3.31.70 \(\int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx\)

Optimal. Leaf size=496 \[ -\frac {\log \left (2 x^2+7 x+3\right )}{5 \sqrt [3]{10}}+\frac {\log \left (4 x^4+28 x^3+61 x^2+42 x+9\right )}{10 \sqrt [3]{10}}+\frac {\log \left (2 \sqrt [3]{10} x^3+3 \sqrt [3]{10} x^2+5 \sqrt [3]{8 x^8+84 x^7+338 x^6+679 x^5+825 x^4+784 x^3+522 x^2+189 x+27}-11 \sqrt [3]{10} x-6 \sqrt [3]{10}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (4\ 10^{2/3} x^6+12\ 10^{2/3} x^5-35\ 10^{2/3} x^4-90\ 10^{2/3} x^3+85\ 10^{2/3} x^2+25 \left (8 x^8+84 x^7+338 x^6+679 x^5+825 x^4+784 x^3+522 x^2+189 x+27\right )^{2/3}+\left (-10 \sqrt [3]{10} x^3-15 \sqrt [3]{10} x^2+55 \sqrt [3]{10} x+30 \sqrt [3]{10}\right ) \sqrt [3]{8 x^8+84 x^7+338 x^6+679 x^5+825 x^4+784 x^3+522 x^2+189 x+27}+132\ 10^{2/3} x+36\ 10^{2/3}\right )}{10 \sqrt [3]{10}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {5 \sqrt {3} \sqrt [3]{8 x^8+84 x^7+338 x^6+679 x^5+825 x^4+784 x^3+522 x^2+189 x+27}}{-4 \sqrt [3]{10} x^3-6 \sqrt [3]{10} x^2+5 \sqrt [3]{8 x^8+84 x^7+338 x^6+679 x^5+825 x^4+784 x^3+522 x^2+189 x+27}+22 \sqrt [3]{10} x+12 \sqrt [3]{10}}\right )}{5 \sqrt [3]{10}} \]

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Rubi [F]  time = 0.34, 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 {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3),x]

[Out]

((1 + x^2)^(1/3)*(3 + 7*x + 2*x^2)*Defer[Int][(1 + x)/((1 + x^2)^(1/3)*(3 + 7*x + 2*x^2)), x])/((1 + x^2)*(3 +
7*x + 2*x^2)^3)^(1/3)

Rubi steps

\begin {align*} \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx &=\int \frac {1+x}{\sqrt [3]{\left (1+x^2\right ) \left (3+7 x+2 x^2\right )^3}} \, dx\\ &=\frac {\left (\sqrt [3]{1+x^2} \left (3+7 x+2 x^2\right )\right ) \int \frac {1+x}{\sqrt [3]{1+x^2} \left (3+7 x+2 x^2\right )} \, dx}{\sqrt [3]{\left (1+x^2\right ) \left (3+7 x+2 x^2\right )^3}}\\ \end {align*}

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Mathematica [F]  time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3),x]

[Out]

Integrate[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3), x]

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IntegrateAlgebraic [A]  time = 1.05, size = 496, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {5 \sqrt {3} \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}{12 \sqrt [3]{10}+22 \sqrt [3]{10} x-6 \sqrt [3]{10} x^2-4 \sqrt [3]{10} x^3+5 \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (3+7 x+2 x^2\right )}{5 \sqrt [3]{10}}+\frac {\log \left (9+42 x+61 x^2+28 x^3+4 x^4\right )}{10 \sqrt [3]{10}}+\frac {\log \left (-6 \sqrt [3]{10}-11 \sqrt [3]{10} x+3 \sqrt [3]{10} x^2+2 \sqrt [3]{10} x^3+5 \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (36\ 10^{2/3}+132\ 10^{2/3} x+85\ 10^{2/3} x^2-90\ 10^{2/3} x^3-35\ 10^{2/3} x^4+12\ 10^{2/3} x^5+4\ 10^{2/3} x^6+\left (30 \sqrt [3]{10}+55 \sqrt [3]{10} x-15 \sqrt [3]{10} x^2-10 \sqrt [3]{10} x^3\right ) \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}+25 \left (27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8\right )^{2/3}\right )}{10 \sqrt [3]{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/
3),x]

[Out]

(Sqrt[3]*ArcTan[(5*Sqrt[3]*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/
3))/(12*10^(1/3) + 22*10^(1/3)*x - 6*10^(1/3)*x^2 - 4*10^(1/3)*x^3 + 5*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x
^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3))])/(5*10^(1/3)) - Log[3 + 7*x + 2*x^2]/(5*10^(1/3)) + Log[9 + 4
2*x + 61*x^2 + 28*x^3 + 4*x^4]/(10*10^(1/3)) + Log[-6*10^(1/3) - 11*10^(1/3)*x + 3*10^(1/3)*x^2 + 2*10^(1/3)*x
^3 + 5*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3)]/(5*10^(1/3)) - L
og[36*10^(2/3) + 132*10^(2/3)*x + 85*10^(2/3)*x^2 - 90*10^(2/3)*x^3 - 35*10^(2/3)*x^4 + 12*10^(2/3)*x^5 + 4*10
^(2/3)*x^6 + (30*10^(1/3) + 55*10^(1/3)*x - 15*10^(1/3)*x^2 - 10*10^(1/3)*x^3)*(27 + 189*x + 522*x^2 + 784*x^3
+ 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8)^(1/3) + 25*(27 + 189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^
5 + 338*x^6 + 84*x^7 + 8*x^8)^(2/3)]/(10*10^(1/3))

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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((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(1/3),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 {x + 1}{{\left (8 \, x^{8} + 84 \, x^{7} + 338 \, x^{6} + 679 \, x^{5} + 825 \, x^{4} + 784 \, x^{3} + 522 \, x^{2} + 189 \, x + 27\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(1/3),x, algorithm="giac")

[Out]

integrate((x + 1)/(8*x^8 + 84*x^7 + 338*x^6 + 679*x^5 + 825*x^4 + 784*x^3 + 522*x^2 + 189*x + 27)^(1/3), x)

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maple [C]  time = 20.44, size = 5476, normalized size = 11.04

method	result	size
trager	\(\text {Expression too large to display}\)	\(5476\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(1/3),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{{\left (8 \, x^{8} + 84 \, x^{7} + 338 \, x^{6} + 679 \, x^{5} + 825 \, x^{4} + 784 \, x^{3} + 522 \, x^{2} + 189 \, x + 27\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(1/3),x, algorithm="maxima")

[Out]

integrate((x + 1)/(8*x^8 + 84*x^7 + 338*x^6 + 679*x^5 + 825*x^4 + 784*x^3 + 522*x^2 + 189*x + 27)^(1/3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x+1}{{\left (8\,x^8+84\,x^7+338\,x^6+679\,x^5+825\,x^4+784\,x^3+522\,x^2+189\,x+27\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8 + 27)^(1/3),x)

[Out]

int((x + 1)/(189*x + 522*x^2 + 784*x^3 + 825*x^4 + 679*x^5 + 338*x^6 + 84*x^7 + 8*x^8 + 27)^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(8*x**8+84*x**7+338*x**6+679*x**5+825*x**4+784*x**3+522*x**2+189*x+27)**(1/3),x)

[Out]

Integral((x + 1)/((x + 3)**3*(2*x + 1)**3*(x**2 + 1))**(1/3), x)

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