Optimal. Leaf size=193 \[ -\frac {d (b c-a d) C(a+b x)}{\pi b^3}-\frac {(b c-a d)^3 S(a+b x)}{3 b^3 d}+\frac {(b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac {d (a+b x) (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}-\frac {2 d^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}+\frac {d^2 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}+\frac {(c+d x)^3 S(a+b x)}{3 d} \]
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Rubi [A] time = 0.23, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6428, 3433, 3351, 3379, 2638, 3385, 3352, 3296, 2637} \[ -\frac {d (b c-a d) \text {FresnelC}(a+b x)}{\pi b^3}-\frac {(b c-a d)^3 S(a+b x)}{3 b^3 d}+\frac {(b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac {d (a+b x) (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^3}-\frac {2 d^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}+\frac {d^2 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}+\frac {(c+d x)^3 S(a+b x)}{3 d} \]
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 (c+d x)^2 S(a+b x) \, dx &=\frac {(c+d x)^3 S(a+b x)}{3 d}-\frac {b \int (c+d x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx}{3 d}\\ &=\frac {(c+d x)^3 S(a+b x)}{3 d}-\frac {\operatorname {Subst}\left (\int \left (b^3 c^3 \left (1-\frac {a d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 c^3}\right ) \sin \left (\frac {\pi x^2}{2}\right )+3 b^2 c^2 d \left (1+\frac {a d (-2 b c+a d)}{b^2 c^2}\right ) x \sin \left (\frac {\pi x^2}{2}\right )+3 b c d^2 \left (1-\frac {a d}{b c}\right ) x^2 \sin \left (\frac {\pi x^2}{2}\right )+d^3 x^3 \sin \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{3 b^3 d}\\ &=\frac {(c+d x)^3 S(a+b x)}{3 d}-\frac {d^2 \operatorname {Subst}\left (\int x^3 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3}-\frac {(d (b c-a d)) \operatorname {Subst}\left (\int x^2 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac {(b c-a d)^2 \operatorname {Subst}\left (\int x \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3 d}\\ &=\frac {d (b c-a d) (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {(b c-a d)^3 S(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 S(a+b x)}{3 d}-\frac {d^2 \operatorname {Subst}\left (\int x \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{6 b^3}-\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^3}-\frac {(d (b c-a d)) \operatorname {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3 \pi }\\ &=\frac {(b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {d (b c-a d) (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {d^2 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }-\frac {d (b c-a d) C(a+b x)}{b^3 \pi }-\frac {(b c-a d)^3 S(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 S(a+b x)}{3 d}-\frac {d^2 \operatorname {Subst}\left (\int \cos \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{3 b^3 \pi }\\ &=\frac {(b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {d (b c-a d) (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {d^2 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }-\frac {d (b c-a d) C(a+b x)}{b^3 \pi }-\frac {(b c-a d)^3 S(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 S(a+b x)}{3 d}-\frac {2 d^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 = 0.55, size = 236, normalized size = 1.22 \[ \frac {\pi a^2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi ^2 S(a+b x) \left (a^3 d^2-3 a^2 b c d+3 a b^2 c^2+b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )\right )+3 \pi b^2 c^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+3 \pi b^2 c d x \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi b^2 d^2 x^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+3 \pi d (a d-b c) C(a+b x)-3 \pi a b c d \cos \left (\frac {1}{2} \pi (a+b x)^2\right )-2 d^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-\pi a b d^2 x \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\rm fresnels}\left (b x + a\right ), x\right ) \]
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 \[ \int {\left (d x + c\right )}^{2} {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 251, normalized size = 1.30 \[ \frac {\frac {\mathrm {S}\left (b x +a \right ) \left (\left (b x +a \right ) d -a d +b c \right )^{3}}{3 b^{2} d}-\frac {-\frac {d^{3} \left (b x +a \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {2 d^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}-\frac {\left (-3 a \,d^{3}+3 b c \,d^{2}\right ) \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {\left (-3 a \,d^{3}+3 b c \,d^{2}\right ) \FresnelC \left (b x +a \right )}{\pi }-\frac {\left (3 a^{2} d^{3}-6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-a^{3} d^{3} \mathrm {S}\left (b x +a \right )+3 a^{2} b c \,d^{2} \mathrm {S}\left (b x +a \right )-3 a \,b^{2} c^{2} d \,\mathrm {S}\left (b x +a \right )+b^{3} c^{3} \mathrm {S}\left (b x +a \right )}{3 b^{2} d}}{b} \]
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 \[ \int {\left (d x + c\right )}^{2} {\rm fresnels}\left (b x + a\right )\,{d x} \]
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 \[ \int \mathrm {FresnelS}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2 \,d x \]
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 \[ \int \left (c + d x\right )^{2} S\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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