Optimal. Leaf size=140 \[ \frac{\sqrt{\frac{\pi }{2}} \sin \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{2 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}-\frac{e \cos \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0558215, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3461, 3447, 3351, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \sin \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )}{2 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}-\frac{e \cos \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 3461
Rule 3447
Rule 3351
Rule 3352
Rubi steps
\begin{align*} \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx &=-\frac{e \cos \left (a+b x+c x^2\right )}{2 c}+\frac{(2 c d-b e) \int \sin \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=-\frac{e \cos \left (a+b x+c x^2\right )}{2 c}+\frac{\left ((2 c d-b e) \cos \left (a-\frac{b^2}{4 c}\right )\right ) \int \sin \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}+\frac{\left ((2 c d-b e) \sin \left (a-\frac{b^2}{4 c}\right )\right ) \int \cos \left (\frac{(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}\\ &=-\frac{e \cos \left (a+b x+c x^2\right )}{2 c}+\frac{(2 c d-b e) \sqrt{\frac{\pi }{2}} \cos \left (a-\frac{b^2}{4 c}\right ) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )}{2 c^{3/2}}+\frac{(2 c d-b e) \sqrt{\frac{\pi }{2}} C\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right ) \sin \left (a-\frac{b^2}{4 c}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.668156, size = 128, normalized size = 0.91 \[ \frac{\sqrt{2 \pi } \sin \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e) \text{FresnelC}\left (\frac{b+2 c x}{\sqrt{2 \pi } \sqrt{c}}\right )+\sqrt{2 \pi } \cos \left (a-\frac{b^2}{4 c}\right ) (2 c d-b e) S\left (\frac{b+2 c x}{\sqrt{c} \sqrt{2 \pi }}\right )-2 \sqrt{c} e \cos (a+x (b+c x))}{4 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 184, normalized size = 1.3 \begin{align*} -{\frac{e\cos \left ( c{x}^{2}+bx+a \right ) }{2\,c}}-{\frac{be\sqrt{2}\sqrt{\pi }}{4} \left ( \cos \left ({\frac{1}{c} \left ({\frac{{b}^{2}}{4}}-ca \right ) } \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( cx+{\frac{b}{2}} \right ){\frac{1}{\sqrt{c}}}} \right ) -\sin \left ({\frac{1}{c} \left ({\frac{{b}^{2}}{4}}-ca \right ) } \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( cx+{\frac{b}{2}} \right ){\frac{1}{\sqrt{c}}}} \right ) \right ){c}^{-{\frac{3}{2}}}}+{\frac{\sqrt{2}\sqrt{\pi }d}{2} \left ( \cos \left ({\frac{1}{c} \left ({\frac{{b}^{2}}{4}}-ca \right ) } \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( cx+{\frac{b}{2}} \right ){\frac{1}{\sqrt{c}}}} \right ) -\sin \left ({\frac{1}{c} \left ({\frac{{b}^{2}}{4}}-ca \right ) } \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( cx+{\frac{b}{2}} \right ){\frac{1}{\sqrt{c}}}} \right ) \right ){\frac{1}{\sqrt{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.71412, size = 1702, normalized size = 12.16 \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 = 1.74135, size = 359, normalized size = 2.56 \begin{align*} \frac{\sqrt{2} \pi{\left (2 \, c d - b e\right )} \sqrt{\frac{c}{\pi }} \cos \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) + \sqrt{2} \pi{\left (2 \, c d - b e\right )} \sqrt{\frac{c}{\pi }} \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, c x + b\right )} \sqrt{\frac{c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - 2 \, c e \cos \left (c x^{2} + b x + a\right )}{4 \, c^{2}} \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 \left (d + e x\right ) \sin{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.28206, size = 441, normalized size = 3.15 \begin{align*} \frac{i \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{1}{4} \, \sqrt{2}{\left (2 \, x + \frac{b}{c}\right )}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}\right ) e^{\left (-\frac{i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{4 \,{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}} - \frac{i \, \sqrt{2} \sqrt{\pi } d \operatorname{erf}\left (-\frac{1}{4} \, \sqrt{2}{\left (2 \, x + \frac{b}{c}\right )}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}\right ) e^{\left (-\frac{-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{4 \,{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}} - \frac{\frac{i \, \sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{4} \, \sqrt{2}{\left (2 \, x + \frac{b}{c}\right )}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}\right ) e^{\left (-\frac{i \, b^{2} - 4 i \, a c - 4 \, c}{4 \, c}\right )}}{{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}} + 2 \, e^{\left (i \, c x^{2} + i \, b x + i \, a + 1\right )}}{8 \, c} - \frac{-\frac{i \, \sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{4} \, \sqrt{2}{\left (2 \, x + \frac{b}{c}\right )}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}\right ) e^{\left (-\frac{-i \, b^{2} + 4 i \, a c - 4 \, c}{4 \, c}\right )}}{{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )} \sqrt{{\left | c \right |}}} + 2 \, e^{\left (-i \, c x^{2} - i \, b x - i \, a + 1\right )}}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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