3.5.100 \(\int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx\) [500]

Optimal. Leaf size=236 \[ -\frac {4 x^{-3 n/4}}{3 b n}+\frac {\sqrt {2} c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac {\sqrt {2} c^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac {c^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n} \]

[Out]

-4/3/b/n/(x^(3/4*n))+1/2*c^(3/4)*ln(-b^(1/4)*c^(1/4)*x^(1/4*n)*2^(1/2)+b^(1/2)+x^(1/2*n)*c^(1/2))/b^(7/4)/n*2^
(1/2)-1/2*c^(3/4)*ln(b^(1/4)*c^(1/4)*x^(1/4*n)*2^(1/2)+b^(1/2)+x^(1/2*n)*c^(1/2))/b^(7/4)/n*2^(1/2)-c^(3/4)*ar
ctan(-1+c^(1/4)*x^(1/4*n)*2^(1/2)/b^(1/4))*2^(1/2)/b^(7/4)/n-c^(3/4)*arctan(1+c^(1/4)*x^(1/4*n)*2^(1/2)/b^(1/4
))*2^(1/2)/b^(7/4)/n

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Rubi [A]
time = 0.15, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1598, 369, 352, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt {2} c^{3/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac {\sqrt {2} c^{3/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}+1\right )}{b^{7/4} n}+\frac {c^{3/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {b}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {b}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {4 x^{-3 n/4}}{3 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]

[Out]

-4/(3*b*n*x^((3*n)/4)) + (Sqrt[2]*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x^(n/4))/b^(1/4)])/(b^(7/4)*n) - (Sqrt[2
]*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x^(n/4))/b^(1/4)])/(b^(7/4)*n) + (c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*
c^(1/4)*x^(n/4) + Sqrt[c]*x^(n/2)])/(Sqrt[2]*b^(7/4)*n) - (c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*x^(n/
4) + Sqrt[c]*x^(n/2)])/(Sqrt[2]*b^(7/4)*n)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 352

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 369

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[x^(m + 1)/(a*(m + 1)), x] - Dist[b/a, Int[x^Simplify
[m + n]/(a + b*x^n), x], x] /; FreeQ[{a, b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, n]

Rule 631

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]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x^{-1+\frac {n}{4}}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-\frac {3 n}{4}}}{b+c x^n} \, dx\\ &=-\frac {4 x^{-3 n/4}}{3 b n}-\frac {c \int \frac {x^{\frac {1}{4} (-4+n)}}{b+c x^n} \, dx}{b}\\ &=-\frac {4 x^{-3 n/4}}{3 b n}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b n}\\ &=-\frac {4 x^{-3 n/4}}{3 b n}-\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}-\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}\\ &=-\frac {4 x^{-3 n/4}}{3 b n}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{b^{3/2} n}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} b^{7/4} n}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,x^{1+\frac {1}{4} (-4+n)}\right )}{\sqrt {2} b^{7/4} n}\\ &=-\frac {4 x^{-3 n/4}}{3 b n}+\frac {c^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {\left (\sqrt {2} c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac {\left (\sqrt {2} c^{3/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x^{1+\frac {1}{4} (-4+n)}}{\sqrt [4]{b}}\right )}{b^{7/4} n}\\ &=-\frac {4 x^{-3 n/4}}{3 b n}+\frac {\sqrt {2} c^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}-\frac {\sqrt {2} c^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{b}}\right )}{b^{7/4} n}+\frac {c^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}-\frac {c^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} x^{n/4}+\sqrt {c} x^{n/2}\right )}{\sqrt {2} b^{7/4} n}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.03, size = 34, normalized size = 0.14 \begin {gather*} -\frac {4 x^{-3 n/4} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\frac {c x^n}{b}\right )}{3 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n/4)/(b*x^n + c*x^(2*n)),x]

[Out]

(-4*Hypergeometric2F1[-3/4, 1, 1/4, -((c*x^n)/b)])/(3*b*n*x^((3*n)/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.25, size = 54, normalized size = 0.23

method	result	size
risch	\(-\frac {4 x^{-\frac {3 n}{4}}}{3 b n}+\left (\munderset {\textit {\_R} =\RootOf \left (b^{7} n^{4} \textit {\_Z}^{4}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{4}}-\frac {b^{2} n \textit {\_R}}{c}\right )\right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+1/4*n)/(b*x^n+c*x^(2*n)),x,method=_RETURNVERBOSE)

