Optimal. Leaf size=497 \[ \frac{i d (a+b x)^2 (b c-a d) \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi (a+b x)^2\right )}{4 \pi b^3}-\frac{i d (a+b x)^2 (b c-a d) \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi (a+b x)^2\right )}{4 \pi b^3}-\frac{d (b c-a d) \text{FresnelC}(a+b x) S(a+b x)}{\pi b^3}+\frac{d (a+b x)^2 (b c-a d) S(a+b x)^2}{b^3}+\frac{(a+b x) (b c-a d)^2 S(a+b x)^2}{b^3}-\frac{(b c-a d)^2 S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b^3}+\frac{2 d (a+b x) (b c-a d) S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac{2 (b c-a d)^2 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac{d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 \pi ^2 b^3}-\frac{5 d^2 \text{FresnelC}\left (\sqrt{2} (a+b x)\right )}{6 \sqrt{2} \pi ^2 b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{4 d^2 S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}+\frac{2 d^2 (a+b x)^2 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}+\frac{d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 \pi ^2 b^3}+\frac{2 d^2 x}{3 \pi ^2 b^2} \]
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Rubi [A] time = 0.410257, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.812, Rules used = {6432, 6420, 6452, 3351, 6430, 6454, 6446, 3379, 2638, 6460, 3357, 3352, 3385} \[ \frac{i d (a+b x)^2 (b c-a d) \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{4 \pi b^3}-\frac{i d (a+b x)^2 (b c-a d) \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{4 \pi b^3}-\frac{d (b c-a d) \text{FresnelC}(a+b x) S(a+b x)}{\pi b^3}+\frac{d (a+b x)^2 (b c-a d) S(a+b x)^2}{b^3}+\frac{(a+b x) (b c-a d)^2 S(a+b x)^2}{b^3}-\frac{(b c-a d)^2 S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b^3}+\frac{2 d (a+b x) (b c-a d) S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac{2 (b c-a d)^2 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^3}+\frac{d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 \pi ^2 b^3}-\frac{5 d^2 \text{FresnelC}\left (\sqrt{2} (a+b x)\right )}{6 \sqrt{2} \pi ^2 b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{4 d^2 S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 \pi ^2 b^3}+\frac{2 d^2 (a+b x)^2 S(a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 \pi b^3}+\frac{d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 \pi ^2 b^3}+\frac{2 d^2 x}{3 \pi ^2 b^2} \]
Antiderivative was successfully verified.
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Rule 6432
Rule 6420
Rule 6452
Rule 3351
Rule 6430
Rule 6454
Rule 6446
Rule 3379
Rule 2638
Rule 6460
Rule 3357
Rule 3352
Rule 3385
Rubi steps
\begin{align*} \int (c+d x)^2 S(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (b^2 c^2 \left (1+\frac{a d (-2 b c+a d)}{b^2 c^2}\right ) S(x)^2+2 b c d \left (1-\frac{a d}{b c}\right ) x S(x)^2+d^2 x^2 S(x)^2\right ) \, dx,x,a+b x\right )}{b^3}\\ &=\frac{d^2 \operatorname{Subst}\left (\int x^2 S(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac{(2 d (b c-a d)) \operatorname{Subst}\left (\int x S(x)^2 \, dx,x,a+b x\right )}{b^3}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int S(x)^2 \, dx,x,a+b x\right )}{b^3}\\ &=\frac{(b c-a d)^2 (a+b x) S(a+b x)^2}{b^3}+\frac{d (b c-a d) (a+b x)^2 S(a+b x)^2}{b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int x^3 S(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3}-\frac{(2 d (b c-a d)) \operatorname{Subst}\left (\int x^2 S(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}-\frac{\left (2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int x S(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^3}\\ &=\frac{2 (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d (b c-a d) (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d^2 (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{3 b^3 \pi }+\frac{(b c-a d)^2 (a+b x) S(a+b x)^2}{b^3}+\frac{d (b c-a d) (a+b x)^2 S(a+b x)^2}{b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{d^2 \operatorname{Subst}\left (\int x^2 \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{3 b^3 \pi }-\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int x \cos \left (\frac{\pi x^2}{2}\right ) S(x) \, dx,x,a+b x\right )}{3 b^3 \pi }-\frac{(d (b c-a d)) \operatorname{Subst}\left (\int x \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b^3 \pi }-\frac{(2 d (b c-a d)) \operatorname{Subst}\left (\int \cos \left (\frac{\pi x^2}{2}\right ) S(x) \, dx,x,a+b x\right )}{b^3 \pi }-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b^3 \pi }\\ &=\frac{d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 b^3 \pi ^2}+\frac{2 (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d (b c-a d) (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d^2 (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{3 b^3 \pi }-\frac{d (b c-a d) C(a+b x) S(a+b x)}{b^3 \pi }+\frac{(b c-a d)^2 (a+b x) S(a+b x)^2}{b^3}+\frac{d (b c-a d) (a+b x)^2 S(a+b x)^2}{b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{(b c-a d)^2 S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^3 \pi }+\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{4 d^2 S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}-\frac{d^2 \operatorname{Subst}\left (\int \cos \left (\pi x^2\right ) \, dx,x,a+b x\right )}{6 b^3 \pi ^2}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \sin ^2\left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{3 b^3 \pi ^2}-\frac{(d (b c-a d)) \operatorname{Subst}\left (\int \sin (\pi x) \, dx,x,(a+b x)^2\right )}{2 b^3 \pi }\\ &=\frac{d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 b^3 \pi ^2}+\frac{d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 b^3 \pi ^2}-\frac{d^2 C\left (\sqrt{2} (a+b x)\right )}{6 \sqrt{2} b^3 \pi ^2}+\frac{2 (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d (b c-a d) (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d^2 (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{3 b^3 \pi }-\frac{d (b c-a d) C(a+b x) S(a+b x)}{b^3 \pi }+\frac{(b c-a d)^2 (a+b x) S(a+b x)^2}{b^3}+\frac{d (b c-a d) (a+b x)^2 S(a+b x)^2}{b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{(b c-a d)^2 S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^3 \pi }+\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{4 d^2 S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}+\frac{\left (4 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{1}{2} \cos \left (\pi x^2\right )\right ) \, dx,x,a+b x\right )}{3 b^3 \pi ^2}\\ &=\frac{2 d^2 x}{3 b^2 \pi ^2}+\frac{d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 b^3 \pi ^2}+\frac{d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 b^3 \pi ^2}-\frac{d^2 C\left (\sqrt{2} (a+b x)\right )}{6 \sqrt{2} b^3 \pi ^2}+\frac{2 (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d (b c-a d) (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d^2 (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{3 b^3 \pi }-\frac{d (b c-a d) C(a+b x) S(a+b x)}{b^3 \pi }+\frac{(b c-a d)^2 (a+b x) S(a+b x)^2}{b^3}+\frac{d (b c-a d) (a+b x)^2 S(a+b x)^2}{b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{(b c-a d)^2 S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^3 \pi }+\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{4 d^2 S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \cos \left (\pi x^2\right ) \, dx,x,a+b x\right )}{3 b^3 \pi ^2}\\ &=\frac{2 d^2 x}{3 b^2 \pi ^2}+\frac{d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 b^3 \pi ^2}+\frac{d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 b^3 \pi ^2}-\frac{d^2 C\left (\sqrt{2} (a+b x)\right )}{6 \sqrt{2} b^3 \pi ^2}-\frac{\sqrt{2} d^2 C\left (\sqrt{2} (a+b x)\right )}{3 b^3 \pi ^2}+\frac{2 (b c-a d)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d (b c-a d) (a+b x) \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{b^3 \pi }+\frac{2 d^2 (a+b x)^2 \cos \left (\frac{1}{2} \pi (a+b x)^2\right ) S(a+b x)}{3 b^3 \pi }-\frac{d (b c-a d) C(a+b x) S(a+b x)}{b^3 \pi }+\frac{(b c-a d)^2 (a+b x) S(a+b x)^2}{b^3}+\frac{d (b c-a d) (a+b x)^2 S(a+b x)^2}{b^3}+\frac{d^2 (a+b x)^3 S(a+b x)^2}{3 b^3}-\frac{(b c-a d)^2 S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^3 \pi }+\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac{4 d^2 S(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}\\ \end{align*}
Mathematica [F] time = 0.664232, size = 0, normalized size = 0. \[ \int (c+d x)^2 S(a+b x)^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.244, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{2} \left ({\it FresnelS} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2}{\rm fresnels}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\rm fresnels}\left (b x + a\right )^{2}, x\right ) \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 (c + d x\right )^{2} S^{2}\left (a + b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2}{\rm fresnels}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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