3.23.14 \(\int \frac {b+a x^6}{\sqrt [3]{x+x^3} (d+c x^6)} \, dx\)

Optimal. Leaf size=164 \[ \frac {(a d-b c) \text {RootSum}\left [\text {$\#$1}^9 d-3 \text {$\#$1}^6 d+3 \text {$\#$1}^3 d+c-d\& ,\frac {\log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 c d}-\frac {a \log \left (\sqrt [3]{x^3+x}-x\right )}{2 c}+\frac {\sqrt {3} a \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{2 c}+\frac {a \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )}{4 c} \]

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Rubi [F]  time = 1.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]

[Out]

(Sqrt[3]*a*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*c*(x + x^3)^(1/3)) -
(3*a*x^(1/3)*(1 + x^2)^(1/3)*Log[x^(2/3) - (1 + x^2)^(1/3)])/(4*c*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 +
x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-d^(1/9) - c^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*
(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-d^(1/9) + (-1)^(1/9)*c^(1
/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)
*Defer[Subst][Defer[Int][1/((-d^(1/9) - (-1)^(2/9)*c^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*
(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-d^(1/9) + (-1)^(1/3)*c^(1
/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)
*Defer[Subst][Defer[Int][1/((-d^(1/9) - (-1)^(4/9)*c^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*
(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-d^(1/9) + (-1)^(5/9)*c^(1
/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)
*Defer[Subst][Defer[Int][1/((-d^(1/9) - (-1)^(2/3)*c^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*
(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-d^(1/9) + (-1)^(7/9)*c^(1
/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*(x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(1 + x^2)^(1/3)
*Defer[Subst][Defer[Int][1/((-d^(1/9) - (-1)^(8/9)*c^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*c*d^(8/9)*
(x + x^3)^(1/3))

Rubi steps

\begin {align*} \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {b+a x^6}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (d+c x^6\right )} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {b+a x^9}{\sqrt [3]{1+x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {a}{c \sqrt [3]{1+x^3}}+\frac {b c-a d}{c \sqrt [3]{1+x^3} \left (d+c x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (3 a \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}}+\frac {\left (3 (b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}}\\ &=\frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x+x^3}}+\frac {\left (3 (b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}}\\ &=\frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}\\ \end {align*}

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Mathematica [F]  time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]

[Out]

Integrate[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 167, normalized size = 1.02 \begin {gather*} \frac {\sqrt {3} a \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 c}-\frac {a \log \left (-c x+c \sqrt [3]{x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + a*x^6)/((x + x^3)^(1/3)*(d + c*x^6)),x]

[Out]

(Sqrt[3]*a*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^3)^(1/3))])/(2*c) - (a*Log[-(c*x) + c*(x + x^3)^(1/3)])/(2*c) + (a
*Log[x^2 + x*(x + x^3)^(1/3) + (x + x^3)^(2/3)])/(4*c) + ((-(b*c) + a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 +
d*#1^9 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ])/(6*c*d)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x, algorithm="giac")

[Out]

integrate((a*x^6 + b)/((c*x^6 + d)*(x^3 + x)^(1/3)), x)

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}+b}{\left (x^{3}+x \right )^{\frac {1}{3}} \left (c \,x^{6}+d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x)

[Out]

int((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/(x^3+x)^(1/3)/(c*x^6+d),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)/((c*x^6 + d)*(x^3 + x)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^6+b}{\left (c\,x^6+d\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + a*x^6)/((d + c*x^6)*(x + x^3)^(1/3)),x)

[Out]

int((b + a*x^6)/((d + c*x^6)*(x + x^3)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (c x^{6} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**6+b)/(x**3+x)**(1/3)/(c*x**6+d),x)

[Out]

Integral((a*x**6 + b)/((x*(x**2 + 1))**(1/3)*(c*x**6 + d)), x)

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