3.7.72 \(\int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx\)

Optimal. Leaf size=53 \[ \frac {4 \left (x^4+x^3\right )^{3/4} \left (262144 x^7-196608 x^6+172032 x^5-157696 x^4+147840 x^3-140448 x^2+134596 x-129789\right )}{4023459 x^{10}} \]

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Rubi [B]  time = 0.22, antiderivative size = 145, normalized size of antiderivative = 2.74, number of steps used = 8, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2016, 2014} \begin {gather*} \frac {1048576 \left (x^4+x^3\right )^{3/4}}{4023459 x^3}-\frac {262144 \left (x^4+x^3\right )^{3/4}}{1341153 x^4}-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^4+x^3\right )^{3/4}}{837 x^9}-\frac {896 \left (x^4+x^3\right )^{3/4}}{6417 x^8}+\frac {17920 \left (x^4+x^3\right )^{3/4}}{121923 x^7}-\frac {57344 \left (x^4+x^3\right )^{3/4}}{365769 x^6}+\frac {229376 \left (x^4+x^3\right )^{3/4}}{1341153 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(x^3 + x^4)^(1/4)),x]

[Out]

(-4*(x^3 + x^4)^(3/4))/(31*x^10) + (112*(x^3 + x^4)^(3/4))/(837*x^9) - (896*(x^3 + x^4)^(3/4))/(6417*x^8) + (1
7920*(x^3 + x^4)^(3/4))/(121923*x^7) - (57344*(x^3 + x^4)^(3/4))/(365769*x^6) + (229376*(x^3 + x^4)^(3/4))/(13
41153*x^5) - (262144*(x^3 + x^4)^(3/4))/(1341153*x^4) + (1048576*(x^3 + x^4)^(3/4))/(4023459*x^3)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}-\frac {28}{31} \int \frac {1}{x^7 \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}+\frac {224}{279} \int \frac {1}{x^6 \sqrt [4]{x^3+x^4}} \, dx\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}-\frac {896 \left (x^3+x^4\right )^{3/4}}{6417 x^8}-\frac {4480 \int \frac {1}{x^5 \sqrt [4]{x^3+x^4}} \, dx}{6417}\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}-\frac {896 \left (x^3+x^4\right )^{3/4}}{6417 x^8}+\frac {17920 \left (x^3+x^4\right )^{3/4}}{121923 x^7}+\frac {71680 \int \frac {1}{x^4 \sqrt [4]{x^3+x^4}} \, dx}{121923}\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}-\frac {896 \left (x^3+x^4\right )^{3/4}}{6417 x^8}+\frac {17920 \left (x^3+x^4\right )^{3/4}}{121923 x^7}-\frac {57344 \left (x^3+x^4\right )^{3/4}}{365769 x^6}-\frac {57344 \int \frac {1}{x^3 \sqrt [4]{x^3+x^4}} \, dx}{121923}\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}-\frac {896 \left (x^3+x^4\right )^{3/4}}{6417 x^8}+\frac {17920 \left (x^3+x^4\right )^{3/4}}{121923 x^7}-\frac {57344 \left (x^3+x^4\right )^{3/4}}{365769 x^6}+\frac {229376 \left (x^3+x^4\right )^{3/4}}{1341153 x^5}+\frac {458752 \int \frac {1}{x^2 \sqrt [4]{x^3+x^4}} \, dx}{1341153}\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}-\frac {896 \left (x^3+x^4\right )^{3/4}}{6417 x^8}+\frac {17920 \left (x^3+x^4\right )^{3/4}}{121923 x^7}-\frac {57344 \left (x^3+x^4\right )^{3/4}}{365769 x^6}+\frac {229376 \left (x^3+x^4\right )^{3/4}}{1341153 x^5}-\frac {262144 \left (x^3+x^4\right )^{3/4}}{1341153 x^4}-\frac {262144 \int \frac {1}{x \sqrt [4]{x^3+x^4}} \, dx}{1341153}\\ &=-\frac {4 \left (x^3+x^4\right )^{3/4}}{31 x^{10}}+\frac {112 \left (x^3+x^4\right )^{3/4}}{837 x^9}-\frac {896 \left (x^3+x^4\right )^{3/4}}{6417 x^8}+\frac {17920 \left (x^3+x^4\right )^{3/4}}{121923 x^7}-\frac {57344 \left (x^3+x^4\right )^{3/4}}{365769 x^6}+\frac {229376 \left (x^3+x^4\right )^{3/4}}{1341153 x^5}-\frac {262144 \left (x^3+x^4\right )^{3/4}}{1341153 x^4}+\frac {1048576 \left (x^3+x^4\right )^{3/4}}{4023459 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3 (x+1)\right )^{3/4} \left (262144 x^7-196608 x^6+172032 x^5-157696 x^4+147840 x^3-140448 x^2+134596 x-129789\right )}{4023459 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(x^3 + x^4)^(1/4)),x]

[Out]

(4*(x^3*(1 + x))^(3/4)*(-129789 + 134596*x - 140448*x^2 + 147840*x^3 - 157696*x^4 + 172032*x^5 - 196608*x^6 +
262144*x^7))/(4023459*x^10)

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IntegrateAlgebraic [A]  time = 0.27, size = 53, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3+x^4\right )^{3/4} \left (-129789+134596 x-140448 x^2+147840 x^3-157696 x^4+172032 x^5-196608 x^6+262144 x^7\right )}{4023459 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^8*(x^3 + x^4)^(1/4)),x]

