3.2650 \(\int x^{-1+3 n} \sqrt{a+b x^n} \, dx\)
Optimal. Leaf size=68 \[ \frac{2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac{4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]
[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)
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Rubi [A] time = 0.032652, antiderivative size = 68, 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} \[ \frac{2 a^2 \left (a+b x^n\right )^{3/2}}{3 b^3 n}+\frac{2 \left (a+b x^n\right )^{7/2}}{7 b^3 n}-\frac{4 a \left (a+b x^n\right )^{5/2}}{5 b^3 n} \]
Antiderivative was successfully verified.
[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 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 x^{-1+3 n} \sqrt{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \sqrt{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{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] time = 0.0227448, size = 44, normalized size = 0.65 \[ \frac{2 \left (a+b x^n\right )^{3/2} \left (8 a^2-12 a b x^n+15 b^2 x^{2 n}\right )}{105 b^3 n} \]
Antiderivative was successfully verified.
[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)
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Maple [A] time = 0.02, size = 54, normalized size = 0.8 \begin{align*}{\frac{30\, \left ({x}^{n} \right ) ^{3}{b}^{3}+6\,a \left ({x}^{n} \right ) ^{2}{b}^{2}-8\,{a}^{2}{x}^{n}b+16\,{a}^{3}}{105\,{b}^{3}n}\sqrt{a+b{x}^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(-1+3*n)*(a+b*x^n)^(1/2),x)
[Out]
2/105*(15*(x^n)^3*b^3+3*a*(x^n)^2*b^2-4*a^2*x^n*b+8*a^3)*(a+b*x^n)^(1/2)/b^3/n
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Maxima [A] time = 1.06804, size = 72, normalized size = 1.06 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [A] time = 1.03466, size = 119, normalized size = 1.75 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [B] time = 109.784, size = 1015, normalized size = 14.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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*x**(-n)/b + 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)*s
qrt(a*x**(-n)/b + 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*x**(-n)/b + 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*x**(-n)/b + 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*x**(-n)/b + 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*x**(-n)/b + 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*x**(-n)/b + 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) + 31
5*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))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{n} + a} x^{3 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)