3.2641 \(\int \frac{x^{-1-\frac{3 n}{4}}}{a+b x^n} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.199741, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {362, 345, 211, 1165, 628, 1162, 617, 204} \[ \frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{n/4}+\sqrt{a}+\sqrt{b} x^{n/2}\right )}{\sqrt{2} a^{7/4} n}+\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}\right )}{a^{7/4} n}-\frac{\sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^{n/4}}{\sqrt [4]{a}}+1\right )}{a^{7/4} n}-\frac{4 x^{-3 n/4}}{3 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - (3*n)/4)/(a + b*x^n),x]

[Out]

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

Rule 362

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 345

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 211

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 1165

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 628

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 1162

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 617

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 204

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

Rubi steps

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

Mathematica [C]  time = 0.0072013, size = 34, normalized size = 0.14 \[ -\frac{4 x^{-3 n/4} \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{b x^n}{a}\right )}{3 a n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - (3*n)/4)/(a + b*x^n),x]

[Out]

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

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Maple [C]  time = 0.052, size = 54, normalized size = 0.2 \begin{align*} -{\frac{4}{3\,an} \left ({x}^{{\frac{n}{4}}} \right ) ^{-3}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{7}{n}^{4}{{\it \_Z}}^{4}+{b}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}-{\frac{{a}^{2}n{\it \_R}}{b}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-3/4*n)/(a+b*x^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{x^{\frac{1}{4} \, n}}{a b x x^{n} + a^{2} x}\,{d x} - \frac{4}{3 \, a n x^{\frac{3}{4} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3/4*n)/(a+b*x^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.75162, size = 625, normalized size = 2.65 \begin{align*} -\frac{12 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{2} b^{2} n x^{\frac{1}{3}} x^{-\frac{1}{4} \, n - \frac{1}{3}} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} - a^{2} n x^{\frac{1}{3}} \sqrt{-\frac{a^{3} b^{3} n^{2} x^{\frac{1}{3}} \sqrt{-\frac{b^{3}}{a^{7} n^{4}}} - b^{4} x x^{-\frac{1}{2} \, n - \frac{2}{3}}}{x}} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}}}{b^{3}}\right ) - 3 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{a^{5} n^{3} x^{\frac{2}{3}} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}} + b^{2} x x^{-\frac{1}{4} \, n - \frac{1}{3}}}{x}\right ) + 3 \, a n \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{a^{5} n^{3} x^{\frac{2}{3}} \left (-\frac{b^{3}}{a^{7} n^{4}}\right )^{\frac{3}{4}} - b^{2} x x^{-\frac{1}{4} \, n - \frac{1}{3}}}{x}\right ) + 4 \, x x^{-\frac{3}{4} \, n - 1}}{3 \, a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3/4*n)/(a+b*x^n),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 5.41957, size = 267, normalized size = 1.13 \begin{align*} \frac{x^{- \frac{3 n}{4}} \Gamma \left (- \frac{3}{4}\right )}{a n \Gamma \left (\frac{1}{4}\right )} - \frac{3 b^{\frac{3}{4}} e^{- \frac{i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{3}{4}\right )}{4 a^{\frac{7}{4}} n \Gamma \left (\frac{1}{4}\right )} + \frac{3 i b^{\frac{3}{4}} e^{- \frac{i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{3 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{3}{4}\right )}{4 a^{\frac{7}{4}} n \Gamma \left (\frac{1}{4}\right )} + \frac{3 b^{\frac{3}{4}} e^{- \frac{i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{5 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{3}{4}\right )}{4 a^{\frac{7}{4}} n \Gamma \left (\frac{1}{4}\right )} - \frac{3 i b^{\frac{3}{4}} e^{- \frac{i \pi }{4}} \log{\left (1 - \frac{\sqrt [4]{b} x^{\frac{n}{4}} e^{\frac{7 i \pi }{4}}}{\sqrt [4]{a}} \right )} \Gamma \left (- \frac{3}{4}\right )}{4 a^{\frac{7}{4}} n \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-3/4*n)/(a+b*x**n),x)

[Out]

x**(-3*n/4)*gamma(-3/4)/(a*n*gamma(1/4)) - 3*b**(3/4)*exp(-I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_polar(I*pi/4)
/a**(1/4))*gamma(-3/4)/(4*a**(7/4)*n*gamma(1/4)) + 3*I*b**(3/4)*exp(-I*pi/4)*log(1 - b**(1/4)*x**(n/4)*exp_pol
ar(3*I*pi/4)/a**(1/4))*gamma(-3/4)/(4*a**(7/4)*n*gamma(1/4)) + 3*b**(3/4)*exp(-I*pi/4)*log(1 - b**(1/4)*x**(n/
4)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(-3/4)/(4*a**(7/4)*n*gamma(1/4)) - 3*I*b**(3/4)*exp(-I*pi/4)*log(1 - b**
(1/4)*x**(n/4)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(-3/4)/(4*a**(7/4)*n*gamma(1/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{3}{4} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3/4*n)/(a+b*x^n),x, algorithm="giac")

[Out]

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