\(\int x^{-1+3 n} \sqrt {a+b x^n} \, dx\) [2650]

Optimal result

Integrand size = 19, antiderivative size = 68 \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\frac {2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}-\frac {4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n}+\frac {2 \left (a+b x^n\right )^{7/2}}{7 b^3 n} \]

[Out]

2/3*a^2*(a+b*x^n)^(3/2)/b^3/n-4/5*a*(a+b*x^n)^(5/2)/b^3/n+2/7*(a+b*x^n)^(7/2)/b^3/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, 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} \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\frac {2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac {2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac {4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]

[In]

Int[x^(-1 + 3*n)*Sqrt[a + b*x^n],x]

[Out]

(2*a^2*(a + b*x^n)^(3/2))/(3*b^3*n) - (4*a*(a + b*x^n)^(5/2))/(5*b^3*n) + (2*(a + b*x^n)^(7/2))/(7*b^3*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^2 \sqrt {a+b x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}-\frac {4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n}+\frac {2 \left (a+b x^n\right )^{7/2}}{7 b^3 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.65 \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\frac {2 \left (a+b x^n\right )^{3/2} \left (8 a^2-12 a b x^n+15 b^2 x^{2 n}\right )}{105 b^3 n} \]

[In]

Integrate[x^(-1 + 3*n)*Sqrt[a + b*x^n],x]

[Out]

(2*(a + b*x^n)^(3/2)*(8*a^2 - 12*a*b*x^n + 15*b^2*x^(2*n)))/(105*b^3*n)

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {2 \left (15 b^{3} x^{3 n}+3 a \,b^{2} x^{2 n}-4 a^{2} b \,x^{n}+8 a^{3}\right ) \sqrt {a +b \,x^{n}}}{105 b^{3} n}\)	\(54\)

[In]

int(x^(-1+3*n)*(a+b*x^n)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/105*(15*b^3*(x^n)^3+3*a*b^2*(x^n)^2-4*a^2*b*x^n+8*a^3)*(a+b*x^n)^(1/2)/b^3/n

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\frac {2 \, {\left (15 \, b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{105 \, b^{3} n} \]

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*b^3*x^(3*n) + 3*a*b^2*x^(2*n) - 4*a^2*b*x^n + 8*a^3)*sqrt(b*x^n + a)/(b^3*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (60) = 120\).

Time = 2.99 (sec) , antiderivative size = 1015, normalized size of antiderivative = 14.93 \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\frac {16 a^{\frac {19}{2}} b^{\frac {9}{2}} x^{\frac {9 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {40 a^{\frac {17}{2}} b^{\frac {11}{2}} x^{\frac {11 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {30 a^{\frac {15}{2}} b^{\frac {13}{2}} x^{\frac {13 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {40 a^{\frac {13}{2}} b^{\frac {15}{2}} x^{\frac {15 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {100 a^{\frac {11}{2}} b^{\frac {17}{2}} x^{\frac {17 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {96 a^{\frac {9}{2}} b^{\frac {19}{2}} x^{\frac {19 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} + \frac {30 a^{\frac {7}{2}} b^{\frac {21}{2}} x^{\frac {21 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {16 a^{10} b^{4} x^{4 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {48 a^{9} b^{5} x^{5 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {48 a^{8} b^{6} x^{6 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} - \frac {16 a^{7} b^{7} x^{7 n}}{105 a^{\frac {13}{2}} b^{7} n x^{4 n} + 315 a^{\frac {11}{2}} b^{8} n x^{5 n} + 315 a^{\frac {9}{2}} b^{9} n x^{6 n} + 105 a^{\frac {7}{2}} b^{10} n x^{7 n}} \]

[In]

integrate(x**(-1+3*n)*(a+b*x**n)**(1/2),x)

[Out]

16*a**(19/2)*b**(9/2)*x**(9*n/2)*sqrt(a/(b*x**n) + 1)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x*
*(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n)) + 40*a**(17/2)*b**(11/2)*x**(11*n/2)*sq
rt(a/(b*x**n) + 1)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*
n) + 105*a**(7/2)*b**10*n*x**(7*n)) + 30*a**(15/2)*b**(13/2)*x**(13*n/2)*sqrt(a/(b*x**n) + 1)/(105*a**(13/2)*b
**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n))
+ 40*a**(13/2)*b**(15/2)*x**(15*n/2)*sqrt(a/(b*x**n) + 1)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8
*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n)) + 100*a**(11/2)*b**(17/2)*x**(17*n
/2)*sqrt(a/(b*x**n) + 1)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*
x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n)) + 96*a**(9/2)*b**(19/2)*x**(19*n/2)*sqrt(a/(b*x**n) + 1)/(105*a**(13
/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(
7*n)) + 30*a**(7/2)*b**(21/2)*x**(21*n/2)*sqrt(a/(b*x**n) + 1)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*
b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n)) - 16*a**10*b**4*x**(4*n)/(105*
a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*
n*x**(7*n)) - 48*a**9*b**5*x**(5*n)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9
/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n)) - 48*a**8*b**6*x**(6*n)/(105*a**(13/2)*b**7*n*x**(4*n) +
315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a**(7/2)*b**10*n*x**(7*n)) - 16*a**7*b**7*x
**(7*n)/(105*a**(13/2)*b**7*n*x**(4*n) + 315*a**(11/2)*b**8*n*x**(5*n) + 315*a**(9/2)*b**9*n*x**(6*n) + 105*a*
*(7/2)*b**10*n*x**(7*n))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\frac {2 \, {\left (15 \, b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{105 \, b^{3} n} \]

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

2/105*(15*b^3*x^(3*n) + 3*a*b^2*x^(2*n) - 4*a^2*b*x^n + 8*a^3)*sqrt(b*x^n + a)/(b^3*n)

Giac [F]

\[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\int { \sqrt {b x^{n} + a} x^{3 \, n - 1} \,d x } \]

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a)*x^(3*n - 1), x)

Mupad [F(-1)]

Timed out. \[ \int x^{-1+3 n} \sqrt {a+b x^n} \, dx=\int x^{3\,n-1}\,\sqrt {a+b\,x^n} \,d x \]

[In]

int(x^(3*n - 1)*(a + b*x^n)^(1/2),x)

[Out]

int(x^(3*n - 1)*(a + b*x^n)^(1/2), x)