\(\int x^{-1+4 n} \sqrt {a+b x^n} \, dx\) [2649]
Optimal result
Integrand size = 19, antiderivative size = 92 \[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=-\frac {2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac {6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}-\frac {6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n}+\frac {2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}
\]
[Out]
-2/3*a^3*(a+b*x^n)^(3/2)/b^4/n+6/5*a^2*(a+b*x^n)^(5/2)/b^4/n-6/7*a*(a+b*x^n)^(7/2)/b^4/n+2/9*(a+b*x^n)^(9/2)/b
^4/n
Rubi [A] (verified)
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {272, 45}
\[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=-\frac {2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac {6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}+\frac {2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}-\frac {6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n}
\]
[In]
Int[x^(-1 + 4*n)*Sqrt[a + b*x^n],x]
[Out]
(-2*a^3*(a + b*x^n)^(3/2))/(3*b^4*n) + (6*a^2*(a + b*x^n)^(5/2))/(5*b^4*n) - (6*a*(a + b*x^n)^(7/2))/(7*b^4*n)
+ (2*(a + b*x^n)^(9/2))/(9*b^4*n)
Rule 45
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])
Rule 272
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]]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int x^3 \sqrt {a+b x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac {6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}-\frac {6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n}+\frac {2 \left (a+b x^n\right )^{9/2}}{9 b^4 n} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.62
\[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=\frac {2 \left (a+b x^n\right )^{3/2} \left (-16 a^3+24 a^2 b x^n-30 a b^2 x^{2 n}+35 b^3 x^{3 n}\right )}{315 b^4 n}
\]
[In]
Integrate[x^(-1 + 4*n)*Sqrt[a + b*x^n],x]
[Out]
(2*(a + b*x^n)^(3/2)*(-16*a^3 + 24*a^2*b*x^n - 30*a*b^2*x^(2*n) + 35*b^3*x^(3*n)))/(315*b^4*n)
Maple [A] (verified)
Time = 3.86 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.73
| | |
method | result | size |
| | |
risch |
\(-\frac {2 \left (-35 x^{4 n} b^{4}-5 a \,x^{3 n} b^{3}+6 a^{2} x^{2 n} b^{2}-8 a^{3} x^{n} b +16 a^{4}\right ) \sqrt {a +b \,x^{n}}}{315 b^{4} n}\) |
\(67\) |
| | |
|
|
|
[In]
int(x^(-1+4*n)*(a+b*x^n)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-2/315*(-35*(x^n)^4*b^4-5*a*(x^n)^3*b^3+6*a^2*(x^n)^2*b^2-8*a^3*x^n*b+16*a^4)*(a+b*x^n)^(1/2)/b^4/n
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72
\[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{4 \, n} + 5 \, a b^{3} x^{3 \, n} - 6 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )} \sqrt {b x^{n} + a}}{315 \, b^{4} n}
\]
[In]
integrate(x^(-1+4*n)*(a+b*x^n)^(1/2),x, algorithm="fricas")
[Out]
2/315*(35*b^4*x^(4*n) + 5*a*b^3*x^(3*n) - 6*a^2*b^2*x^(2*n) + 8*a^3*b*x^n - 16*a^4)*sqrt(b*x^n + a)/(b^4*n)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2572 vs. \(2 (82) = 164\).
Time = 4.80 (sec) , antiderivative size = 2572, normalized size of antiderivative = 27.96
\[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=\text {Too large to display}
\]
[In]
integrate(x**(-1+4*n)*(a+b*x**n)**(1/2),x)
[Out]
-32*a**(29/2)*b**(23/2)*x**(23*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**
16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*
x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) - 176*a**(27/2)*b**(25/2)*x**(2
5*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/
2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b*
*20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) - 396*a**(25/2)*b**(27/2)*x**(27*n/2)*sqrt(a/(b*x**n) + 1)/(
315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a
**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2
)*b**21*n*x**(17*n)) - 462*a**(23/2)*b**(29/2)*x**(29*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*
n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) +
4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) - 210*a*
*(21/2)*b**(31/2)*x**(31*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x
**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15
*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 378*a**(19/2)*b**(33/2)*x**(33*n/2)
*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**
17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*
x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 1134*a**(17/2)*b**(35/2)*x**(35*n/2)*sqrt(a/(b*x**n) + 1)/(315*a
**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15
/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**
21*n*x**(17*n)) + 1494*a**(15/2)*b**(37/2)*x**(37*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) +
1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725
*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 1098*a**(1
3/2)*b**(39/2)*x**(39*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(
12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n)
+ 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 430*a**(11/2)*b**(41/2)*x**(41*n/2)*sq
rt(a/(b*x**n) + 1)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*
n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**
(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 70*a**(9/2)*b**(43/2)*x**(43*n/2)*sqrt(a/(b*x**n) + 1)/(315*a**(21/
2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b*
*18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x
**(17*n)) + 32*a**15*b**11*x**(11*n)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 472
5*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**
(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 192*a**14*b**12*x**(12*n)/(315*a**(21/2)*b**15*n*
x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(1
4*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) +
480*a**13*b**13*x**(13*n)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2
)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**
20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 640*a**12*b**14*x**(14*n)/(315*a**(21/2)*b**15*n*x**(11*n)
+ 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 472
5*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 480*a**11
*b**15*x**(15*n)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*
x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(1
6*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 192*a**10*b**16*x**(16*n)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**
(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) + 6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2
)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a**(9/2)*b**21*n*x**(17*n)) + 32*a**9*b**17*x**(1
7*n)/(315*a**(21/2)*b**15*n*x**(11*n) + 1890*a**(19/2)*b**16*n*x**(12*n) + 4725*a**(17/2)*b**17*n*x**(13*n) +
6300*a**(15/2)*b**18*n*x**(14*n) + 4725*a**(13/2)*b**19*n*x**(15*n) + 1890*a**(11/2)*b**20*n*x**(16*n) + 315*a
**(9/2)*b**21*n*x**(17*n))
Maxima [A] (verification not implemented)
none
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.72
\[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{4 \, n} + 5 \, a b^{3} x^{3 \, n} - 6 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )} \sqrt {b x^{n} + a}}{315 \, b^{4} n}
\]
[In]
integrate(x^(-1+4*n)*(a+b*x^n)^(1/2),x, algorithm="maxima")
[Out]
2/315*(35*b^4*x^(4*n) + 5*a*b^3*x^(3*n) - 6*a^2*b^2*x^(2*n) + 8*a^3*b*x^n - 16*a^4)*sqrt(b*x^n + a)/(b^4*n)
Giac [F]
\[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=\int { \sqrt {b x^{n} + a} x^{4 \, n - 1} \,d x }
\]
[In]
integrate(x^(-1+4*n)*(a+b*x^n)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*x^n + a)*x^(4*n - 1), x)
Mupad [F(-1)]
Timed out. \[
\int x^{-1+4 n} \sqrt {a+b x^n} \, dx=\int x^{4\,n-1}\,\sqrt {a+b\,x^n} \,d x
\]
[In]
int(x^(4*n - 1)*(a + b*x^n)^(1/2),x)
[Out]
int(x^(4*n - 1)*(a + b*x^n)^(1/2), x)