Optimal. Leaf size=279 \[ \frac{i d (a+b x)^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}-\frac{i d (a+b x)^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}+\frac{(a+b x) (b c-a d) \text{FresnelC}(a+b x)^2}{b^2}-\frac{2 (b c-a d) \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b^2}+\frac{d \text{FresnelC}(a+b x) S(a+b x)}{2 \pi b^2}+\frac{d (a+b x)^2 \text{FresnelC}(a+b x)^2}{2 b^2}-\frac{d (a+b x) \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{d \cos \left (\pi (a+b x)^2\right )}{4 \pi ^2 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.185127, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6433, 6421, 6453, 3351, 6431, 6455, 6447, 3379, 2638} \[ \frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}-\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{8 \pi b^2}+\frac{(a+b x) (b c-a d) \text{FresnelC}(a+b x)^2}{b^2}-\frac{2 (b c-a d) \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} \pi b^2}+\frac{d \text{FresnelC}(a+b x) S(a+b x)}{2 \pi b^2}+\frac{d (a+b x)^2 \text{FresnelC}(a+b x)^2}{2 b^2}-\frac{d (a+b x) \text{FresnelC}(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b^2}-\frac{d \cos \left (\pi (a+b x)^2\right )}{4 \pi ^2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6433
Rule 6421
Rule 6453
Rule 3351
Rule 6431
Rule 6455
Rule 6447
Rule 3379
Rule 2638
Rubi steps
\begin{align*} \int (c+d x) C(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (b c \left (1-\frac{a d}{b c}\right ) C(x)^2+d x C(x)^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{d \operatorname{Subst}\left (\int x C(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac{(b c-a d) \operatorname{Subst}\left (\int C(x)^2 \, dx,x,a+b x\right )}{b^2}\\ &=\frac{(b c-a d) (a+b x) C(a+b x)^2}{b^2}+\frac{d (a+b x)^2 C(a+b x)^2}{2 b^2}-\frac{d \operatorname{Subst}\left (\int x^2 \cos \left (\frac{\pi x^2}{2}\right ) C(x) \, dx,x,a+b x\right )}{b^2}-\frac{(2 (b c-a d)) \operatorname{Subst}\left (\int x \cos \left (\frac{\pi x^2}{2}\right ) C(x) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{(b c-a d) (a+b x) C(a+b x)^2}{b^2}+\frac{d (a+b x)^2 C(a+b x)^2}{2 b^2}-\frac{2 (b c-a d) C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }-\frac{d (a+b x) C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac{d \operatorname{Subst}\left (\int x \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{2 b^2 \pi }+\frac{d \operatorname{Subst}\left (\int C(x) \sin \left (\frac{\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2 \pi }+\frac{(b c-a d) \operatorname{Subst}\left (\int \sin \left (\pi x^2\right ) \, dx,x,a+b x\right )}{b^2 \pi }\\ &=\frac{(b c-a d) (a+b x) C(a+b x)^2}{b^2}+\frac{d (a+b x)^2 C(a+b x)^2}{2 b^2}+\frac{d C(a+b x) S(a+b x)}{2 b^2 \pi }+\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^2 \pi }+\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{2 (b c-a d) C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }-\frac{d (a+b x) C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac{d \operatorname{Subst}\left (\int \sin (\pi x) \, dx,x,(a+b x)^2\right )}{4 b^2 \pi }\\ &=-\frac{d \cos \left (\pi (a+b x)^2\right )}{4 b^2 \pi ^2}+\frac{(b c-a d) (a+b x) C(a+b x)^2}{b^2}+\frac{d (a+b x)^2 C(a+b x)^2}{2 b^2}+\frac{d C(a+b x) S(a+b x)}{2 b^2 \pi }+\frac{(b c-a d) S\left (\sqrt{2} (a+b x)\right )}{\sqrt{2} b^2 \pi }+\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{i d (a+b x)^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i \pi (a+b x)^2\right )}{8 b^2 \pi }-\frac{2 (b c-a d) C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }-\frac{d (a+b x) C(a+b x) \sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b^2 \pi }\\ \end{align*}
Mathematica [F] time = 0.548492, size = 0, normalized size = 0. \[ \int (c+d x) \text{FresnelC}(a+b x)^2 \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) \left ({\it FresnelC} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\rm fresnelc}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}{\rm fresnelc}\left (b x + a\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) C^{2}\left (a + b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}{\rm fresnelc}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]