Optimal. Leaf size=196 \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}+\frac{16 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{15 e}-\frac{64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x))}{15 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
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Rubi [A] time = 0.133782, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3113, 3112} \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2}}{5 e}+\frac{16 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{15 e}-\frac{64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x))}{15 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
Antiderivative was successfully verified.
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Rule 3113
Rule 3112
Rubi steps
\begin{align*} \int \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{5/2} \, 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 )^{3/2}}{5 e}-\frac{1}{5} \left (8 \sqrt{b^2+c^2}\right ) \int \left (-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right )^{3/2} \, dx\\ &=\frac{16 \sqrt{b^2+c^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)}}{15 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 )^{3/2}}{5 e}+\frac{1}{15} \left (32 \left (b^2+c^2\right )\right ) \int \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx\\ &=-\frac{64 \left (b^2+c^2\right ) (c \cos (d+e x)-b \sin (d+e x))}{15 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}+\frac{16 \sqrt{b^2+c^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)}}{15 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 )^{3/2}}{5 e}\\ \end{align*}
Mathematica [C] time = 34.2309, size = 11602, normalized size = 59.19 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.753, size = 204, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -2 \right ) \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) \left ( 3\, \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{2}{b}^{2}+3\,{c}^{2} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) ^{2}-14\,{b}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -14\,{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +43\,{b}^{2}+43\,{c}^{2} \right ) }{15\,\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}\sqrt{{b}^{2}+{c}^{2}}{\frac{1}{\sqrt{{({b}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -{b}^{2}-{c}^{2}){\frac{1}{\sqrt{{b}^{2}+{c}^{2}}}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89513, size = 463, normalized size = 2.36 \begin{align*} \frac{2 \,{\left (3 \,{\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{3} +{\left (29 \, b^{3} + 38 \, b c^{2}\right )} \cos \left (e x + d\right ) +{\left (29 \, b^{2} c + 32 \, c^{3} + 3 \,{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right ) -{\left (22 \, b c \cos \left (e x + d\right ) \sin \left (e x + d\right ) + 11 \,{\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} - 43 \, b^{2} - 32 \, c^{2}\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}}}}{15 \,{\left (c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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