Optimal. Leaf size=113 \[ -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {266, 47, 51, 63, 208} \[ \frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int x^{-1-3 n} \sqrt {a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^n\right )}{6 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^n\right )}{8 a n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{16 a^2 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{8 a^2 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 42, normalized size = 0.37 \[ \frac {2 b^3 \left (a+b x^n\right )^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {b x^n}{a}+1\right )}{3 a^4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 183, normalized size = 1.62 \[ \left [\frac {3 \, \sqrt {a} b^{3} x^{3 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{48 \, a^{3} n x^{3 \, n}}, \frac {3 \, \sqrt {-a} b^{3} x^{3 \, n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{24 \, a^{3} n x^{3 \, n}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} x^{-3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,x^{n}+a}\, x^{-3 n -1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} x^{-3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,x^n}}{x^{3\,n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________