Optimal. Leaf size=292 \[ \frac{\sqrt{6 \pi } b^{3/2} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{2 \pi } b^{3/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{2 \pi } b^{3/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{\sqrt{6 \pi } b^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{4 b \sin ^2(a+b x) \cos (a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.710384, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3314, 3306, 3305, 3351, 3304, 3352, 3312} \[ \frac{\sqrt{6 \pi } b^{3/2} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{2 \pi } b^{3/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{\sqrt{2 \pi } b^{3/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{\sqrt{6 \pi } b^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}-\frac{4 b \sin ^2(a+b x) \cos (a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3314
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rule 3312
Rubi steps
\begin{align*} \int \frac{\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx &=-\frac{4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (8 b^2\right ) \int \frac{\sin (a+b x)}{\sqrt{c+d x}} \, dx}{d^2}-\frac{\left (12 b^2\right ) \int \frac{\sin ^3(a+b x)}{\sqrt{c+d x}} \, dx}{d^2}\\ &=-\frac{4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}-\frac{\left (12 b^2\right ) \int \left (\frac{3 \sin (a+b x)}{4 \sqrt{c+d x}}-\frac{\sin (3 a+3 b x)}{4 \sqrt{c+d x}}\right ) \, dx}{d^2}+\frac{\left (8 b^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{d^2}+\frac{\left (8 b^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{d^2}\\ &=-\frac{4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (3 b^2\right ) \int \frac{\sin (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{d^2}-\frac{\left (9 b^2\right ) \int \frac{\sin (a+b x)}{\sqrt{c+d x}} \, dx}{d^2}+\frac{\left (16 b^2 \cos \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{\left (16 b^2 \sin \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=\frac{8 b^{3/2} \sqrt{2 \pi } \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{8 b^{3/2} \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{d^{5/2}}-\frac{4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (3 b^2 \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{3 b c}{d}+3 b x\right )}{\sqrt{c+d x}} \, dx}{d^2}-\frac{\left (9 b^2 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{d^2}+\frac{\left (3 b^2 \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{3 b c}{d}+3 b x\right )}{\sqrt{c+d x}} \, dx}{d^2}-\frac{\left (9 b^2 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{\sqrt{c+d x}} \, dx}{d^2}\\ &=\frac{8 b^{3/2} \sqrt{2 \pi } \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{8 b^{3/2} \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{d^{5/2}}-\frac{4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac{\left (6 b^2 \cos \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}-\frac{\left (18 b^2 \cos \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{\left (6 b^2 \sin \left (3 a-\frac{3 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}-\frac{\left (18 b^2 \sin \left (a-\frac{b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=-\frac{b^{3/2} \sqrt{2 \pi } \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{b^{3/2} \sqrt{6 \pi } \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{b^{3/2} \sqrt{6 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (3 a-\frac{3 b c}{d}\right )}{d^{5/2}}-\frac{b^{3/2} \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (a-\frac{b c}{d}\right )}{d^{5/2}}-\frac{4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt{c+d x}}-\frac{2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.42588, size = 496, normalized size = 1.7 \[ \frac{6 \sqrt{6 \pi } b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )+6 \sqrt{6 \pi } b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-6 \sqrt{2 \pi } b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-6 \sqrt{2 \pi } b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-6 \sqrt{2 \pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )+6 \sqrt{6 \pi } b \sqrt{\frac{b}{d}} (c+d x)^{3/2} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right )-6 b c \cos (a+b x)+6 b c \cos (3 (a+b x))-3 d \sin (a+b x)+d \sin (3 (a+b x))-6 b d x \cos (a+b x)+6 b d x \cos (3 (a+b x))}{6 d^2 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 368, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{d} \left ( -1/4\,{\frac{1}{ \left ( dx+c \right ) ^{3/2}}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }+1/2\,{\frac{b}{d} \left ( -{\frac{1}{\sqrt{dx+c}}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }-{\frac{b\sqrt{2}\sqrt{\pi }}{d} \left ( \cos \left ({\frac{da-cb}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ({\frac{da-cb}{d}} \right ){\it FresnelC} \left ({\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 ) }+1/12\,{\frac{1}{ \left ( dx+c \right ) ^{3/2}}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }-1/2\,{\frac{b}{d} \left ( -{\frac{1}{\sqrt{dx+c}}\cos \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }-{\frac{b\sqrt{2}\sqrt{\pi }\sqrt{3}}{d} \left ( \cos \left ( 3\,{\frac{da-cb}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ( 3\,{\frac{da-cb}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.51922, size = 1265, normalized size = 4.33 \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 = 2.66328, size = 963, normalized size = 3.3 \begin{align*} \frac{3 \, \sqrt{6}{\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 3 \, \sqrt{2}{\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{b c - a d}{d}\right ) \operatorname{S}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 3 \, \sqrt{2}{\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (\sqrt{2} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{b c - a d}{d}\right ) + 3 \, \sqrt{6}{\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + 2 \,{\left (6 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - 6 \,{\left (b d x + b c\right )} \cos \left (b x + a\right ) +{\left (d \cos \left (b x + a\right )^{2} - d\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{3 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac{5}{2}}}\, dx \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 \frac{\sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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