Optimal. Leaf size=256 \[ -\frac{2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}-\frac{2 \left (-46 a^2 b^2+71 a^4-25 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 d \sqrt{a+b \sin (c+d x)}}+\frac{32 a \left (11 a^2+13 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d} \]
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Rubi [A] time = 0.382149, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2656, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}-\frac{2 \left (-46 a^2 b^2+71 a^4-25 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 d \sqrt{a+b \sin (c+d x)}}+\frac{32 a \left (11 a^2+13 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{105 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d} \]
Antiderivative was successfully verified.
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Rule 2656
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int (a+b \sin (c+d x))^{7/2} \, dx &=-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac{2}{7} \int (a+b \sin (c+d x))^{3/2} \left (\frac{1}{2} \left (7 a^2+5 b^2\right )+6 a b \sin (c+d x)\right ) \, dx\\ &=-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac{4}{35} \int \sqrt{a+b \sin (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 ) \sin (c+d x)\right ) \, dx\\ &=-\frac{2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{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 ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx\\ &=-\frac{2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac{1}{105} \left (16 a \left (11 a^2+13 b^2\right )\right ) \int \sqrt{a+b \sin (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 \sin (c+d x)}} \, dx\\ &=-\frac{2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac{\left (16 a \left (11 a^2+13 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}{105 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (\left (-71 a^4+46 a^2 b^2+25 b^4\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}{105 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 b \left (71 a^2+25 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{105 d}-\frac{24 a b \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 d}-\frac{2 b \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac{32 a \left (11 a^2+13 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)}}{105 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \left (71 a^4-46 a^2 b^2-25 b^4\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}}}{105 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.969083, size = 220, normalized size = 0.86 \[ \frac{-b \cos (c+d x) \left (b \left (752 a^2+145 b^2\right ) \sin (c+d x)+488 a^3-162 a b^2 \cos (2 (c+d x))+262 a b^2-15 b^3 \sin (3 (c+d x))\right )+4 \left (-46 a^2 b^2+71 a^4-25 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )-64 a \left (11 a^2 b+11 a^3+13 a b^2+13 b^3\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{210 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.199, size = 1040, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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