Optimal. Leaf size=75 \[ \frac {(d x)^{m+1} \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{d (m+1)}-\frac {2 b c (d x)^{m+3} \, _2F_1\left (1,\frac {m+3}{4};\frac {m+7}{4};-c^2 x^4\right )}{d^3 (m+1) (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5033, 16, 364} \[ \frac {(d x)^{m+1} \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{d (m+1)}-\frac {2 b c (d x)^{m+3} \, _2F_1\left (1,\frac {m+3}{4};\frac {m+7}{4};-c^2 x^4\right )}{d^3 (m+1) (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 364
Rule 5033
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {(2 b c) \int \frac {x (d x)^{1+m}}{1+c^2 x^4} \, dx}{d (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {(2 b c) \int \frac {(d x)^{2+m}}{1+c^2 x^4} \, dx}{d^2 (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {2 b c (d x)^{3+m} \, _2F_1\left (1,\frac {3+m}{4};\frac {7+m}{4};-c^2 x^4\right )}{d^3 (1+m) (3+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 65, normalized size = 0.87 \[ -\frac {x (d x)^m \left (2 b c x^2 \, _2F_1\left (1,\frac {m+3}{4};\frac {m+7}{4};-c^2 x^4\right )-(m+3) \left (a+b \tan ^{-1}\left (c x^2\right )\right )\right )}{(m+1) (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arctan \left (c x^{2}\right ) + a\right )} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x^{2}\right ) + a\right )} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctan \left (c \,x^{2}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d^{m} x x^{m} \arctan \left (c x^{2}\right ) - 2 \, {\left (c d^{m} m + c d^{m}\right )} \int \frac {x^{2} x^{m}}{{\left (c^{2} m + c^{2}\right )} x^{4} + m + 1}\,{d x}\right )} b}{m + 1} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________