3.3.67 \(\int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx\)

Optimal. Leaf size=75 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n} \]

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Rubi [A]  time = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2028, 2029, 206} \begin {gather*} \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - j/2)*Sqrt[a*x^j + b*x^n],x]

[Out]

(-2*Sqrt[a*x^j + b*x^n])/((j - n)*x^(j/2)) + (2*Sqrt[a]*ArcTanh[(Sqrt[a]*x^(j/2))/Sqrt[a*x^j + b*x^n]])/(j - n
)

Rule 206

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

Rule 2028

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*p*(n - j)), x] + Dist[a/c^j, Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c,
j, m, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps

\begin {align*} \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx &=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+a \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx\\ &=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}\\ &=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 104, normalized size = 1.39 \begin {gather*} -\frac {2 x^{-j/2} \left (-\sqrt {a} \sqrt {b} x^{\frac {j+n}{2}} \sqrt {\frac {a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )+a x^j+b x^n\right )}{(j-n) \sqrt {a x^j+b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - j/2)*Sqrt[a*x^j + b*x^n],x]

[Out]

(-2*(a*x^j + b*x^n - Sqrt[a]*Sqrt[b]*x^((j + n)/2)*Sqrt[1 + (a*x^(j - n))/b]*ArcSinh[(Sqrt[a]*x^((j - n)/2))/S
qrt[b]]))/((j - n)*x^(j/2)*Sqrt[a*x^j + b*x^n])

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IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^(-1 - j/2)*Sqrt[a*x^j + b*x^n],x]

[Out]

Defer[IntegrateAlgebraic][x^(-1 - j/2)*Sqrt[a*x^j + b*x^n], 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(x^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x^{j} + b x^{n}} x^{-\frac {1}{2} \, j - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^j + b*x^n)*x^(-1/2*j - 1), x)

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maple [F]  time = 0.87, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \,x^{j}+b \,x^{n}}\, x^{-\frac {j}{2}-1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x)

[Out]

int(x^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x^{j} + b x^{n}} x^{-\frac {1}{2} \, j - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^j + b*x^n)*x^(-1/2*j - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a\,x^j+b\,x^n}}{x^{\frac {j}{2}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^j + b*x^n)^(1/2)/x^(j/2 + 1),x)

[Out]

int((a*x^j + b*x^n)^(1/2)/x^(j/2 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- \frac {j}{2} - 1} \sqrt {a x^{j} + b x^{n}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-1/2*j)*(a*x**j+b*x**n)**(1/2),x)

[Out]

Integral(x**(-j/2 - 1)*sqrt(a*x**j + b*x**n), x)

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