Optimal. Leaf size=154 \[ -\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac{b}{a}\right .\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.167102, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ -\frac{b \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{a (a+b) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (e+f x\left |-\frac{b}{a}\right .\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{3 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3180
Rule 3172
Rule 3178
Rule 3177
Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=-\frac{b \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{1}{3} \int \frac{a (3 a+b)+2 b (2 a+b) \sin ^2(e+f x)}{\sqrt{a+b \sin ^2(e+f x)}} \, dx\\ &=-\frac{b \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{1}{3} (a (a+b)) \int \frac{1}{\sqrt{a+b \sin ^2(e+f x)}} \, dx+\frac{1}{3} (2 (2 a+b)) \int \sqrt{a+b \sin ^2(e+f x)} \, dx\\ &=-\frac{b \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left (2 (2 a+b) \sqrt{a+b \sin ^2(e+f x)}\right ) \int \sqrt{1+\frac{b \sin ^2(e+f x)}{a}} \, dx}{3 \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{\left (a (a+b) \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}\right ) \int \frac{1}{\sqrt{1+\frac{b \sin ^2(e+f x)}{a}}} \, dx}{3 \sqrt{a+b \sin ^2(e+f x)}}\\ &=-\frac{b \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{2 (2 a+b) E\left (e+f x\left |-\frac{b}{a}\right .\right ) \sqrt{a+b \sin ^2(e+f x)}}{3 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{a (a+b) F\left (e+f x\left |-\frac{b}{a}\right .\right ) \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}{3 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.767113, size = 156, normalized size = 1.01 \[ \frac{b \sin (2 (e+f x)) (-2 a+b \cos (2 (e+f x))-b)-2 \sqrt{2} a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )+4 \sqrt{2} a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{6 \sqrt{2} f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.317, size = 266, normalized size = 1.7 \begin{align*}{\frac{1}{f\cos \left ( fx+e \right ) } \left ( -{\frac{{a}^{2}}{3}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }-{\frac{ab}{3}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }+{\frac{4\,{a}^{2}}{3}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }+{\frac{2\,ab}{3}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) }+{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{5}{b}^{2}}{3}}+{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{3}ab}{3}}-{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{3}{b}^{2}}{3}}-{\frac{\sin \left ( fx+e \right ) ab}{3}} \right ){\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac{3}{2}}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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