Optimal. Leaf size=291 \[ \frac {x \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {i 2^{\frac {1}{2}-\frac {i n}{2}} \left (1-n^2\right ) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \, _2F_1\left (\frac {1}{2} (i n-1),\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1}{2} (1-i a x)\right )}{a^3 \left (n^2+1\right ) \sqrt {a^2 c x^2+c}}-\frac {(1+i n) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{2 a^3 (n+i) \sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5085, 5082, 90, 79, 69} \[ -\frac {i 2^{\frac {1}{2}-\frac {i n}{2}} \left (1-n^2\right ) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \, _2F_1\left (\frac {1}{2} (i n-1),\frac {1}{2} (i n+1);\frac {1}{2} (i n+3);\frac {1}{2} (1-i a x)\right )}{a^3 \left (n^2+1\right ) \sqrt {a^2 c x^2+c}}+\frac {x \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {(1+i n) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{2 a^3 (n+i) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 79
Rule 90
Rule 5082
Rule 5085
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)} x^2}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)} x^2}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int x^2 (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {x (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{2 a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \int (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} (-1-a n x) \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {(1+i n) (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{2 a^3 (i+n) \sqrt {c+a^2 c x^2}}+\frac {x (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{2 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (\left (1-n^2\right ) \sqrt {1+a^2 x^2}\right ) \int (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2} i (i+n)} \, dx}{2 a^2 (1-i n) \sqrt {c+a^2 c x^2}}\\ &=-\frac {(1+i n) (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{2 a^3 (i+n) \sqrt {c+a^2 c x^2}}+\frac {x (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{2 a^2 \sqrt {c+a^2 c x^2}}-\frac {i 2^{\frac {1}{2}-\frac {i n}{2}} \left (1-n^2\right ) (1-i a x)^{\frac {1}{2} (1+i n)} \sqrt {1+a^2 x^2} \, _2F_1\left (\frac {1}{2} (-1+i n),\frac {1}{2} (1+i n);\frac {1}{2} (3+i n);\frac {1}{2} (1-i a x)\right )}{a^3 \left (1+n^2\right ) \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 206, normalized size = 0.71 \[ \frac {2^{-1-\frac {i n}{2}} \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2 i \sqrt {2} \left (n^2-1\right ) (1+i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {1}{2} (i n+1),\frac {1}{2} i (n+i);\frac {1}{2} (i n+3);\frac {1}{2} (1-i a x)\right )+2^{\frac {i n}{2}} (n-i) \sqrt {1+i a x} (n (a x-i)+i a x-1)\right )}{a^3 \left (n^2+1\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{2}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________