Integrand size = 32, antiderivative size = 126 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=-\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e} \]
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Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3192, 3191} \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=-\frac {2 \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} (c \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
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Rule 3191
Rule 3192
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e}+\frac {1}{3} \left (4 \sqrt {b^2+c^2}\right ) \int \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx \\ & = -\frac {8 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 17.98 (sec) , antiderivative size = 5279, normalized size of antiderivative = 41.90 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\text {Result too large to show} \]
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Time = 1.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {2 \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+1\right ) \left (b^{2}+c^{2}\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+5\right )}{3 \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\) | \(126\) |
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none
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.99 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\frac {2 \, {\left (2 \, b c \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} - 5 \, b^{2} - 4 \, c^{2} + 4 \, \sqrt {b^{2} + c^{2}} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right )\right )}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}}}{3 \, {\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \]
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\[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\int \left (b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} + \sqrt {b^{2} + c^{2}}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx=\int {\left (b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}\right )}^{3/2} \,d x \]
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