Optimal. Leaf size=200 \[ -\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 c^2 d}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^4 d}+\frac{3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^5 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2}}{16 c \sqrt{d-c^2 d x^2}}+\frac{3 b x^2 \sqrt{1-c^2 x^2}}{16 c^3 \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.249813, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {4707, 4643, 4641, 30} \[ -\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 c^2 d}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^4 d}+\frac{3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^5 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2}}{16 c \sqrt{d-c^2 d x^2}}+\frac{3 b x^2 \sqrt{1-c^2 x^2}}{16 c^3 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4707
Rule 4643
Rule 4641
Rule 30
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 c^2 d}+\frac{3 \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx}{4 c^2}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int x^3 \, dx}{4 c \sqrt{d-c^2 d x^2}}\\ &=\frac{b x^4 \sqrt{1-c^2 x^2}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 c^2 d}+\frac{3 \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d-c^2 d x^2}} \, dx}{8 c^4}+\frac{\left (3 b \sqrt{1-c^2 x^2}\right ) \int x \, dx}{8 c^3 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b x^2 \sqrt{1-c^2 x^2}}{16 c^3 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 c^2 d}+\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{3 b x^2 \sqrt{1-c^2 x^2}}{16 c^3 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 c^2 d}+\frac{3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c^5 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.804621, size = 161, normalized size = 0.8 \[ \frac{-\frac{16 a c x \left (2 c^2 x^2+3\right ) \sqrt{d-c^2 d x^2}}{d}-\frac{48 a \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )}{\sqrt{d}}+\frac{b \sqrt{1-c^2 x^2} \left (4 \sin ^{-1}(c x) \left (6 \sin ^{-1}(c x)-8 \sin \left (2 \sin ^{-1}(c x)\right )+\sin \left (4 \sin ^{-1}(c x)\right )\right )-16 \cos \left (2 \sin ^{-1}(c x)\right )+\cos \left (4 \sin ^{-1}(c x)\right )\right )}{\sqrt{d-c^2 d x^2}}}{128 c^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.332, size = 400, normalized size = 2. \begin{align*} -{\frac{a{x}^{3}}{4\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{3\,ax}{8\,{c}^{4}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{3\,a}{8\,{c}^{4}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{b\arcsin \left ( cx \right ){x}^{5}}{4\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b\arcsin \left ( cx \right ){x}^{3}}{8\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{3\,b\arcsin \left ( cx \right ) x}{8\,{c}^{4}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{15\,b}{128\,d{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,b \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{16\,d{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{x}^{4}}{16\,dc \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,b{x}^{2}}{16\,{c}^{3}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}} \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 (-\frac{{\left (b x^{4} \arcsin \left (c x\right ) + a x^{4}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \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 \frac{{\left (b \arcsin \left (c x\right ) + a\right )} x^{4}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]