Integrand size = 29, antiderivative size = 144 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\left (g+c^2 f x\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g) \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^2 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g) \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^2 d \sqrt {d-c^2 d x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4861, 651, 4845, 12, 647, 31} \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\left (c^2 f x+g\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f+g) \log (1-c x)}{2 c^2 d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} (c f-g) \log (c x+1)}{2 c^2 d \sqrt {d-c^2 d x^2}} \]
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Rule 12
Rule 31
Rule 647
Rule 651
Rule 4845
Rule 4861
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (g+c^2 f x\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {g+c^2 f x}{c^2 \left (1-c^2 x^2\right )} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (g+c^2 f x\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {g+c^2 f x}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (g+c^2 f x\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f-g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{-c-c^2 x} \, dx}{2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b (c f+g) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c-c^2 x} \, dx}{2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {\left (g+c^2 f x\right ) (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g) \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^2 d \sqrt {d-c^2 d x^2}}+\frac {b (c f-g) \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^2 d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-c^2 x^2} \left ((c f-g) \left (-\left ((a+b \arcsin (c x)) \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 b \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+(c f+g) \left (2 b \log \left (\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+(a+b \arcsin (c x)) \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{2 c^2 d \sqrt {d-c^2 d x^2}} \]
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Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.12
method | result | size |
default | \(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \arcsin \left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}\, c f +c^{2} f x +g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f -g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f +g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(305\) |
parts | \(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (\frac {2 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, f \arcsin \left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}\, c f +c^{2} f x +g \right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f -g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c f +g \right ) \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(305\) |
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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