Optimal. Leaf size=353 \[ -\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 d \sqrt {c+d x} \cos (a+b x)}{8 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b} \]
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Rubi [A] time = 0.57, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {3 d \sqrt {c+d x} \cos (a+b x)}{8 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{4} (c+d x)^{3/2} \cos (a+b x)-\frac {1}{4} (c+d x)^{3/2} \cos (3 a+3 b x)\right ) \, dx\\ &=\frac {1}{4} \int (c+d x)^{3/2} \cos (a+b x) \, dx-\frac {1}{4} \int (c+d x)^{3/2} \cos (3 a+3 b x) \, dx\\ &=\frac {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac {d \int \sqrt {c+d x} \sin (3 a+3 b x) \, dx}{8 b}-\frac {(3 d) \int \sqrt {c+d x} \sin (a+b x) \, dx}{8 b}\\ &=\frac {3 d \sqrt {c+d x} \cos (a+b x)}{8 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac {d^2 \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{48 b^2}-\frac {\left (3 d^2\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{16 b^2}\\ &=\frac {3 d \sqrt {c+d x} \cos (a+b x)}{8 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac {\left (d^2 \cos \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 (3 d^2 \cos \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 (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 (3 d^2 \sin \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 {3 d \sqrt {c+d x} \cos (a+b x)}{8 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{24 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}+\frac {\left (d \cos \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 (3 d \cos \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 (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 (3 d \sin \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 {3 d \sqrt {c+d x} \cos (a+b x)}{8 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{24 b^2}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \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 {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \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 {d^{3/2} \sqrt {\frac {\pi }{6}} S\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 {3 d^{3/2} \sqrt {\frac {\pi }{2}} S\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 {(c+d x)^{3/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{12 b}\\ \end {align*}
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Mathematica [C] time = 8.95, size = 677, normalized size = 1.92 \[ \frac {d \left (\sqrt {2 \pi } \sqrt {\frac {b}{d}} C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \sin \left (a-\frac {b c}{d}\right )-3 d \cos \left (a-\frac {b c}{d}\right )\right )+\sqrt {2 \pi } \sqrt {\frac {b}{d}} S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (3 d \sin \left (a-\frac {b c}{d}\right )+2 b c \cos \left (a-\frac {b c}{d}\right )\right )+2 b \sqrt {c+d x} (2 b x \sin (a+b x)+3 \cos (a+b x))\right )}{16 b^3}-\frac {d \left (\sqrt {2 \pi } \sqrt {\frac {b}{d}} C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \sin \left (3 a-\frac {3 b c}{d}\right )-d \cos \left (3 a-\frac {3 b c}{d}\right )\right )+\sqrt {2 \pi } \sqrt {\frac {b}{d}} S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (d \sin \left (3 a-\frac {3 b c}{d}\right )+2 b c \cos \left (3 a-\frac {3 b c}{d}\right )\right )+2 \sqrt {3} b \sqrt {c+d x} (2 b x \sin (3 (a+b x))+\cos (3 (a+b x)))\right )}{48 \sqrt {3} b^3}-\frac {c \left (-\sqrt {2 \pi } \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )-\sqrt {2 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+2 \sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (3 (a+b x))\right )}{24 \sqrt {3} b \sqrt {\frac {b}{d}}}-\frac {i c \sqrt {c+d x} e^{-\frac {i (a d+b c)}{d}} \left (\frac {e^{2 i a} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 298, normalized size = 0.84 \[ \frac {\sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 27 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 27 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 24 \, {\left (b d \cos \left (b x + a\right )^{3} - 3 \, b d \cos \left (b x + a\right ) - 2 \, {\left (b^{2} d x + b^{2} c - {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{144 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 6.94, size = 1529, normalized size = 4.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 386, normalized size = 1.09 \[ \frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{4 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}+\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{4 b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.61, size = 495, normalized size = 1.40 \[ -\frac {{\left (\frac {48 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {144 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}{d} + 24 \, \sqrt {d x + c} b^{2} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 216 \, \sqrt {d x + c} b^{2} \cos \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 (27 i - 27\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 (27 i + 27\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 (27 i + 27\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 (27 i - 27\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 \[ \int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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 \[ \int \left (c + d x\right )^{\frac {3}{2}} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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