Optimal. Leaf size=354 \[ \frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}-\frac {9 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}-\frac {9 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
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Rubi [A] time = 0.97, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3311, 3296, 3306, 3305, 3351, 3304, 3352, 3312} \[ \frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}-\frac {9 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}-\frac {9 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3311
Rule 3312
Rule 3351
Rule 3352
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx &=-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {2}{3} \int (c+d x)^{3/2} \sin (a+b x) \, dx-\frac {d^2 \int \frac {\sin ^3(a+b x)}{\sqrt {c+d x}} \, dx}{12 b^2}\\ &=-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {d \int \sqrt {c+d x} \cos (a+b x) \, dx}{b}-\frac {d^2 \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}{12 b^2}\\ &=-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {d^2 \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{48 b^2}-\frac {d^2 \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{16 b^2}-\frac {d^2 \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{2 b^2}\\ &=-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {\left (d^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}{48 b^2}-\frac {\left (d^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}{16 b^2}-\frac {\left (d^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}{2 b^2}+\frac {\left (d^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}{48 b^2}-\frac {\left (d^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}{16 b^2}-\frac {\left (d^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}{2 b^2}\\ &=-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {\left (d \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 )}{24 b^2}-\frac {\left (d \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 )}{8 b^2}-\frac {\left (d \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 )}{b^2}+\frac {\left (d \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 )}{24 b^2}-\frac {\left (d \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 )}{8 b^2}-\frac {\left (d \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 )}{b^2}\\ &=-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}-\frac {9 d^{3/2} \sqrt {\frac {\pi }{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 )}{8 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \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 )}{24 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{6}} 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 )}{24 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{8 b^{5/2}}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 389, normalized size = 1.10 \[ \frac {\sqrt {6 \pi } d \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )-81 \sqrt {2 \pi } d \sin \left (a-\frac {b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )-81 \sqrt {2 \pi } d \cos \left (a-\frac {b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )+\sqrt {6 \pi } d \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+162 d \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (a+b x)-6 d \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (3 (a+b x))-108 b d x \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (a+b x)-108 b c \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (a+b x)+12 b d x \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (3 (a+b x))+12 b c \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (3 (a+b x))}{144 b^2 \sqrt {\frac {b}{d}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 300, normalized size = 0.85 \[ \frac {\sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left (2 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{3} - 6 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right ) - {\left (b d \cos \left (b x + a\right )^{2} - 7 \, b d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{144 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 3.21, size = 1538, normalized size = 4.34 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 384, normalized size = 1.08 \[ \frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{4 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{4 b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.87, size = 499, normalized size = 1.41 \[ \frac {{\left (\frac {48 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {432 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}{d} - 24 \, \sqrt {d x + c} b^{2} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 648 \, \sqrt {d x + c} b^{2} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - {\left (-\left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) - {\left (\left (81 i + 81\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (81 i - 81\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - {\left (-\left (81 i - 81\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (81 i + 81\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) - {\left (\left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right )\right )} d}{576 \, b^{4}} \]
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 \[ \int {\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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