Optimal. Leaf size=255 \[ \frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {d (c x+1) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b d \left (1-c^2 x^2\right )^{5/2}}{6 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {b d \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b d \left (1-c^2 x^2\right )^{5/2} \tanh ^{-1}(c x)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
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Rubi [A] time = 0.25, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4673, 639, 191, 4761, 627, 44, 207, 260} \[ \frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {d (c x+1) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b d \left (1-c^2 x^2\right )^{5/2}}{6 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {b d \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b d \left (1-c^2 x^2\right )^{5/2} \tanh ^{-1}(c x)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 191
Rule 207
Rule 260
Rule 627
Rule 639
Rule 4673
Rule 4761
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} (f-c f x)^{5/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(d+c d x) \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac {d (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac {d (1+c x)}{3 c \left (1-c^2 x^2\right )^2}+\frac {2 d x}{3 \left (1-c^2 x^2\right )}\right ) \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac {d (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b d \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1+c x}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (2 b c d \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac {d (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b d \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b d \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{(1-c x)^2 (1+c x)} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=\frac {d (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b d \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b d \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac {1}{2 (-1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {b d \left (1-c^2 x^2\right )^{5/2}}{6 c (1-c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {d (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b d \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b d \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{6 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {b d \left (1-c^2 x^2\right )^{5/2}}{6 c (1-c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {d (1+c x) \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {2 d x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {b d \left (1-c^2 x^2\right )^{5/2} \tanh ^{-1}(c x)}{6 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {b d \left (1-c^2 x^2\right )^{5/2} \log \left (1-c^2 x^2\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 184, normalized size = 0.72 \[ \frac {\sqrt {c d x+d} \left (8 a c^2 x^2-8 a c x-4 a+5 b c x \sqrt {1-c^2 x^2} \log (f-c f x)+3 b (c x-1) \sqrt {1-c^2 x^2} \log (-f (c x+1))-5 b \sqrt {1-c^2 x^2} \log (f-c f x)+2 b \sqrt {1-c^2 x^2}+4 b \left (2 c^2 x^2-2 c x-1\right ) \sin ^{-1}(c x)\right )}{12 c d^2 f^2 \left (c^2 x^2-1\right ) \sqrt {f-c f x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c d x + d} \sqrt {-c f x + f} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{5} d^{2} f^{3} x^{5} - c^{4} d^{2} f^{3} x^{4} - 2 \, c^{3} d^{2} f^{3} x^{3} + 2 \, c^{2} d^{2} f^{3} x^{2} + c d^{2} f^{3} x - d^{2} f^{3}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c f x + f\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arcsin \left (c x \right )}{\left (c d x +d \right )^{\frac {3}{2}} \left (-c f x +f \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 237, normalized size = 0.93 \[ \frac {1}{12} \, b c {\left (\frac {2 \, \sqrt {d} \sqrt {f}}{c^{3} d^{2} f^{3} x - c^{2} d^{2} f^{3}} + \frac {3 \, \log \left (c x + 1\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}} + \frac {5 \, \log \left (c x - 1\right )}{c^{2} d^{\frac {3}{2}} f^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {1}{\sqrt {-c^{2} d f x^{2} + d f} c^{2} d f^{2} x - \sqrt {-c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {-c^{2} d f x^{2} + d f} d f^{2}}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {1}{\sqrt {-c^{2} d f x^{2} + d f} c^{2} d f^{2} x - \sqrt {-c^{2} d f x^{2} + d f} c d f^{2}} - \frac {2 \, x}{\sqrt {-c^{2} d f x^{2} + d f} d f^{2}}\right )} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (f-c\,f\,x\right )}^{5/2}} \,d x \]
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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