Optimal. Leaf size=238 \[ \frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{6 d}+\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.39, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2691, 2861, 2752, 2663, 2661, 2655, 2653} \[ \frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{6 d}+\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2691
Rule 2752
Rule 2861
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {1}{3} \int \sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (-2 a^2+\frac {3 b^2}{2}-\frac {1}{2} a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}+\frac {1}{3} \int \frac {-\frac {a b^2}{4}-\frac {1}{4} b \left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}+\frac {1}{3} \left (a \left (a^2-b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx+\frac {1}{12} \left (-4 a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx\\ &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}+\frac {\left (\left (-4 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{12 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{3 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {\left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (a^2-b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d}\\ \end {align*}
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Mathematica [A] time = 3.52, size = 259, normalized size = 1.09 \[ \frac {-4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\left (4 a^3+4 a^2 b-3 a b^2-3 b^3\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\frac {1}{8} \sec ^3(c+d x) \left (24 a^3 \sin (c+d x)+8 a^3 \sin (3 (c+d x))-4 \left (3 a^2 b+2 b^3\right ) \cos (2 (c+d x))+\left (3 b^3-4 a^2 b\right ) \cos (4 (c+d x))+40 a^2 b+40 a b^2 \sin (c+d x)-8 a b^2 \sin (3 (c+d x))+5 b^3\right )}{6 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, a b \sec \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sec \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
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 = 0.91, size = 1249, normalized size = 5.25 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^4} \,d x \]
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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