1/2)^(2/3)*a^2*d^2*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(2/3) + 4*(b*x^3

+ a)^(2/3)) - 1/6*(1/2)^(1/3)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*log(4*(1/2)^(1/3)*(b*x^3 +

a)^(1/3)*a^3*d^4*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 4*(1/2)^(2/3)*a^2*d^2

*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(2/3) + 4*(b*x^3 + a)^(2/3)) + 1/3*(1/2)^(1/3)*(-(3*a^2*d^3*sqr

t(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*log((1/2)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1

/3)*sqrt(1/(a^4*d^6)) + (b*x^3 + a)^(1/3)) + 1/3*(1/2)^(1/3)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/

3)*log(-(1/2)^(1/3)*a^3*d^4*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + (b*x^3 + a

)^(1/3))

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Algebraic extensions not allowed in a ro

otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio