Optimal. Leaf size=308 \[ -\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b d}+\frac {4 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d} \]
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Rubi [A] time = 0.51, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2791, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b d}+\frac {4 a \left (-62 a^2 b^2+5 a^4+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-279 a^2 b^2+10 a^4-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2791
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx &=\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {2 \int \left (\frac {7 b}{2}-a \cos (c+d x)\right ) (a+b \cos (c+d x))^{5/2} \, dx}{9 b}\\ &=-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {4 \int (a+b \cos (c+d x))^{3/2} \left (\frac {39 a b}{4}-\frac {1}{4} \left (10 a^2-49 b^2\right ) \cos (c+d x)\right ) \, dx}{63 b}\\ &=-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {8 \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{8} b \left (55 a^2+49 b^2\right )-\frac {3}{4} a \left (5 a^2-57 b^2\right ) \cos (c+d x)\right ) \, dx}{315 b}\\ &=-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {16 \int \frac {\frac {3}{16} a b \left (155 a^2+261 b^2\right )-\frac {3}{16} \left (10 a^4-279 a^2 b^2-147 b^4\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{945 b}\\ &=-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}-\frac {\left (10 a^4-279 a^2 b^2-147 b^4\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{315 b^2}+\frac {\left (2 a \left (5 a^4-62 a^2 b^2+57 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^2}\\ &=-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}-\frac {\left (\left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{315 b^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (2 a \left (5 a^4-62 a^2 b^2+57 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}{315 b^2 \sqrt {a+b \cos (c+d x)}}\\ &=-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a \left (5 a^4-62 a^2 b^2+57 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 )}{315 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}\\ \end {align*}
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Mathematica [A] time = 1.38, size = 263, normalized size = 0.85 \[ \frac {16 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b \sin (c+d x) \left (40 a^4+4 a b \left (160 a^2+619 b^2\right ) \cos (c+d x)+1984 a^2 b^2+8 \left (85 a^2 b^2+42 b^4\right ) \cos (2 (c+d x))+260 a b^3 \cos (3 (c+d x))+35 b^4 \cos (4 (c+d x))+301 b^4\right )-8 \left (10 a^5+10 a^4 b-279 a^3 b^2-279 a^2 b^3-147 a b^4-147 b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1260 b^2 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \cos \left (d x + c\right )^{4} + 2 \, a b \cos \left (d x + c\right )^{3} + a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.95, size = 995, normalized size = 3.23 \[ \text {result too large to display} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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