Optimal. Leaf size=130 \[ \frac{8 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e} \]
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Rubi [A] time = 0.0807832, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3113, 3112} \[ \frac{8 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e} \]
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 )^{3/2} \, dx &=-\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*}
Mathematica [C] time = 21.2547, size = 11512, normalized size = 88.55 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.591, size = 130, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -2 \right ) \left ({b}^{2}+{c}^{2} \right ) \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -5 \right ) }{3\,\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}{\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 = 1.88337, size = 313, normalized size = 2.41 \begin{align*} \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 )}} \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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