Integrand size = 21, antiderivative size = 202 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \arcsin (c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}-\frac {b c \left (2 e^2 g-c^2 d (e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {37, 4837, 12, 1665, 858, 222, 739, 210} \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (d+e x)^2 (e f-d g)}+\frac {b g^2 \arcsin (c x)}{2 e^2 (e f-d g)}-\frac {b c \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 e^2 g-c^2 d (d g+e f)\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} (e f-d g)}{2 e \left (c^2 d^2-e^2\right ) (d+e x)} \]
[In]
[Out]
Rule 12
Rule 37
Rule 210
Rule 222
Rule 739
Rule 858
Rule 1665
Rule 4837
Rubi steps \begin{align*} \text {integral}& = -\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}-(b c) \int -\frac {(f+g x)^2}{2 (e f-d g) (d+e x)^2 \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}+\frac {(b c) \int \frac {(f+g x)^2}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 (e f-d g)} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}+\frac {(b c) \int \frac {c^2 d f^2-g (2 e f-d g)+\left (\frac {c^2 d^2}{e}-e\right ) g^2 x}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 \left (c^2 d^2-e^2\right ) (e f-d g)} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}+\frac {\left (b c g^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 e^2 (e f-d g)}-\frac {\left (b c \left (2 e^2 g-c^2 d (e f+d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 e^2 \left (c^2 d^2-e^2\right )} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \arcsin (c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}+\frac {\left (b c \left (2 e^2 g-c^2 d (e f+d g)\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )} \\ & = \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \arcsin (c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}-\frac {b c \left (2 e^2 g-c^2 d (e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.30 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {\frac {a (-e f+d g)}{(d+e x)^2}-\frac {2 a g}{d+e x}-\frac {b c e (e f-d g) \sqrt {1-c^2 x^2}}{\left (-c^2 d^2+e^2\right ) (d+e x)}-\frac {b (d g+e (f+2 g x)) \arcsin (c x)}{(d+e x)^2}+\frac {b c \left (-2 e^2 g+c^2 d (e f+d g)\right ) \log (d+e x)}{(c d-e) (c d+e) \sqrt {-c^2 d^2+e^2}}+\frac {b c \left (-2 e^2 g+c^2 d (e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e) (c d+e) \sqrt {-c^2 d^2+e^2}}}{2 e^2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(534\) vs. \(2(186)=372\).
Time = 2.56 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.65
method | result | size |
parts | \(a \left (-\frac {g}{e^{2} \left (e x +d \right )}-\frac {-d g +e f}{2 e^{2} \left (e x +d \right )^{2}}\right )+\frac {b \left (\frac {c^{3} \arcsin \left (c x \right ) d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {c^{3} \arcsin \left (c x \right ) f}{2 e \left (c e x +d c \right )^{2}}-\frac {c^{2} \arcsin \left (c x \right ) g}{e^{2} \left (c e x +d c \right )}+\frac {c^{2} \left (-\frac {2 g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}\right )}{2 e^{2}}\right )}{c}\) | \(535\) |
derivativedivides | \(\frac {a \,c^{2} \left (\frac {c \left (d g -e f \right )}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {g}{e^{2} \left (c e x +d c \right )}\right )+b \,c^{2} \left (\frac {\arcsin \left (c x \right ) c d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {\arcsin \left (c x \right ) c f}{2 e \left (c e x +d c \right )^{2}}-\frac {\arcsin \left (c x \right ) g}{e^{2} \left (c e x +d c \right )}+\frac {-\frac {2 g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}}{2 e^{2}}\right )}{c}\) | \(539\) |
default | \(\frac {a \,c^{2} \left (\frac {c \left (d g -e f \right )}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {g}{e^{2} \left (c e x +d c \right )}\right )+b \,c^{2} \left (\frac {\arcsin \left (c x \right ) c d g}{2 e^{2} \left (c e x +d c \right )^{2}}-\frac {\arcsin \left (c x \right ) c f}{2 e \left (c e x +d c \right )^{2}}-\frac {\arcsin \left (c x \right ) g}{e^{2} \left (c e x +d c \right )}+\frac {-\frac {2 g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}}{2 e^{2}}\right )}{c}\) | \(539\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (185) = 370\).
Time = 3.24 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.86 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (e x + d\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^3} \,d x \]
[In]
[Out]