Optimal. Leaf size=249 \[ \frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{21 d}+\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d} \]
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Rubi [A] time = 0.36, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (3 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{21 d}-\frac {2 \left (2 a^2 b^2+3 a^4-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx &=\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {2}{7} \int \left (\frac {5 b}{2}+\frac {5}{2} a \cos (c+d x)\right ) (a+b \cos (c+d x))^{3/2} \, dx\\ &=\frac {2 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {a+b \cos (c+d x)} \left (10 a b+\frac {5}{4} \left (3 a^2+5 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {8}{105} \int \frac {\frac {5}{8} b \left (27 a^2+5 b^2\right )+\frac {5}{8} a \left (3 a^2+29 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (a \left (3 a^2+29 b^2\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{21 b}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{21 b}\\ &=\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{21 b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{21 b \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 214, normalized size = 0.86 \[ \frac {-4 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b \sin (c+d x) \left (36 a^3+b \left (72 a^2+29 b^2\right ) \cos (c+d x)+24 a b^2 \cos (2 (c+d x))+44 a b^2+3 b^3 \cos (3 (c+d x))\right )+4 a \left (3 a^3+3 a^2 b+29 a b^2+29 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{42 b d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \cos \left (d x + c\right )^{3} + 2 \, a b \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.85, size = 827, normalized size = 3.32 \[ \text {result too large to display} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \cos \left (c+d\,x\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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