Integrand size = 32, antiderivative size = 258 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=-\frac {256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{35 e}-\frac {24 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e} \]
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Time = 0.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3192, 3191} \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}-\frac {24 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac {64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x)) \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{35 e}-\frac {256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
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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)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}+\frac {1}{7} \left (12 \sqrt {b^2+c^2}\right ) \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2} \, dx \\ & = -\frac {24 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}+\frac {1}{35} \left (96 \left (b^2+c^2\right )\right ) \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx \\ & = -\frac {64 \left (b^2+c^2\right ) (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)}}{35 e}-\frac {24 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e}+\frac {1}{35} \left (128 \left (b^2+c^2\right )^{3/2}\right ) \int \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx \\ & = -\frac {256 \left (b^2+c^2\right )^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac {64 \left (b^2+c^2\right ) (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)}}{35 e}-\frac {24 \sqrt {b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{35 e}-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2}}{7 e} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 37.94 (sec) , antiderivative size = 5490, normalized size of antiderivative = 21.28 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Result too large to show} \]
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Time = 1.64 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {2 \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+1\right ) \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \left (5 b^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{3}+10 b^{2} c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{3}+5 c^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{3}+27 b^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{2}+54 b^{2} c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{2}+27 c^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )^{2}+71 b^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+142 b^{2} c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+71 c^{4} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+177 b^{4}+354 b^{2} c^{2}+177 c^{4}\right )}{35 \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\) | \(306\) |
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Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.04 \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\frac {2 \, {\left (5 \, {\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{4} - 177 \, b^{4} - 310 \, b^{2} c^{2} - 128 \, c^{4} + 2 \, {\left (22 \, b^{4} + 15 \, b^{2} c^{2} - 27 \, c^{4}\right )} \cos \left (e x + d\right )^{2} + 4 \, {\left (5 \, {\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{3} + {\left (22 \, b^{3} c + 27 \, b c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 2 \, {\left (11 \, {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{3} + {\left (53 \, b^{3} + 86 \, b c^{2}\right )} \cos \left (e x + d\right ) + {\left (53 \, b^{2} c + 64 \, c^{3} + 11 \, {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )\right )} \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}}}{35 \, {\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \]
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Timed out. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{7/2} \, dx=\int {\left (b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}\right )}^{7/2} \,d x \]
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