Optimal. Leaf size=167 \[ -\frac {d x^{1+m} \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1+m}{n};\frac {1}{2},1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}}+\frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2187, 371, 525,
524} \begin {gather*} \frac {c x^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right )}-\frac {d x^{m+1} \sqrt {\frac {b x^n}{a}+1} F_1\left (\frac {m+1}{n};\frac {1}{2},1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 524
Rule 525
Rule 2187
Rubi steps
\begin {align*} \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx &=(a c) \int \frac {x^m}{a^2 c^2-a d^2+a b c^2 x^n} \, dx-(a d) \int \frac {x^m}{\sqrt {a+b x^n} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx\\ &=\frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)}-\frac {\left (a d \sqrt {1+\frac {b x^n}{a}}\right ) \int \frac {x^m}{\sqrt {1+\frac {b x^n}{a}} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx}{\sqrt {a+b x^n}}\\ &=-\frac {d x^{1+m} \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1+m}{n};\frac {1}{2},1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}}+\frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.36, size = 156, normalized size = 0.93 \begin {gather*} \frac {x^{1+m} \left (-d \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1+m}{n};\frac {1}{2},1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )+c \sqrt {a+b x^n} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{a c +b c \,x^{n}+d \sqrt {a +b \,x^{n}}}\, dx\]
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]
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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m}}{a c + b c x^{n} + d \sqrt {a + 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 = -0.01 \begin {gather*} \int \frac {x^m}{a\,c+d\,\sqrt {a+b\,x^n}+b\,c\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________