Optimal. Leaf size=54 \[ -\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{\sqrt {a} c (n+1) \sqrt {c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2031, 2029, 206} \[ -\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{\sqrt {a} c (n+1) \sqrt {c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{3/2} \sqrt {\frac {a}{x}+b x^n}} \, dx &=\frac {\sqrt {x} \int \frac {1}{x^{3/2} \sqrt {\frac {a}{x}+b x^n}} \, dx}{c \sqrt {c x}}\\ &=-\frac {\left (2 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{c (1+n) \sqrt {c x}}\\ &=-\frac {2 \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {x} \sqrt {\frac {a}{x}+b x^n}}\right )}{\sqrt {a} c (1+n) \sqrt {c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 68, normalized size = 1.26 \[ -\frac {2 x \sqrt {a+b x^{n+1}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+1}}}{\sqrt {a}}\right )}{\sqrt {a} (n+1) (c x)^{3/2} \sqrt {\frac {a}{x}+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{n} + \frac {a}{x}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {3}{2}} \sqrt {b \,x^{n}+\frac {a}{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{n} + \frac {a}{x}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{3/2}\,\sqrt {b\,x^n+\frac {a}{x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x\right )^{\frac {3}{2}} \sqrt {\frac {a}{x} + b x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________