Optimal. Leaf size=621 \[ -\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {15}{64} b^2 d f x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {2 b^2 d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {16 b^2 d g \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {8 b^2 d g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \]
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Rubi [A] time = 0.72, antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1247, 698} \[ -\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}-\frac {15}{64} b^2 d f x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {2 b^2 d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {16 b^2 d g \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {8 b^2 d g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 195
Rule 216
Rule 321
Rule 698
Rule 1247
Rule 4627
Rule 4641
Rule 4645
Rule 4647
Rule 4649
Rule 4677
Rule 4763
Rule 4777
Rubi steps
\begin {align*} \int (f+g x) \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2+g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2 \, 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 \sin ^{-1}(c x)\right )^2-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{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 \sin ^{-1}(c x)\right )^2 \, 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 ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (3 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d f \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d f \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}\\ &=\frac {16 b^2 d g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f x \sqrt {d-c^2 d x^2}+\frac {8 b^2 d g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {1}{32} b^2 d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {2 b^2 d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {9 b^2 d f \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {2 b d g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c d g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b d f \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 395, normalized size = 0.64 \[ \frac {d \sqrt {d-c^2 d x^2} \left (9000 a^3 c f+15 b \sin ^{-1}(c x) \left (1800 a^2 c f-240 a b \sqrt {1-c^2 x^2} \left (5 c^2 f x \left (2 c^2 x^2-5\right )+8 g \left (c^2 x^2-1\right )^2\right )+b^2 c \left (75 f \left (8 c^4 x^4-40 c^2 x^2+17\right )+128 g x \left (3 c^4 x^4-10 c^2 x^2+15\right )\right )\right )-1800 a^2 b \sqrt {1-c^2 x^2} \left (5 c^2 f x \left (2 c^2 x^2-5\right )+8 g \left (c^2 x^2-1\right )^2\right )+1800 b^2 \sin ^{-1}(c x)^2 \left (15 a c f+b \sqrt {1-c^2 x^2} \left (5 c^2 f x \left (5-2 c^2 x^2\right )-8 g \left (c^2 x^2-1\right )^2\right )\right )+120 a b^2 c x \left (75 c^2 f x \left (c^2 x^2-5\right )+16 g \left (3 c^4 x^4-10 c^2 x^2+15\right )\right )+b^3 \sqrt {1-c^2 x^2} \left (1125 c^2 f x \left (2 c^2 x^2-17\right )+128 g \left (9 c^4 x^4-38 c^2 x^2+149\right )\right )+9000 b^3 c f \sin ^{-1}(c x)^3\right )}{72000 b c^2 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d g x^{3} + a^{2} c^{2} d f x^{2} - a^{2} d g x - a^{2} d f + {\left (b^{2} c^{2} d g x^{3} + b^{2} c^{2} d f x^{2} - b^{2} d g x - b^{2} d f\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d g x^{3} + a b c^{2} d f x^{2} - a b d g x - a b d f\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.94, size = 5957, normalized size = 9.59 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a^{2} f - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2} g}{5 \, c^{2} d} + \sqrt {d} \int -{\left ({\left (b^{2} c^{2} d g x^{3} + b^{2} c^{2} d f x^{2} - b^{2} d g x - b^{2} d f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d g x^{3} + a b c^{2} d f x^{2} - a b d g x - a b d f\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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