3.2694 \(\int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx\)

Optimal. Leaf size=120 \[ \frac {32 b^3 x^{-n/2} \sqrt {a+b x^n}}{35 a^4 n}-\frac {16 b^2 x^{-3 n/2} \sqrt {a+b x^n}}{35 a^3 n}+\frac {12 b x^{-5 n/2} \sqrt {a+b x^n}}{35 a^2 n}-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n} \]

[Out]

-2/7*(a+b*x^n)^(1/2)/a/n/(x^(7/2*n))+12/35*b*(a+b*x^n)^(1/2)/a^2/n/(x^(5/2*n))-16/35*b^2*(a+b*x^n)^(1/2)/a^3/n
/(x^(3/2*n))+32/35*b^3*(a+b*x^n)^(1/2)/a^4/n/(x^(1/2*n))

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Rubi [A]  time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {271, 264} \[ -\frac {16 b^2 x^{-3 n/2} \sqrt {a+b x^n}}{35 a^3 n}+\frac {32 b^3 x^{-n/2} \sqrt {a+b x^n}}{35 a^4 n}+\frac {12 b x^{-5 n/2} \sqrt {a+b x^n}}{35 a^2 n}-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x^n])/(7*a*n*x^((7*n)/2)) + (12*b*Sqrt[a + b*x^n])/(35*a^2*n*x^((5*n)/2)) - (16*b^2*Sqrt[a + b*
x^n])/(35*a^3*n*x^((3*n)/2)) + (32*b^3*Sqrt[a + b*x^n])/(35*a^4*n*x^(n/2))

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]

Rule 271

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

Rubi steps

\begin {align*} \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}-\frac {(6 b) \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx}{7 a}\\ &=-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}+\frac {12 b x^{-5 n/2} \sqrt {a+b x^n}}{35 a^2 n}+\frac {\left (24 b^2\right ) \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx}{35 a^2}\\ &=-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}+\frac {12 b x^{-5 n/2} \sqrt {a+b x^n}}{35 a^2 n}-\frac {16 b^2 x^{-3 n/2} \sqrt {a+b x^n}}{35 a^3 n}-\frac {\left (16 b^3\right ) \int \frac {x^{-1-\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx}{35 a^3}\\ &=-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}+\frac {12 b x^{-5 n/2} \sqrt {a+b x^n}}{35 a^2 n}-\frac {16 b^2 x^{-3 n/2} \sqrt {a+b x^n}}{35 a^3 n}+\frac {32 b^3 x^{-n/2} \sqrt {a+b x^n}}{35 a^4 n}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 64, normalized size = 0.53 \[ -\frac {2 x^{-7 n/2} \sqrt {a+b x^n} \left (5 a^3-6 a^2 b x^n+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[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*n*x^((7*n)/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {7 n}{2}-1}}{\sqrt {b \,x^{n}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{\frac {7\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 6.51, size = 605, normalized size = 5.04 \[ - \frac {10 a^{6} b^{\frac {19}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {18 a^{5} b^{\frac {21}{2}} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {10 a^{4} b^{\frac {23}{2}} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {10 a^{3} b^{\frac {25}{2}} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {60 a^{2} b^{\frac {27}{2}} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {80 a b^{\frac {29}{2}} x^{5 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {32 b^{\frac {31}{2}} x^{6 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*a**6*b**(19/2)*sqrt(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)) - 18*a**5*b**(21/2)*x**n*sqrt(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)) - 10*a**4*b**(23/2)*x**(
2*n)*sqrt(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)) + 10*a**3*b**(25/2)*x**(3*n)*sqrt(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)) + 60*a**2*b**(27/2)*x**(4*n)*sqrt(
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)) + 80*a*b**(29/2)*x**(5*n)*sqrt(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)) + 32*b**(31/2)*x**(6*n)*sqrt(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))

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