[Out]

-4/3/b/n/(x^(1/4*n))^3+sum(_R*ln(x^(1/4*n)-b^2*n/c*_R),_R=RootOf(_Z^4*b^7*n^4+c^3))

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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(x^(-1+1/4*n)/(b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

-c*integrate(x^(1/4*n)/(b*c*x*x^n + b^2*x), x) - 4/3/(b*n*x^(3/4*n))

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Fricas [A]
time = 0.40, size = 272, normalized size = 1.15 \begin {gather*} -\frac {12 \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b^{5} c n^{3} x x^{\frac {1}{4} \, n - 1} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {3}{4}} - b^{5} n^{3} x \sqrt {\frac {b^{4} n^{2} \sqrt {-\frac {c^{3}}{b^{7} n^{4}}} + c^{2} x^{2} x^{\frac {1}{2} \, n - 2}}{x^{2}}} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {3}{4}}}{c^{3}}\right ) + 3 \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} n \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} + c x x^{\frac {1}{4} \, n - 1}}{x}\right ) - 3 \, b n x^{3} x^{\frac {3}{4} \, n - 3} \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} n \left (-\frac {c^{3}}{b^{7} n^{4}}\right )^{\frac {1}{4}} - c x x^{\frac {1}{4} \, n - 1}}{x}\right ) + 4}{3 \, b n x^{3} x^{\frac {3}{4} \, n - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/4*n)/(b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

-1/3*(12*b*n*x^3*x^(3/4*n - 3)*(-c^3/(b^7*n^4))^(1/4)*arctan(-(b^5*c*n^3*x*x^(1/4*n - 1)*(-c^3/(b^7*n^4))^(3/4
) - b^5*n^3*x*sqrt((b^4*n^2*sqrt(-c^3/(b^7*n^4)) + c^2*x^2*x^(1/2*n - 2))/x^2)*(-c^3/(b^7*n^4))^(3/4))/c^3) +
3*b*n*x^3*x^(3/4*n - 3)*(-c^3/(b^7*n^4))^(1/4)*log((b^2*n*(-c^3/(b^7*n^4))^(1/4) + c*x*x^(1/4*n - 1))/x) - 3*b
*n*x^3*x^(3/4*n - 3)*(-c^3/(b^7*n^4))^(1/4)*log(-(b^2*n*(-c^3/(b^7*n^4))^(1/4) - c*x*x^(1/4*n - 1))/x) + 4)/(b
*n*x^3*x^(3/4*n - 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+1/4*n)/(b*x**n+c*x**(2*n)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4374 deep

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Giac [A]
time = 5.04, size = 203, normalized size = 0.86 \begin {gather*} -\frac {\frac {6 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, {\left (x^{n}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {6 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, {\left (x^{n}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{2}} + \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (x^{\frac {1}{2} \, n} + \sqrt {2} {\left (x^{n}\right )}^{\frac {1}{4}} \left (\frac {b}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{c}}\right )}{b^{2}} - \frac {3 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (x^{\frac {1}{2} \, n} - \sqrt {2} {\left (x^{n}\right )}^{\frac {1}{4}} \left (\frac {b}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{c}}\right )}{b^{2}} + \frac {8}{b x^{\frac {3}{4} \, n}}}{6 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/4*n)/(b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

-1/6*(6*sqrt(2)*(b*c^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*(x^n)^(1/4))/(b/c)^(1/4))/b^2 + 6*sq
rt(2)*(b*c^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*(x^n)^(1/4))/(b/c)^(1/4))/b^2 + 3*sqrt(2)*(b*
c^3)^(1/4)*log(x^(1/2*n) + sqrt(2)*(x^n)^(1/4)*(b/c)^(1/4) + sqrt(b/c))/b^2 - 3*sqrt(2)*(b*c^3)^(1/4)*log(x^(1
/2*n) - sqrt(2)*(x^n)^(1/4)*(b/c)^(1/4) + sqrt(b/c))/b^2 + 8/(b*x^(3/4*n)))/n

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{\frac {n}{4}-1}}{b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n/4 - 1)/(b*x^n + c*x^(2*n)),x)

[Out]

int(x^(n/4 - 1)/(b*x^n + c*x^(2*n)), x)

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