3.2788 \(\int \frac{(c x)^{-1-\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=129 \[ \frac{32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac{16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((7*n)/2)) + (4*(a + b*x^n)^(3/2))/(a^2*c*n*(c*x)^((7*n)/2)) - (16*(a + b*x^
n)^(5/2))/(5*a^3*c*n*(c*x)^((7*n)/2)) + (32*(a + b*x^n)^(7/2))/(35*a^4*c*n*(c*x)^((7*n)/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0502062, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac{32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac{16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((7*n)/2)) + (4*(a + b*x^n)^(3/2))/(a^2*c*n*(c*x)^((7*n)/2)) - (16*(a + b*x^
n)^(5/2))/(5*a^3*c*n*(c*x)^((7*n)/2)) + (32*(a + b*x^n)^(7/2))/(35*a^4*c*n*(c*x)^((7*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{7 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n}-\frac{6 \int (c x)^{-1-\frac{7 n}{2}} \sqrt{a+b x^n} \, dx}{a}\\ &=-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}+\frac{8 \int (c x)^{-1-\frac{7 n}{2}} \left (a+b x^n\right )^{3/2} \, dx}{a^2}\\ &=-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac{16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}-\frac{16 \int (c x)^{-1-\frac{7 n}{2}} \left (a+b x^n\right )^{5/2} \, dx}{5 a^3}\\ &=-\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n}}{a c n}+\frac{4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac{16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac{32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}\\ \end{align*}

Mathematica [A]  time = 0.0247761, size = 69, normalized size = 0.53 \[ -\frac{2 (c x)^{-7 n/2} \sqrt{a+b x^n} \left (-6 a^2 b x^n+5 a^3+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 c n} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n]*(5*a^3 - 6*a^2*b*x^n + 8*a*b^2*x^(2*n) - 16*b^3*x^(3*n)))/(35*a^4*c*n*(c*x)^((7*n)/2))

________________________________________________________________________________________

Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1-{\frac{7\,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)^(-1-7/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1-7/2*n)/(a+b*x^n)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{7}{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)^(-1-7/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x)^(-7/2*n - 1)/sqrt(b*x^n + a), x)

________________________________________________________________________________________

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)^(-1-7/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [B]  time = 13.6101, size = 665, normalized size = 5.16 \begin{align*} - \frac{10 a^{6} b^{\frac{19}{2}} c^{- \frac{7 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} - \frac{18 a^{5} b^{\frac{21}{2}} c^{- \frac{7 n}{2}} x^{n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} - \frac{10 a^{4} b^{\frac{23}{2}} c^{- \frac{7 n}{2}} x^{2 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac{10 a^{3} b^{\frac{25}{2}} c^{- \frac{7 n}{2}} x^{3 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac{60 a^{2} b^{\frac{27}{2}} c^{- \frac{7 n}{2}} x^{4 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac{80 a b^{\frac{29}{2}} c^{- \frac{7 n}{2}} x^{5 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} + \frac{32 b^{\frac{31}{2}} c^{- \frac{7 n}{2}} x^{6 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{c \left (35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(-1-7/2*n)/(a+b*x**n)**(1/2),x)

[Out]

-10*a**6*b**(19/2)*c**(-7*n/2)*sqrt(a*x**(-n)/b + 1)/(c*(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) +
105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n))) - 18*a**5*b**(21/2)*c**(-7*n/2)*x**n*sqrt(a*x**(-n)/b
+ 1)/(c*(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**
(6*n))) - 10*a**4*b**(23/2)*c**(-7*n/2)*x**(2*n)*sqrt(a*x**(-n)/b + 1)/(c*(35*a**7*b**9*n*x**(3*n) + 105*a**6*
b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n))) + 10*a**3*b**(25/2)*c**(-7*n/2)*x**(
3*n)*sqrt(a*x**(-n)/b + 1)/(c*(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n)
+ 35*a**4*b**12*n*x**(6*n))) + 60*a**2*b**(27/2)*c**(-7*n/2)*x**(4*n)*sqrt(a*x**(-n)/b + 1)/(c*(35*a**7*b**9*
n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n))) + 80*a*b**(29/
2)*c**(-7*n/2)*x**(5*n)*sqrt(a*x**(-n)/b + 1)/(c*(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a*
*5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n))) + 32*b**(31/2)*c**(-7*n/2)*x**(6*n)*sqrt(a*x**(-n)/b + 1)/(c*
(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{-\frac{7}{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)^(-1-7/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x)^(-7/2*n - 1)/sqrt(b*x^n + a), x)