Optimal. Leaf size=62 \[ \frac {2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{\sqrt {a} c (j-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2056, 2054,
212} \begin {gather*} \frac {2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{\sqrt {a} c (j-n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2054
Rule 2056
Rubi steps
\begin {align*} \int \frac {(c x)^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx &=\frac {\left (x^{-j/2} (c x)^{j/2}\right ) \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx}{c}\\ &=\frac {\left (2 x^{-j/2} (c x)^{j/2}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)}\\ &=\frac {2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{\sqrt {a} c (j-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.13, size = 98, normalized size = 1.58 \begin {gather*} \frac {2 \sqrt {b} x^{\frac {1}{2} (-j+n)} (c x)^{j/2} \sqrt {1+\frac {a x^{j-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} c (j-n) \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{-1+\frac {j}{2}}}{\sqrt {a \,x^{j}+b \,x^{n}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x\right )^{\frac {j}{2} - 1}}{\sqrt {a x^{j} + b x^{n}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c\,x\right )}^{\frac {j}{2}-1}}{\sqrt {a\,x^j+b\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________