Optimal. Leaf size=99 \[ \frac {2 \sqrt {a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)}-\frac {2 (c x)^{-j/2} \sqrt {a x^j+b x^n}}{c (j-n)} \]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2031, 2028, 2029, 206} \begin {gather*} \frac {2 \sqrt {a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)}-\frac {2 (c x)^{-j/2} \sqrt {a x^j+b x^n}}{c (j-n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2028
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int (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 x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx}{c}\\ &=-\frac {2 (c x)^{-j/2} \sqrt {a x^j+b x^n}}{c (j-n)}+\frac {\left (a 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 {2 (c x)^{-j/2} \sqrt {a x^j+b x^n}}{c (j-n)}+\frac {\left (2 a x^{j/2} (c x)^{-j/2}\right ) \operatorname {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 (c x)^{-j/2} \sqrt {a x^j+b x^n}}{c (j-n)}+\frac {2 \sqrt {a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 109, normalized size = 1.10 \begin {gather*} -\frac {2 (c x)^{-j/2} \left (-\sqrt {a} \sqrt {b} x^{\frac {j+n}{2}} \sqrt {\frac {a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )+a x^j+b x^n\right )}{c (j-n) \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx \end {gather*}
Verification is not applicable to the result.
[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]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x^{j} + b x^{n}} \left (c x\right )^{-\frac {1}{2} \, j - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.95, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \,x^{j}+b \,x^{n}}\, \left (c x \right )^{-\frac {j}{2}-1}\, dx \end {gather*}
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*} \int \sqrt {a x^{j} + b x^{n}} \left (c x\right )^{-\frac {1}{2} \, j - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a\,x^j+b\,x^n}}{{\left (c\,x\right )}^{\frac {j}{2}+1}} \,d x \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 \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]
________________________________________________________________________________________