[Out]

(4*(x^3 + x^4)^(3/4)*(-129789 + 134596*x - 140448*x^2 + 147840*x^3 - 157696*x^4 + 172032*x^5 - 196608*x^6 + 26
2144*x^7))/(4023459*x^10)

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fricas [A]  time = 0.44, size = 49, normalized size = 0.92 \begin {gather*} \frac {4 \, {\left (262144 \, x^{7} - 196608 \, x^{6} + 172032 \, x^{5} - 157696 \, x^{4} + 147840 \, x^{3} - 140448 \, x^{2} + 134596 \, x - 129789\right )} {\left (x^{4} + x^{3}\right )}^{\frac {3}{4}}}{4023459 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(x^4+x^3)^(1/4),x, algorithm="fricas")

[Out]

4/4023459*(262144*x^7 - 196608*x^6 + 172032*x^5 - 157696*x^4 + 147840*x^3 - 140448*x^2 + 134596*x - 129789)*(x
^4 + x^3)^(3/4)/x^10

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giac [A]  time = 0.35, size = 73, normalized size = 1.38 \begin {gather*} -\frac {4}{31} \, {\left (\frac {1}{x} + 1\right )}^{\frac {31}{4}} + \frac {28}{27} \, {\left (\frac {1}{x} + 1\right )}^{\frac {27}{4}} - \frac {84}{23} \, {\left (\frac {1}{x} + 1\right )}^{\frac {23}{4}} + \frac {140}{19} \, {\left (\frac {1}{x} + 1\right )}^{\frac {19}{4}} - \frac {28}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {15}{4}} + \frac {84}{11} \, {\left (\frac {1}{x} + 1\right )}^{\frac {11}{4}} - 4 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(x^4+x^3)^(1/4),x, algorithm="giac")

[Out]

-4/31*(1/x + 1)^(31/4) + 28/27*(1/x + 1)^(27/4) - 84/23*(1/x + 1)^(23/4) + 140/19*(1/x + 1)^(19/4) - 28/3*(1/x
+ 1)^(15/4) + 84/11*(1/x + 1)^(11/4) - 4*(1/x + 1)^(7/4) + 4/3*(1/x + 1)^(3/4)

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maple [C]  time = 0.09, size = 15, normalized size = 0.28

method	result	size
meijerg	\(-\frac {4 \hypergeom \left (\left [-\frac {31}{4}, \frac {1}{4}\right ], \left [-\frac {27}{4}\right ], -x \right )}{31 x^{\frac {31}{4}}}\)	\(15\)
trager	\(\frac {4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}} \left (262144 x^{7}-196608 x^{6}+172032 x^{5}-157696 x^{4}+147840 x^{3}-140448 x^{2}+134596 x -129789\right )}{4023459 x^{10}}\)	\(50\)
gosper	\(\frac {4 \left (1+x \right ) \left (262144 x^{7}-196608 x^{6}+172032 x^{5}-157696 x^{4}+147840 x^{3}-140448 x^{2}+134596 x -129789\right )}{4023459 x^{7} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\)	\(53\)
risch	\(\frac {-\frac {4}{31}+\frac {4}{837} x -\frac {112}{19251} x^{2}+\frac {896}{121923} x^{3}-\frac {3584}{365769} x^{4}+\frac {57344}{4023459} x^{5}-\frac {32768}{1341153} x^{6}+\frac {262144}{4023459} x^{7}+\frac {1048576}{4023459} x^{8}}{x^{7} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(x^4+x^3)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-4/31/x^(31/4)*hypergeom([-31/4,1/4],[-27/4],-x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(x^4+x^3)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((x^4 + x^3)^(1/4)*x^8), x)

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mupad [B]  time = 0.58, size = 113, normalized size = 2.13 \begin {gather*} \frac {1048576\,{\left (x^4+x^3\right )}^{3/4}}{4023459\,x^3}-\frac {262144\,{\left (x^4+x^3\right )}^{3/4}}{1341153\,x^4}+\frac {229376\,{\left (x^4+x^3\right )}^{3/4}}{1341153\,x^5}-\frac {57344\,{\left (x^4+x^3\right )}^{3/4}}{365769\,x^6}+\frac {17920\,{\left (x^4+x^3\right )}^{3/4}}{121923\,x^7}-\frac {896\,{\left (x^4+x^3\right )}^{3/4}}{6417\,x^8}+\frac {112\,{\left (x^4+x^3\right )}^{3/4}}{837\,x^9}-\frac {4\,{\left (x^4+x^3\right )}^{3/4}}{31\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(x^3 + x^4)^(1/4)),x)

[Out]

(1048576*(x^3 + x^4)^(3/4))/(4023459*x^3) - (262144*(x^3 + x^4)^(3/4))/(1341153*x^4) + (229376*(x^3 + x^4)^(3/
4))/(1341153*x^5) - (57344*(x^3 + x^4)^(3/4))/(365769*x^6) + (17920*(x^3 + x^4)^(3/4))/(121923*x^7) - (896*(x^
3 + x^4)^(3/4))/(6417*x^8) + (112*(x^3 + x^4)^(3/4))/(837*x^9) - (4*(x^3 + x^4)^(3/4))/(31*x^10)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{8} \sqrt [4]{x^{3} \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(x**4+x**3)**(1/4),x)

[Out]

Integral(1/(x**8*(x**3*(x + 1))**(1/4)), x)

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