Optimal. Leaf size=121 \[ \frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {1+\frac {b \cos ^2(x)}{a}}}-\frac {a (a+b) \sqrt {1+\frac {b \cos ^2(x)}{a}} F\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x) \]
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Rubi [A]
time = 0.11, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3259, 3251,
3257, 3256, 3262, 3261} \begin {gather*} \frac {1}{3} b \sin (x) \cos (x) \sqrt {a+b \cos ^2(x)}-\frac {a (a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} F\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{3 \sqrt {\frac {b \cos ^2(x)}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3251
Rule 3256
Rule 3257
Rule 3259
Rule 3261
Rule 3262
Rubi steps
\begin {align*} \int \left (a+b \cos ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)+\frac {1}{3} \int \frac {a (3 a+b)+2 b (2 a+b) \cos ^2(x)}{\sqrt {a+b \cos ^2(x)}} \, dx\\ &=\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)-\frac {1}{3} (a (a+b)) \int \frac {1}{\sqrt {a+b \cos ^2(x)}} \, dx+\frac {1}{3} (2 (2 a+b)) \int \sqrt {a+b \cos ^2(x)} \, dx\\ &=\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)+\frac {\left (2 (2 a+b) \sqrt {a+b \cos ^2(x)}\right ) \int \sqrt {1+\frac {b \cos ^2(x)}{a}} \, dx}{3 \sqrt {1+\frac {b \cos ^2(x)}{a}}}-\frac {\left (a (a+b) \sqrt {1+\frac {b \cos ^2(x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \cos ^2(x)}{a}}} \, dx}{3 \sqrt {a+b \cos ^2(x)}}\\ &=\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {1+\frac {b \cos ^2(x)}{a}}}-\frac {a (a+b) \sqrt {1+\frac {b \cos ^2(x)}{a}} F\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x)\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 123, normalized size = 1.02 \begin {gather*} \frac {8 \left (2 a^2+3 a b+b^2\right ) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} E\left (x\left |\frac {b}{a+b}\right .\right )-4 a (a+b) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} F\left (x\left |\frac {b}{a+b}\right .\right )+\sqrt {2} b (2 a+b+b \cos (2 x)) \sin (2 x)}{12 \sqrt {2 a+b+b \cos (2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 192, normalized size = 1.59
method | result | size |
default | \(-\frac {-\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\left (x \right )\right )}{a}}\, \EllipticF \left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}-\frac {a \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\left (x \right )\right )}{a}}\, \EllipticF \left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b}{3}+\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\left (x \right )\right )}{a}}\, \EllipticE \left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}}{3}+\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\left (x \right )\right )}{a}}\, \EllipticE \left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a b}{3}+\frac {\left (\cos ^{5}\left (x \right )\right ) b^{2}}{3}+\frac {a b \left (\cos ^{3}\left (x \right )\right )}{3}-\frac {b^{2} \left (\cos ^{3}\left (x \right )\right )}{3}-\frac {a b \cos \left (x \right )}{3}}{\sin \left (x \right ) \sqrt {a +b \left (\cos ^{2}\left (x \right )\right )}}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.13, size = 12, normalized size = 0.10 \begin {gather*} {\rm integral}\left ({\left (b \cos \left (x\right )^{2} + a\right )}^{\frac {3}{2}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \cos ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\cos \left (x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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