Optimal. Leaf size=376 \[ \frac {1}{4} c^2 f^2 x^3 \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 f^2 \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {2 f^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {3}{8} f^2 x \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )-\frac {3 b c f^2 x^2 \sqrt {c d x+d} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b f^2 x \sqrt {c d x+d} \sqrt {f-c f x}}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f^2 x^3 \sqrt {c d x+d} \sqrt {f-c f x}}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 f^2 x^4 \sqrt {c d x+d} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.54, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4673, 4763, 4647, 4641, 30, 4677, 4697, 4707} \[ \frac {1}{4} c^2 f^2 x^3 \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 f^2 \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}+\frac {2 f^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {3}{8} f^2 x \sqrt {c d x+d} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c^3 f^2 x^4 \sqrt {c d x+d} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f^2 x^3 \sqrt {c d x+d} \sqrt {f-c f x}}{9 \sqrt {1-c^2 x^2}}-\frac {3 b c f^2 x^2 \sqrt {c d x+d} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}-\frac {2 b f^2 x \sqrt {c d x+d} \sqrt {f-c f x}}{3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4641
Rule 4647
Rule 4673
Rule 4677
Rule 4697
Rule 4707
Rule 4763
Rubi steps
\begin {align*} \int \sqrt {d+c d x} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {f-c f x}\right ) \int (f-c f x)^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (\sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \left (f^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-2 c f^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+c^2 f^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 c f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} f^2 x \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {\left (f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (b c f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int x^3 \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b f^2 x \sqrt {d+c d x} \sqrt {f-c f x}}{3 \sqrt {1-c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d+c d x} \sqrt {f-c f x}}{4 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x}}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 f^2 x^4 \sqrt {d+c d x} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt {1-c^2 x^2}}+\frac {\left (f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c f^2 \sqrt {d+c d x} \sqrt {f-c f x}\right ) \int x \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b f^2 x \sqrt {d+c d x} \sqrt {f-c f x}}{3 \sqrt {1-c^2 x^2}}-\frac {3 b c f^2 x^2 \sqrt {d+c d x} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x}}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 f^2 x^4 \sqrt {d+c d x} \sqrt {f-c f x}}{16 \sqrt {1-c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} c^2 f^2 x^3 \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )+\frac {2 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {5 f^2 \sqrt {d+c d x} \sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )^2}{16 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.35, size = 293, normalized size = 0.78 \[ \frac {f^2 \sqrt {c d x+d} \sqrt {f-c f x} \left (48 a \sqrt {1-c^2 x^2} \left (6 c^3 x^3-16 c^2 x^2+9 c x+16\right )+256 b c x \left (c^2 x^2-3\right )+144 b \cos \left (2 \sin ^{-1}(c x)\right )-9 b \cos \left (4 \sin ^{-1}(c x)\right )\right )-720 a \sqrt {d} f^{5/2} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (c^2 x^2-1\right )}\right )-12 b f^2 \sqrt {c d x+d} \sqrt {f-c f x} \sin ^{-1}(c x) \left (-64 \left (1-c^2 x^2\right )^{3/2}-24 \sin \left (2 \sin ^{-1}(c x)\right )+3 \sin \left (4 \sin ^{-1}(c x)\right )\right )+360 b f^2 \sqrt {c d x+d} \sqrt {f-c f x} \sin ^{-1}(c x)^2}{1152 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{2} f^{2} x^{2} - 2 \, a c f^{2} x + a f^{2} + {\left (b c^{2} f^{2} x^{2} - 2 \, b c f^{2} x + b f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c f x + f}, x\right ) \]
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 \[ \int \sqrt {c d x + d} {\left (-c f x + f\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}\,{d x} \]
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 \[ \int \sqrt {c d x +d}\, \left (-c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )\, dx \]
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 \[ b \sqrt {d} \sqrt {f} \int {\left (c^{2} f^{2} x^{2} - 2 \, c f^{2} x + f^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} + \frac {1}{24} \, {\left (15 \, \sqrt {-c^{2} d f x^{2} + d f} f^{2} x + \frac {15 \, d f^{3} \arcsin \left (c x\right )}{\sqrt {d f} c} - \frac {6 \, {\left (-c^{2} d f x^{2} + d f\right )}^{\frac {3}{2}} f x}{d} + \frac {16 \, {\left (-c^{2} d f x^{2} + d f\right )}^{\frac {3}{2}} f}{c d}\right )} a \]
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 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d+c\,d\,x}\,{\left (f-c\,f\,x\right )}^{5/2} \,d x \]
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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