Optimal. Leaf size=337 \[ \frac{b x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac{3 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}+\frac{b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.484204, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {4707, 4643, 4641, 4627, 321, 216} \[ \frac{b x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac{3 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 c^4 d}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}+\frac{b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4707
Rule 4643
Rule 4641
Rule 4627
Rule 321
Rule 216
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{\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 )^2}{4 c^2 d}+\frac{3 \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx}{4 c^2}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 c \sqrt{d-c^2 d x^2}}\\ &=\frac{b x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 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 )^2}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac{3 \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx}{8 c^4}-\frac{\left (b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b \sqrt{1-c^2 x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 c^3 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 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 )^2}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{8 c^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt{d-c^2 d x^2}}+\frac{b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 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 )^2}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{64 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{16 c^4 \sqrt{d-c^2 d x^2}}\\ &=\frac{15 b^2 x \left (1-c^2 x^2\right )}{64 c^4 \sqrt{d-c^2 d x^2}}+\frac{b^2 x^3 \left (1-c^2 x^2\right )}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}}+\frac{3 b x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}+\frac{b x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 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 )^2}{8 c^4 d}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.3866, size = 283, normalized size = 0.84 \[ \frac{32 a^2 c \sqrt{d} x \left (c^2 x^2-1\right ) \left (2 c^2 x^2+3\right )-96 a^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-4 a b \sqrt{d} \sqrt{1-c^2 x^2} \left (-4 \sin ^{-1}(c x) \left (6 \sin ^{-1}(c x)-8 \sin \left (2 \sin ^{-1}(c x)\right )+\sin \left (4 \sin ^{-1}(c x)\right )\right )+16 \cos \left (2 \sin ^{-1}(c x)\right )-\cos \left (4 \sin ^{-1}(c x)\right )\right )+b^2 \sqrt{d} \sqrt{1-c^2 x^2} \left (32 \sin ^{-1}(c x)^3+8 \left (\sin \left (4 \sin ^{-1}(c x)\right )-8 \sin \left (2 \sin ^{-1}(c x)\right )\right ) \sin ^{-1}(c x)^2+32 \sin \left (2 \sin ^{-1}(c x)\right )-\sin \left (4 \sin ^{-1}(c x)\right )+4 \sin ^{-1}(c x) \left (\cos \left (4 \sin ^{-1}(c x)\right )-16 \cos \left (2 \sin ^{-1}(c x)\right )\right )\right )}{256 c^5 \sqrt{d} \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.476, size = 871, normalized size = 2.6 \begin{align*} \text{result too large to display} \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^{2} x^{4} \arcsin \left (c x\right )^{2} + 2 \, a b x^{4} \arcsin \left (c x\right ) + a^{2} 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 )^{2}}{\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 )}^{2} 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]