Optimal. Leaf size=414 \[ \frac {3 d x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}+\frac {3 d (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac {1}{4} d x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5 b c d x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^2 d x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^4 d x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^3 d x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4673, 4763, 4649, 4647, 4641, 30, 14, 4677, 194} \[ \frac {3 d x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}+\frac {3 d (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac {1}{4} d x (c d x+d)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {b c^4 d x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^3 d x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^2 d x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {5 b c d x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 194
Rule 4641
Rule 4647
Rule 4649
Rule 4673
Rule 4677
Rule 4763
Rubi steps
\begin {align*} \int (d+c d x)^{5/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left ((d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int (d+c d x) \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {\left ((d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+c d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {\left (d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}+\frac {\left (c d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} d x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {d (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac {\left (3 d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (b d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^2 \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} d x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3 d x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac {\left (3 d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (b d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b c d (d+c d x)^{3/2} (f-c f x)^{3/2}\right ) \int x \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {b d x (d+c d x)^{3/2} (f-c f x)^{3/2}}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {5 b c d x^2 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c^2 d x^3 (d+c d x)^{3/2} (f-c f x)^{3/2}}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^3 d x^4 (d+c d x)^{3/2} (f-c f x)^{3/2}}{16 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c^4 d x^5 (d+c d x)^{3/2} (f-c f x)^{3/2}}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} d x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3 d x (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (f-c f x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{5 c}+\frac {3 d (d+c d x)^{3/2} (f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.74, size = 305, normalized size = 0.74 \[ \frac {d^2 f \left (\sqrt {c d x+d} \sqrt {f-c f x} \left (-240 a \sqrt {1-c^2 x^2} \left (8 c^4 x^4+10 c^3 x^3-16 c^2 x^2-25 c x+8\right )+128 b c x \left (3 c^4 x^4-10 c^2 x^2+15\right )+1200 b \cos \left (2 \sin ^{-1}(c x)\right )+75 b \cos \left (4 \sin ^{-1}(c x)\right )\right )-3600 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (c^2 x^2-1\right )}\right )-60 b \sqrt {c d x+d} \sqrt {f-c f x} \left (32 \left (1-c^2 x^2\right )^{5/2}-40 \sin \left (2 \sin ^{-1}(c x)\right )-5 \sin \left (4 \sin ^{-1}(c x)\right )\right ) \sin ^{-1}(c x)+1800 b \sqrt {c d x+d} \sqrt {f-c f x} \sin ^{-1}(c x)^2\right )}{9600 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{3} d^{2} f x^{3} + a c^{2} d^{2} f x^{2} - a c d^{2} f x - a d^{2} f + {\left (b c^{3} d^{2} f x^{3} + b c^{2} d^{2} f x^{2} - b c d^{2} f x - b d^{2} f\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c f x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d x + d\right )}^{\frac {5}{2}} {\left (-c f x + f\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )\, dx \]
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 \[ b \sqrt {d} \sqrt {f} \int -{\left (c^{3} d^{2} f x^{3} + c^{2} d^{2} f x^{2} - c d^{2} f x - d^{2} f\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} + \frac {1}{40} \, {\left (15 \, \sqrt {-c^{2} d f x^{2} + d f} d^{2} f x + \frac {15 \, d^{3} f^{2} \arcsin \left (c x\right )}{\sqrt {d f} c} + 10 \, {\left (-c^{2} d f x^{2} + d f\right )}^{\frac {3}{2}} d x - \frac {8 \, {\left (-c^{2} d f x^{2} + d f\right )}^{\frac {5}{2}}}{c f}\right )} a \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{5/2}\,{\left (f-c\,f\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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