Optimal. Leaf size=228 \[ -\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{7/2}}{28 d} \]
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Rubi [A] time = 0.398063, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}-\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^{7/2}}{28 d} \]
Antiderivative was successfully verified.
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Rule 4406
Rule 3296
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x)^{5/2}-\frac{1}{8} (c+d x)^{5/2} \cos (4 a+4 b x)\right ) \, dx\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{1}{8} \int (c+d x)^{5/2} \cos (4 a+4 b x) \, dx\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac{(5 d) \int (c+d x)^{3/2} \sin (4 a+4 b x) \, dx}{64 b}\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \cos (4 a+4 b x) \, dx}{512 b^2}\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}-\frac{\left (15 d^3\right ) \int \frac{\sin (4 a+4 b x)}{\sqrt{c+d x}} \, dx}{4096 b^3}\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}-\frac{\left (15 d^3 \cos \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{4 b c}{d}+4 b x\right )}{\sqrt{c+d x}} \, dx}{4096 b^3}-\frac{\left (15 d^3 \sin \left (4 a-\frac{4 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{4 b c}{d}+4 b x\right )}{\sqrt{c+d x}} \, dx}{4096 b^3}\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}-\frac{\left (15 d^2 \cos \left (4 a-\frac{4 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{2048 b^3}-\frac{\left (15 d^2 \sin \left (4 a-\frac{4 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{2048 b^3}\\ &=\frac{(c+d x)^{7/2}}{28 d}-\frac{5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac{15 d^{5/2} \sqrt{\frac{\pi }{2}} \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{4096 b^{7/2}}-\frac{15 d^{5/2} \sqrt{\frac{\pi }{2}} C\left (\frac{2 \sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (4 a-\frac{4 b c}{d}\right )}{4096 b^{7/2}}+\frac{15 d^2 \sqrt{c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac{(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}\\ \end{align*}
Mathematica [A] time = 3.65687, size = 206, normalized size = 0.9 \[ \frac{\sqrt{\frac{b}{d}} \left (4 \sqrt{\frac{b}{d}} \sqrt{c+d x} \left (-7 d \sin (4 (a+b x)) \left (64 b^2 (c+d x)^2-15 d^2\right )-280 b d^2 (c+d x) \cos (4 (a+b x))+512 b^3 (c+d x)^3\right )-105 \sqrt{2 \pi } d^3 \sin \left (4 a-\frac{4 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-105 \sqrt{2 \pi } d^3 \cos \left (4 a-\frac{4 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )\right )}{57344 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 251, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ({\frac{ \left ( dx+c \right ) ^{7/2}}{56}}-{\frac{d \left ( dx+c \right ) ^{5/2}}{64\,b}\sin \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }+{\frac{5\,d}{64\,b} \left ( -1/8\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\cos \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }+3/8\,{\frac{d}{b} \left ( 1/8\,{\frac{d\sqrt{dx+c}}{b}\sin \left ( 4\,{\frac{ \left ( dx+c \right ) b}{d}}+4\,{\frac{ad-bc}{d}} \right ) }-1/32\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ( 4\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.08834, size = 922, normalized size = 4.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.651637, size = 840, normalized size = 3.68 \begin{align*} -\frac{105 \, \sqrt{2} \pi d^{4} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 105 \, \sqrt{2} \pi d^{4} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (2 \, \sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{4 \,{\left (b c - a d\right )}}{d}\right ) - 16 \,{\left (128 \, b^{4} d^{3} x^{3} + 384 \, b^{4} c d^{2} x^{2} + 128 \, b^{4} c^{3} - 70 \, b^{2} c d^{2} - 560 \,{\left (b^{2} d^{3} x + b^{2} c d^{2}\right )} \cos \left (b x + a\right )^{4} + 560 \,{\left (b^{2} d^{3} x + b^{2} c d^{2}\right )} \cos \left (b x + a\right )^{2} + 2 \,{\left (192 \, b^{4} c^{2} d - 35 \, b^{2} d^{3}\right )} x - 7 \,{\left (2 \,{\left (64 \, b^{3} d^{3} x^{2} + 128 \, b^{3} c d^{2} x + 64 \, b^{3} c^{2} d - 15 \, b d^{3}\right )} \cos \left (b x + a\right )^{3} -{\left (64 \, b^{3} d^{3} x^{2} + 128 \, b^{3} c d^{2} x + 64 \, b^{3} c^{2} d - 15 \, b d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{57344 \, b^{4} d} \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 [C] time = 1.68562, size = 1467, normalized size = 6.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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