3.31.70 \(\int \frac {1+x}{\sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}} \, dx\) [3070]
Optimal. Leaf size=496 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {5 \sqrt {3} \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}{12 \sqrt [3]{10}+22 \sqrt [3]{10} x-6 \sqrt [3]{10} x^2-4 \sqrt [3]{10} x^3+5 \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (3+7 x+2 x^2\right )}{5 \sqrt [3]{10}}+\frac {\log \left (9+42 x+61 x^2+28 x^3+4 x^4\right )}{10 \sqrt [3]{10}}+\frac {\log \left (-6 \sqrt [3]{10}-11 \sqrt [3]{10} x+3 \sqrt [3]{10} x^2+2 \sqrt [3]{10} x^3+5 \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}\right )}{5 \sqrt [3]{10}}-\frac {\log \left (36\ 10^{2/3}+132\ 10^{2/3} x+85\ 10^{2/3} x^2-90\ 10^{2/3} x^3-35\ 10^{2/3} x^4+12\ 10^{2/3} x^5+4\ 10^{2/3} x^6+\left (30 \sqrt [3]{10}+55 \sqrt [3]{10} x-15 \sqrt [3]{10} x^2-10 \sqrt [3]{10} x^3\right ) \sqrt [3]{27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8}+25 \left (27+189 x+522 x^2+784 x^3+825 x^4+679 x^5+338 x^6+84 x^7+8 x^8\right )^{2/3}\right )}{10 \sqrt [3]{10}} \]
[Out]
1/50*3^(1/2)*arctan(5*3^(1/2)*(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)/(12*10^(1/
3)+22*10^(1/3)*x-6*10^(1/3)*x^2-4*10^(1/3)*x^3+5*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+2
7)^(1/3)))*10^(2/3)-1/50*ln(2*x^2+7*x+3)*10^(2/3)+1/100*ln(4*x^4+28*x^3+61*x^2+42*x+9)*10^(2/3)+1/50*ln(-6*10^
(1/3)-11*10^(1/3)*x+3*10^(1/3)*x^2+2*10^(1/3)*x^3+5*(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))*10^(2/3)-1/100*ln(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*1
0^(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)*(8*x^8+84*x^7+338*x^6+6
79*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(1/3)+25*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+
27)^(2/3))*10^(2/3)
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Rubi [F]
time = 0.23, 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 [A]
time = 0.24, size = 189, normalized size = 0.38 \begin {gather*} -\frac {\sqrt [3]{1+x^2} \left (3+7 x+2 x^2\right ) \left (2 \sqrt {3} \text {ArcTan}\left (\frac {4 \sqrt [3]{10}-2 \sqrt [3]{10} x+5 \sqrt [3]{1+x^2}}{5 \sqrt {3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 \sqrt [3]{10}+\sqrt [3]{10} x+5 \sqrt [3]{1+x^2}\right )+\log \left (4\ 10^{2/3}-4\ 10^{2/3} x+10^{2/3} x^2-5 \sqrt [3]{10} (-2+x) \sqrt [3]{1+x^2}+25 \left (1+x^2\right )^{2/3}\right )\right )}{10 \sqrt [3]{10} \sqrt [3]{\left (1+x^2\right ) \left (3+7 x+2 x^2\right )^3}} \end {gather*}
Antiderivative was successfully verified.
