Optimal. Leaf size=450 \[ \frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}-\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}+\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.529749, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {4778, 4764, 4648, 4642, 30, 4678, 4698, 4708} \[ \frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}-\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}+\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4778
Rule 4764
Rule 4648
Rule 4642
Rule 30
Rule 4678
Rule 4698
Rule 4708
Rubi steps
\begin{align*} \int (f+g x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\sqrt{d-c^2 d x^2} \int (f+g x)^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{d-c^2 d x^2} \int \left (f^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )+2 f g x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )+g^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (f^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (2 f g \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (g^2 \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}+\frac{\left (f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b c f^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (2 b f g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 c \sqrt{1-c^2 x^2}}+\frac{\left (g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (b c g^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}+\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}-\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{\left (g^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (b g^2 \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}+\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}+\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{g^2 x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{2 f g \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}-\frac{f^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{g^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.3145, size = 320, normalized size = 0.71 \[ \frac{48 a c \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (12 c^2 f^2 x+16 f g \left (c^2 x^2-1\right )+3 g^2 x \left (2 c^2 x^2-1\right )\right )-144 a \sqrt{d} \sqrt{1-c^2 x^2} \left (4 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+144 b c^2 f^2 \sqrt{d-c^2 d x^2} \left (\cos \left (2 \cos ^{-1}(c x)\right )+2 \cos ^{-1}(c x) \left (\sin \left (2 \cos ^{-1}(c x)\right )-\cos ^{-1}(c x)\right )\right )-64 b c f g \sqrt{d-c^2 d x^2} \left (12 \left (1-c^2 x^2\right )^{3/2} \cos ^{-1}(c x)+9 c x-\cos \left (3 \cos ^{-1}(c x)\right )\right )+9 b g^2 \sqrt{d-c^2 d x^2} \left (-8 \cos ^{-1}(c x)^2+\cos \left (4 \cos ^{-1}(c x)\right )+4 \cos ^{-1}(c x) \sin \left (4 \cos ^{-1}(c x)\right )\right )}{1152 c^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.533, size = 912, normalized size = 2. \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 (\sqrt{-c^{2} d x^{2} + d}{\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \arccos \left (c x\right )\right )}, 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 \sqrt{-c^{2} d x^{2} + d}{\left (g x + f\right )}^{2}{\left (b \arccos \left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]