Optimal. Leaf size=166 \[ \frac{3 c^2 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{40 b \sqrt{\sin (2 a+2 b x)}}+\frac{c d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{20 b}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{11/2}}{7 b d}+\frac{3 c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{70 b} \]
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Rubi [A] time = 0.235413, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2568, 2569, 2572, 2639} \[ \frac{3 c^2 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{40 b \sqrt{\sin (2 a+2 b x)}}+\frac{c d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{20 b}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{11/2}}{7 b d}+\frac{3 c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{70 b} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2569
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx &=-\frac{c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac{1}{14} \left (3 c^2\right ) \int (d \cos (a+b x))^{9/2} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac{c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac{1}{20} \left (3 c^2 d^2\right ) \int (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac{3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac{c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac{1}{40} \left (3 c^2 d^4\right ) \int \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)} \, dx\\ &=\frac{c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac{3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac{c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac{\left (3 c^2 d^4 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{40 \sqrt{\sin (2 a+2 b x)}}\\ &=\frac{c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac{3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac{c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac{3 c^2 d^4 \sqrt{d \cos (a+b x)} E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{c \sin (a+b x)}}{40 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [C] time = 0.172968, size = 72, normalized size = 0.43 \[ \frac{2 \sqrt [4]{\cos ^2(a+b x)} \sec ^5(a+b x) (c \sin (a+b x))^{7/2} (d \cos (a+b x))^{9/2} \, _2F_1\left (-\frac{7}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right )}{7 b c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 545, normalized size = 3.3 \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 (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac{5}{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 (c^{2} d^{4} \cos \left (b x + a\right )^{6} - c^{2} d^{4} \cos \left (b x + a\right )^{4}\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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