Optimal. Leaf size=370 \[ \frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{b c^3 d f x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{5 b c d f x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b c^3 d g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{2 b c d g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}-\frac{b d g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.334807, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {4778, 4764, 4650, 4648, 4642, 30, 14, 4678, 194} \[ \frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{3 d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{b c^3 d f x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{5 b c d f x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b c^3 d g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{2 b c d g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}-\frac{b d g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4778
Rule 4764
Rule 4650
Rule 4648
Rule 4642
Rule 30
Rule 14
Rule 4678
Rule 194
Rubi steps
\begin{align*} \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int (f+g x) \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (f \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )+g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d f \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (d g \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}+\frac{\left (3 d f \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (b c d f \sqrt{d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (b d g \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}+\frac{\left (3 d f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}+\frac{\left (b c d f \sqrt{d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \sqrt{1-c^2 x^2}}+\frac{\left (3 b c d f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 \sqrt{1-c^2 x^2}}-\frac{\left (b d g \sqrt{d-c^2 d x^2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt{1-c^2 x^2}}\\ &=-\frac{b d g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}+\frac{5 b c d f x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{2 b c d g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b c^3 d g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{4} d f x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac{d g \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{5 c^2}-\frac{3 d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{16 b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 1.56584, size = 337, normalized size = 0.91 \[ \frac{-d \sqrt{d-c^2 d x^2} \left (3 \left (80 a \sqrt{1-c^2 x^2} \left (5 c^2 f x \left (2 c^2 x^2-5\right )+8 g \left (c^2 x^2-1\right )^2\right )+25 b c f \cos \left (4 \cos ^{-1}(c x)\right )+400 b c g x+8 b g \cos \left (5 \cos ^{-1}(c x)\right )\right )-1200 b c f \cos \left (2 \cos ^{-1}(c x)\right )-200 b g \cos \left (3 \cos ^{-1}(c x)\right )\right )-3600 a c d^{3/2} f \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+20 b d \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left (160 c^2 g x^2 \sqrt{1-c^2 x^2}-100 g \sqrt{1-c^2 x^2}+120 c f \sin \left (2 \cos ^{-1}(c x)\right )-15 c f \sin \left (4 \cos ^{-1}(c x)\right )-10 g \sin \left (3 \cos ^{-1}(c x)\right )-6 g \sin \left (5 \cos ^{-1}(c x)\right )\right )-1800 b c d f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{9600 c^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.49, size = 698, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a c^{2} d g x^{3} + a c^{2} d f x^{2} - a d g x - a d f +{\left (b c^{2} d g x^{3} + b c^{2} d f x^{2} - b d g x - b d f\right )} \arccos \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}{\left (b \arccos \left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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