3.4.91 \(\int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx\) [391]

Optimal. Leaf size=53 \[ \frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \]

[Out]

2*arctanh(x^(3/2)*a^(1/2)/(a*x^3+b*x^n)^(1/2))*(c*x)^(1/2)/(3-n)/a^(1/2)/x^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2056, 2054, 212} \begin {gather*} \frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x]/Sqrt[a*x^3 + b*x^n],x]

[Out]

(2*Sqrt[c*x]*ArcTanh[(Sqrt[a]*x^(3/2))/Sqrt[a*x^3 + b*x^n]])/(Sqrt[a]*(3 - n)*Sqrt[x])

Rule 212

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

Rule 2054

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]

Rule 2056

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c x}}{\sqrt {a x^3+b x^n}} \, dx &=\frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {a x^3+b x^n}} \, dx}{\sqrt {x}}\\ &=\frac {\left (2 \sqrt {c x}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{(3-n) \sqrt {x}}\\ &=\frac {2 \sqrt {c x} \tanh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{\sqrt {a} (3-n) \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 89, normalized size = 1.68 \begin {gather*} -\frac {2 \sqrt {b} x^{\frac {1}{2} (-1+n)} \sqrt {c x} \sqrt {1+\frac {a x^{3-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {3}{2}-\frac {n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} (-3+n) \sqrt {a x^3+b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x]/Sqrt[a*x^3 + b*x^n],x]

[Out]

(-2*Sqrt[b]*x^((-1 + n)/2)*Sqrt[c*x]*Sqrt[1 + (a*x^(3 - n))/b]*ArcSinh[(Sqrt[a]*x^(3/2 - n/2))/Sqrt[b]])/(Sqrt
[a]*(-3 + n)*Sqrt[a*x^3 + b*x^n])

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c x}}{\sqrt {a \,x^{3}+b \,x^{n}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x)

[Out]

int((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x)

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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((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x)/sqrt(a*x^3 + 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((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x}}{\sqrt {a x^{3} + b x^{n}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(1/2)/(a*x**3+b*x**n)**(1/2),x)

[Out]

Integral(sqrt(c*x)/sqrt(a*x**3 + b*x**n), 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((c*x)^(1/2)/(a*x^3+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x)/sqrt(a*x^3 + b*x^n), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x}}{\sqrt {b\,x^n+a\,x^3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(1/2)/(b*x^n + a*x^3)^(1/2),x)

[Out]

int((c*x)^(1/2)/(b*x^n + a*x^3)^(1/2), x)

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