3.28.86 \(\int \frac {(c x)^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [2786]

Optimal. Leaf size=65 \[ -\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {279, 270} \begin {gather*} \frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((3*n)/2)) + (4*(a + b*x^n)^(3/2))/(3*a^2*c*n*(c*x)^((3*n)/2))

Rule 270

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

Rule 279

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

Rubi steps

\begin {align*} \int \frac {(c x)^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n}-\frac {2 \int (c x)^{-1-\frac {3 n}{2}} \sqrt {a+b x^n} \, dx}{a}\\ &=-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.63 \begin {gather*} -\frac {2 (c x)^{-3 n/2} \left (a-2 b x^n\right ) \sqrt {a+b x^n}}{3 a^2 c n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{-1-\frac {3 n}{2}}}{\sqrt {a +b \,x^{n}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c*x)^(-3/2*n - 1)/sqrt(b*x^n + a), 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)^(-1-3/2*n)/(a+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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Sympy [A]
time = 3.04, size = 68, normalized size = 1.05 \begin {gather*} - \frac {2 \sqrt {b} c^{- \frac {3 n}{2}} x^{- n} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a c n} + \frac {4 b^{\frac {3}{2}} c^{- \frac {3 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{2} c n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x**n) + 1)/(3*a*c*c**(3*n/2)*n*x**n) + 4*b**(3/2)*sqrt(a/(b*x**n) + 1)/(3*a**2*c*c**(3*n/
2)*n)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{\frac {3\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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