3.2658 \(\int \frac{x^{-1+4 n}}{\sqrt{a+b x^n}} \, dx\)
Optimal. Leaf size=88 \[ \frac{2 a^2 \left (a+b x^n\right )^{3/2}}{b^4 n}-\frac{2 a^3 \sqrt{a+b x^n}}{b^4 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^4 n}-\frac{6 a \left (a+b x^n\right )^{5/2}}{5 b^4 n} \]
[Out]
(-2*a^3*Sqrt[a + b*x^n])/(b^4*n) + (2*a^2*(a + b*x^n)^(3/2))/(b^4*n) - (6*a*(a + b*x^n)^(5/2))/(5*b^4*n) + (2*
(a + b*x^n)^(7/2))/(7*b^4*n)
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Rubi [A] time = 0.0410302, antiderivative size = 88, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{266, 43} \[ \frac{2 a^2 \left (a+b x^n\right )^{3/2}}{b^4 n}-\frac{2 a^3 \sqrt{a+b x^n}}{b^4 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^4 n}-\frac{6 a \left (a+b x^n\right )^{5/2}}{5 b^4 n} \]
Antiderivative was successfully verified.
[In]
Int[x^(-1 + 4*n)/Sqrt[a + b*x^n],x]
[Out]
(-2*a^3*Sqrt[a + b*x^n])/(b^4*n) + (2*a^2*(a + b*x^n)^(3/2))/(b^4*n) - (6*a*(a + b*x^n)^(5/2))/(5*b^4*n) + (2*
(a + b*x^n)^(7/2))/(7*b^4*n)
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{x^{-1+4 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt{a+b x}}+\frac{3 a^2 \sqrt{a+b x}}{b^3}-\frac{3 a (a+b x)^{3/2}}{b^3}+\frac{(a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{2 a^3 \sqrt{a+b x^n}}{b^4 n}+\frac{2 a^2 \left (a+b x^n\right )^{3/2}}{b^4 n}-\frac{6 a \left (a+b x^n\right )^{5/2}}{5 b^4 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^4 n}\\ \end{align*}
Mathematica [A] time = 0.0275422, size = 57, normalized size = 0.65 \[ \frac{2 \sqrt{a+b x^n} \left (8 a^2 b x^n-16 a^3-6 a b^2 x^{2 n}+5 b^3 x^{3 n}\right )}{35 b^4 n} \]
Antiderivative was successfully verified.
[In]
Integrate[x^(-1 + 4*n)/Sqrt[a + b*x^n],x]
[Out]
(2*Sqrt[a + b*x^n]*(-16*a^3 + 8*a^2*b*x^n - 6*a*b^2*x^(2*n) + 5*b^3*x^(3*n)))/(35*b^4*n)
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Maple [A] time = 0.021, size = 54, normalized size = 0.6 \begin{align*} -{\frac{-10\, \left ({x}^{n} \right ) ^{3}{b}^{3}+12\,a \left ({x}^{n} \right ) ^{2}{b}^{2}-16\,{a}^{2}{x}^{n}b+32\,{a}^{3}}{35\,{b}^{4}n}\sqrt{a+b{x}^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(-1+4*n)/(a+b*x^n)^(1/2),x)
[Out]
-2/35*(-5*(x^n)^3*b^3+6*a*(x^n)^2*b^2-8*a^2*x^n*b+16*a^3)*(a+b*x^n)^(1/2)/b^4/n
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Maxima [A] time = 0.992495, size = 89, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (5 \, b^{4} x^{4 \, n} - a b^{3} x^{3 \, n} + 2 \, a^{2} b^{2} x^{2 \, n} - 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )}}{35 \, \sqrt{b x^{n} + a} b^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(-1+4*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")
[Out]
2/35*(5*b^4*x^(4*n) - a*b^3*x^(3*n) + 2*a^2*b^2*x^(2*n) - 8*a^3*b*x^n - 16*a^4)/(sqrt(b*x^n + a)*b^4*n)
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Fricas [A] time = 1.01314, size = 117, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{3 \, n} - 6 \, a b^{2} x^{2 \, n} + 8 \, a^{2} b x^{n} - 16 \, a^{3}\right )} \sqrt{b x^{n} + a}}{35 \, b^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(-1+4*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")
[Out]
2/35*(5*b^3*x^(3*n) - 6*a*b^2*x^(2*n) + 8*a^2*b*x^n - 16*a^3)*sqrt(b*x^n + a)/(b^4*n)
________________________________________________________________________________________
Sympy [B] time = 105.634, size = 2422, normalized size = 27.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**(-1+4*n)/(a+b*x**n)**(1/2),x)
[Out]
-32*a**(25/2)*b**(23/2)*x**(23*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**1
6*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(
15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) - 176*a**(23/2)*b**(25/2)*x**(25*n/2)*
sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*
n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*
n) + 35*a**(7/2)*b**21*n*x**(17*n)) - 396*a**(21/2)*b**(27/2)*x**(27*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*
b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*
x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n))
- 462*a**(19/2)*b**(29/2)*x**(29*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b*
*16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x*
*(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) - 280*a**(17/2)*b**(31/2)*x**(31*n/2
)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**1
7*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(1
6*n) + 35*a**(7/2)*b**21*n*x**(17*n)) - 42*a**(15/2)*b**(33/2)*x**(33*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)
*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n
*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n))
+ 84*a**(13/2)*b**(35/2)*x**(35*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b*
*16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x*
*(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 94*a**(11/2)*b**(37/2)*x**(37*n/2)
*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17
*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16
*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 48*a**(9/2)*b**(39/2)*x**(39*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b
**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x
**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) +
10*a**(7/2)*b**(41/2)*x**(41*n/2)*sqrt(a*x**(-n)/b + 1)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16
*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(1
5*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 32*a**13*b**11*x**(11*n)/(35*a**(19/2
)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*
n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)
) + 192*a**12*b**12*x**(12*n)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2
)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n
*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 480*a**11*b**13*x**(13*n)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*
a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2
)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 640*a**10*b**14*x**(14
*n)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*
a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b
**21*n*x**(17*n)) + 480*a**9*b**15*x**(15*n)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n)
+ 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a
**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 192*a**8*b**16*x**(16*n)/(35*a**(19/2)*b**15*n*x*
*(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(13*n) + 700*a**(13/2)*b**18*n*x**(14*n)
+ 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 35*a**(7/2)*b**21*n*x**(17*n)) + 32*a**7*
b**17*x**(17*n)/(35*a**(19/2)*b**15*n*x**(11*n) + 210*a**(17/2)*b**16*n*x**(12*n) + 525*a**(15/2)*b**17*n*x**(
13*n) + 700*a**(13/2)*b**18*n*x**(14*n) + 525*a**(11/2)*b**19*n*x**(15*n) + 210*a**(9/2)*b**20*n*x**(16*n) + 3
5*a**(7/2)*b**21*n*x**(17*n))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(-1+4*n)/(a+b*x^n)^(1/2),x, algorithm="giac")
[Out]
integrate(x^(4*n - 1)/sqrt(b*x^n + a), x)