Optimal. Leaf size=246 \[ \frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 d}+\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d} \]
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Rubi [A] time = 0.37, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2656, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 b \left (71 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 d}-\frac {2 \left (-46 a^2 b^2+71 a^4-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2656
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^{7/2} \, dx &=\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {2}{7} \int (a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} \left (7 a^2+5 b^2\right )+6 a b \cos (c+d x)\right ) \, dx\\ &=\frac {24 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b (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 (\frac {1}{4} a \left (35 a^2+61 b^2\right )+\frac {1}{4} b \left (71 a^2+25 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 b \left (71 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {24 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {8}{105} \int \frac {\frac {1}{8} \left (105 a^4+254 a^2 b^2+25 b^4\right )+2 a b \left (11 a^2+13 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 b \left (71 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {24 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{105} \left (16 a \left (11 a^2+13 b^2\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx+\frac {1}{105} \left (-71 a^4+46 a^2 b^2+25 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 b \left (71 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {24 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (16 a \left (11 a^2+13 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}{105 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (-71 a^4+46 a^2 b^2+25 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}{105 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {32 a \left (11 a^2+13 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (71 a^4-46 a^2 b^2-25 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 )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (71 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {24 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 211, normalized size = 0.86 \[ \frac {-4 \left (71 a^4-46 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b \sin (c+d x) \left (488 a^3+b \left (752 a^2+145 b^2\right ) \cos (c+d x)+162 a b^2 \cos (2 (c+d x))+262 a b^2+15 b^3 \cos (3 (c+d x))\right )+64 a \left (11 a^3+11 a^2 b+13 a b^2+13 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{210 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
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 = 0.92, size = 824, normalized size = 3.35 \[ \text {result too large to display} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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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