\(\int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [2788]

Optimal result

Integrand size = 23, antiderivative size = 129 \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac {16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac {32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n} \]

[Out]

4*(a+b*x^n)^(3/2)/a^2/c/n/((c*x)^(7/2*n))-16/5*(a+b*x^n)^(5/2)/a^3/c/n/((c*x)^(7/2*n))+32/35*(a+b*x^n)^(7/2)/a
^4/c/n/((c*x)^(7/2*n))-2*(a+b*x^n)^(1/2)/a/c/n/((c*x)^(7/2*n))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {279, 270} \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {32 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^4 c n}-\frac {16 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a^3 c n}+\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{a^2 c n}-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n} \]

[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 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*} \text {integral}& = -\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] (verified)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.53 \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n} \left (5 a^3-6 a^2 b x^n+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 c n} \]

[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]

\[\int \frac {\left (c x \right )^{-1-\frac {7 n}{2}}}{\sqrt {a +b \,x^{n}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c*x)^(-1-7/2*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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (109) = 218\).

Time = 1.02 (sec) , antiderivative size = 677, normalized size of antiderivative = 5.25 \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=- \frac {10 a^{6} b^{\frac {19}{2}} c^{- \frac {7 n}{2} - 1} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {18 a^{5} b^{\frac {21}{2}} c^{- \frac {7 n}{2} - 1} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {10 a^{4} b^{\frac {23}{2}} c^{- \frac {7 n}{2} - 1} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {10 a^{3} b^{\frac {25}{2}} c^{- \frac {7 n}{2} - 1} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {60 a^{2} b^{\frac {27}{2}} c^{- \frac {7 n}{2} - 1} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {80 a b^{\frac {29}{2}} c^{- \frac {7 n}{2} - 1} x^{5 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {32 b^{\frac {31}{2}} c^{- \frac {7 n}{2} - 1} x^{6 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} \]

[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 - 1)*sqrt(a/(b*x**n) + 1)/(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 - 1)*x**n*sqrt(a/(b*x**n
) + 1)/(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 - 1)*x**(2*n)*sqrt(a/(b*x**n) + 1)/(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 - 1)*x**
(3*n)*sqrt(a/(b*x**n) + 1)/(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 - 1)*x**(4*n)*sqrt(a/(b*x**n) + 1)/(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 - 1)*x**(5*n)*sqrt(a/(b*x**n) + 1)/(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 - 1)*x**(6*n)*sqrt(a/(b*x**n) + 1)/(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))

Maxima [F]

\[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{\frac {7\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \]

[In]

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

[Out]

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