Optimal. Leaf size=147 \[ \frac {a^3 S(a+b x)}{3 b^3}+\frac {a^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac {a C(a+b x)}{\pi b^3}-\frac {2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}-\frac {a (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac {(a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}+\frac {1}{3} x^3 S(a+b x) \]
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Rubi [A] time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6428, 3433, 3351, 3379, 2638, 3385, 3352, 3296, 2637} \[ \frac {a^3 S(a+b x)}{3 b^3}+\frac {a^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac {a \text {FresnelC}(a+b x)}{\pi b^3}-\frac {2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}-\frac {a (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac {(a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}+\frac {1}{3} x^3 S(a+b x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3351
Rule 3352
Rule 3379
Rule 3385
Rule 3433
Rule 6428
Rubi steps
\begin {align*} \int x^2 S(a+b x) \, dx &=\frac {1}{3} x^3 S(a+b x)-\frac {1}{3} b \int x^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx\\ &=\frac {1}{3} x^3 S(a+b x)-\frac {\operatorname {Subst}\left (\int \left (-a^3 \sin \left (\frac {\pi x^2}{2}\right )+3 a^2 x \sin \left (\frac {\pi x^2}{2}\right )-3 a x^2 \sin \left (\frac {\pi x^2}{2}\right )+x^3 \sin \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac {1}{3} x^3 S(a+b x)-\frac {\operatorname {Subst}\left (\int x^3 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3}+\frac {a \operatorname {Subst}\left (\int x^2 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \operatorname {Subst}\left (\int x \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}+\frac {a^3 \operatorname {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3}\\ &=-\frac {a (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {a^3 S(a+b x)}{3 b^3}+\frac {1}{3} x^3 S(a+b x)-\frac {\operatorname {Subst}\left (\int x \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{6 b^3}-\frac {a^2 \operatorname {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^3}+\frac {a \operatorname {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3 \pi }\\ &=\frac {a^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {a (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {(a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }+\frac {a C(a+b x)}{b^3 \pi }+\frac {a^3 S(a+b x)}{3 b^3}+\frac {1}{3} x^3 S(a+b x)-\frac {\operatorname {Subst}\left (\int \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{3 b^3 \pi }\\ &=\frac {a^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {a (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {(a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }+\frac {a C(a+b x)}{b^3 \pi }+\frac {a^3 S(a+b x)}{3 b^3}+\frac {1}{3} x^3 S(a+b x)-\frac {2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 115, normalized size = 0.78 \[ \frac {\pi ^2 \left (a^3+b^3 x^3\right ) S(a+b x)+\pi a^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi b^2 x^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+3 \pi a C(a+b x)-2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-\pi a b x \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} {\rm fresnels}\left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 121, normalized size = 0.82 \[ \frac {\frac {b^{3} x^{3} \mathrm {S}\left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi }-\frac {2 \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi ^{2}}-\frac {a \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {a \FresnelC \left (b x +a \right )}{\pi }+\frac {a^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {a^{3} \mathrm {S}\left (b x +a \right )}{3}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {FresnelS}\left (a+b\,x\right ) \,d x \]
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 \[ \int x^{2} S\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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