3.2787 \(\int \frac{(c x)^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{16 (c x)^{-5 n/2} \left (a+b x^n\right )^{5/2}}{15 a^3 c n}+\frac{8 (c x)^{-5 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((5*n)/2)) + (8*(a + b*x^n)^(3/2))/(3*a^2*c*n*(c*x)^((5*n)/2)) - (16*(a + b*
x^n)^(5/2))/(15*a^3*c*n*(c*x)^((5*n)/2))

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Rubi [A]  time = 0.0305248, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {273, 264} \[ -\frac{16 (c x)^{-5 n/2} \left (a+b x^n\right )^{5/2}}{15 a^3 c n}+\frac{8 (c x)^{-5 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((5*n)/2)) + (8*(a + b*x^n)^(3/2))/(3*a^2*c*n*(c*x)^((5*n)/2)) - (16*(a + b*
x^n)^(5/2))/(15*a^3*c*n*(c*x)^((5*n)/2))

Rule 273

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] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

Rule 264

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]

Rubi steps

\begin{align*} \int \frac{(c x)^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n}-\frac{4 \int (c x)^{-1-\frac{5 n}{2}} \sqrt{a+b x^n} \, dx}{a}\\ &=-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n}+\frac{8 (c x)^{-5 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}+\frac{8 \int (c x)^{-1-\frac{5 n}{2}} \left (a+b x^n\right )^{3/2} \, dx}{3 a^2}\\ &=-\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n}}{a c n}+\frac{8 (c x)^{-5 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{16 (c x)^{-5 n/2} \left (a+b x^n\right )^{5/2}}{15 a^3 c n}\\ \end{align*}

Mathematica [A]  time = 0.0196091, size = 56, normalized size = 0.57 \[ -\frac{2 (c x)^{-5 n/2} \sqrt{a+b x^n} \left (3 a^2-4 a b x^n+8 b^2 x^{2 n}\right )}{15 a^3 c n} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 - (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n]*(3*a^2 - 4*a*b*x^n + 8*b^2*x^(2*n)))/(15*a^3*c*n*(c*x)^((5*n)/2))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1-{\frac{5\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(-1-5/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1-5/2*n)/(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1-5/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x)^(-5/2*n - 1)/sqrt(b*x^n + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1-5/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 13.2821, size = 396, normalized size = 4.04 \begin{align*} - \frac{6 a^{4} b^{\frac{9}{2}} c^{- \frac{5 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}\right )} - \frac{4 a^{3} b^{\frac{11}{2}} c^{- \frac{5 n}{2}} x^{n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}\right )} - \frac{6 a^{2} b^{\frac{13}{2}} c^{- \frac{5 n}{2}} x^{2 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}\right )} - \frac{24 a b^{\frac{15}{2}} c^{- \frac{5 n}{2}} x^{3 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}\right )} - \frac{16 b^{\frac{17}{2}} c^{- \frac{5 n}{2}} x^{4 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(-1-5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

-6*a**4*b**(9/2)*c**(-5*n/2)*sqrt(a*x**(-n)/b + 1)/(c*(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5*n*x**(3*n) + 15*
a**3*b**6*n*x**(4*n))) - 4*a**3*b**(11/2)*c**(-5*n/2)*x**n*sqrt(a*x**(-n)/b + 1)/(c*(15*a**5*b**4*n*x**(2*n) +
30*a**4*b**5*n*x**(3*n) + 15*a**3*b**6*n*x**(4*n))) - 6*a**2*b**(13/2)*c**(-5*n/2)*x**(2*n)*sqrt(a*x**(-n)/b
+ 1)/(c*(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5*n*x**(3*n) + 15*a**3*b**6*n*x**(4*n))) - 24*a*b**(15/2)*c**(-5
*n/2)*x**(3*n)*sqrt(a*x**(-n)/b + 1)/(c*(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5*n*x**(3*n) + 15*a**3*b**6*n*x*
*(4*n))) - 16*b**(17/2)*c**(-5*n/2)*x**(4*n)*sqrt(a*x**(-n)/b + 1)/(c*(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5*
n*x**(3*n) + 15*a**3*b**6*n*x**(4*n)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1-5/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x)^(-5/2*n - 1)/sqrt(b*x^n + a), x)