[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]
-1/10*((1 + x^2)^(1/3)*(3 + 7*x + 2*x^2)*(2*Sqrt[3]*ArcTan[(4*10^(1/3) - 2*10^(1/3)*x + 5*(1 + x^2)^(1/3))/(5*
Sqrt[3]*(1 + x^2)^(1/3))] - 2*Log[-2*10^(1/3) + 10^(1/3)*x + 5*(1 + x^2)^(1/3)] + Log[4*10^(2/3) - 4*10^(2/3)*
x + 10^(2/3)*x^2 - 5*10^(1/3)*(-2 + x)*(1 + x^2)^(1/3) + 25*(1 + x^2)^(2/3)]))/(10^(1/3)*((1 + x^2)*(3 + 7*x +
2*x^2)^3)^(1/3))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 11.54, size = 3645, normalized size = 7.35
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result |
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\(\text {Expression too large to display}\) |
\(3645\) |
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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]
1/50*RootOf(_Z^3-100)*ln(-(-2150980087500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*Ro
otOf(_Z^3-100)^2*x^6+113015711700*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-
100)^3*x^6-9858658734375*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*
x^5+517988678625*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^5-298747
23437500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^4+156966266250
0*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^4-49293293671875*RootOf
(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^3+2589943393125*RootOf(81*Roo
tOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^3-31189211268750*RootOf(81*RootOf(_Z^3
-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^2+1638727819650*RootOf(81*RootOf(_Z^3-100)^2
+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^2-6452940262500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*R
ootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x+339047135100*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3
-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x+888764972400*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_
Z^2)*RootOf(_Z^3-100)^2*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(2/3)*x-607979150775*(
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)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootO
f(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x^3-10335645563175*(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)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x^2+972766
6412400*(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)*RootOf(81*RootOf(_Z^3-100)^2+450
*_Z*RootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)*x-7055214687000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z
^3-100)+2500*_Z^2)*x^7+46719287500500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^6+4804
85931945750*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^5+909111733981875*RootOf(81*Root
Of(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x^4+473645815267500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*Root
Of(_Z^3-100)+2500*_Z^2)*x^3+1008078327807750*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x
^2+966779510127750*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*x+370691534376*RootOf(_Z^3-
100)*x^7-2454701258124*RootOf(_Z^3-100)*x^6-25245449679546*RootOf(_Z^3-100)*x^5-47766090550005*RootOf(_Z^3-100
)*x^4-24886059716340*RootOf(_Z^3-100)*x^3-52965943445322*RootOf(_Z^3-100)*x^2-50796041780682*RootOf(_Z^3-100)*
x-21051914292450*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(2/3)+1215958301550*(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)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-1
00)+2500*_Z^2)*RootOf(_Z^3-100)*x^4-13029581401893*RootOf(_Z^3-100)+247986494287875*RootOf(81*RootOf(_Z^3-100)
^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)+639910780128*RootOf(_Z^3-100)^2*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+78
4*x^3+522*x^2+189*x+27)^(1/3)*x^4-319955390064*RootOf(_Z^3-100)^2*(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^3-1777529944800*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*Ro
otOf(_Z^3-100)^2*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(2/3)-5439241631088*RootOf(_Z
^3-100)^2*(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^2+5119286241024*RootOf(_Z^3-
100)^2*(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+7295749809300*(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)*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500
*_Z^2)*RootOf(_Z^3-100)+12557301300*RootOf(_Z^3-100)^3*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+25
00*_Z^2)*x^7-238997787500*RootOf(_Z^3-100)^2*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2
*x^7+10525957146225*(8*x^8+84*x^7+338*x^6+679*x^5+825*x^4+784*x^3+522*x^2+189*x+27)^(2/3)*x+3839464680768*Root
Of(_Z^3-100)^2*(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))/(1+2*x)^3/(3+x)^4)+1/9*R
ootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*ln((1255730130000*RootOf(81*RootOf(_Z^3-100)^2+
450*_Z*RootOf(_Z^3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^6-774352831500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*R
ootOf(_Z^3-100)+2500*_Z^2)*RootOf(_Z^3-100)^3*x^6+5755429762500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^
3-100)+2500*_Z^2)^2*RootOf(_Z^3-100)^2*x^5-3549117144375*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+
2500*_Z^2)*RootOf(_Z^3-100)^3*x^5+17440696250000*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^
2)^2*RootOf(_Z^3-100)^2*x^4-10754900437500*RootOf(81*RootOf(_Z^3-100)^2+450*_Z*RootOf(_Z^3-100)+2500*_Z^2)*Roo
tOf(_Z^3-100)^3*x^4+28777148812500*RootOf(81*Ro...
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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