Optimal. Leaf size=293 \[ -\frac{b (f x)^{m+2} \left (\frac{c^4 d^2 (m+3) (m+5)}{m+1}+\frac{e (m+2) \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{(m+3) (m+5)}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{c^3 f^2 (m+2) (m+3) (m+5)}+\frac{d^2 (f x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (m+5)}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^3 f^2 (m+3)^2 (m+5)^2}+\frac{b e^2 \sqrt{1-c^2 x^2} (f x)^{m+4}}{c f^4 (m+5)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.415264, antiderivative size = 272, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {270, 4731, 12, 1267, 459, 364} \[ \frac{d^2 (f x)^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (m+5)}-\frac{b c (f x)^{m+2} \left (\frac{e \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^4 (m+3)^2 (m+5)^2}+\frac{d^2}{m^2+3 m+2}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{f^2}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^3 f^2 (m+3)^2 (m+5)^2}+\frac{b e^2 \sqrt{1-c^2 x^2} (f x)^{m+4}}{c f^4 (m+5)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 4731
Rule 12
Rule 1267
Rule 459
Rule 364
Rubi steps
\begin{align*} \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-(b c) \int \frac{(f x)^{1+m} \left (\frac{d^2}{1+m}+\frac{2 d e x^2}{3+m}+\frac{e^2 x^4}{5+m}\right )}{f \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{(b c) \int \frac{(f x)^{1+m} \left (\frac{d^2}{1+m}+\frac{2 d e x^2}{3+m}+\frac{e^2 x^4}{5+m}\right )}{\sqrt{1-c^2 x^2}} \, dx}{f}\\ &=\frac{b e^2 (f x)^{4+m} \sqrt{1-c^2 x^2}}{c f^4 (5+m)^2}+\frac{d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}+\frac{b \int \frac{(f x)^{1+m} \left (-\frac{c^2 d^2 (5+m)}{1+m}-\frac{e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) x^2}{(3+m) (5+m)}\right )}{\sqrt{1-c^2 x^2}} \, dx}{c f (5+m)}\\ &=\frac{b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \sqrt{1-c^2 x^2}}{c^3 f^2 (3+m)^2 (5+m)^2}+\frac{b e^2 (f x)^{4+m} \sqrt{1-c^2 x^2}}{c f^4 (5+m)^2}+\frac{d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{\left (b \left (\frac{c^4 d^2}{1+m}+\frac{e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m)^2 (5+m)^2}\right )\right ) \int \frac{(f x)^{1+m}}{\sqrt{1-c^2 x^2}} \, dx}{c^3 f}\\ &=\frac{b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \sqrt{1-c^2 x^2}}{c^3 f^2 (3+m)^2 (5+m)^2}+\frac{b e^2 (f x)^{4+m} \sqrt{1-c^2 x^2}}{c f^4 (5+m)^2}+\frac{d^2 (f x)^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{f (1+m)}+\frac{2 d e (f x)^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{f^3 (3+m)}+\frac{e^2 (f x)^{5+m} \left (a+b \sin ^{-1}(c x)\right )}{f^5 (5+m)}-\frac{b \left (\frac{c^4 d^2}{1+m}+\frac{e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m)^2 (5+m)^2}\right ) (f x)^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{c^3 f^2 (2+m)}\\ \end{align*}
Mathematica [A] time = 0.279903, size = 224, normalized size = 0.76 \[ x (f x)^m \left (-\frac{b c d^2 x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )}{m^2+3 m+2}-\frac{2 b c d e x^3 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2}+2,\frac{m}{2}+3,c^2 x^2\right )}{m^2+7 m+12}-\frac{b c e^2 x^5 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2}+3,\frac{m}{2}+4,c^2 x^2\right )}{(m+5) (m+6)}+\frac{a d^2}{m+1}+\frac{2 a d e x^2}{m+3}+\frac{a e^2 x^4}{m+5}+\frac{b d^2 \sin ^{-1}(c x)}{m+1}+\frac{2 b d e x^2 \sin ^{-1}(c x)}{m+3}+\frac{b e^2 x^4 \sin ^{-1}(c x)}{m+5}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 8.931, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) ^{2} \left ( a+b\arcsin \left ( cx \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arcsin \left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]