Optimal. Leaf size=611 \[ -\frac{b^4 f \sin (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \sin (a) (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b^6 f^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^4 f \cos (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \cos (a) (d e-c f)^2 \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f (c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}-\frac{b^3 f \sqrt{c+d x} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^6 f^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{f (c+d x)^2 (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{(c+d x) (d e-c f)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (c+d x)^{3/2} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b \sqrt{c+d x} (d e-c f)^2 \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3} \]
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Rubi [A] time = 0.790798, antiderivative size = 611, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {3431, 3297, 3303, 3299, 3302} \[ -\frac{b^4 f \sin (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \sin (a) (d e-c f)^2 \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b^6 f^2 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^4 f \cos (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 \cos (a) (d e-c f)^2 \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f (c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}-\frac{b^3 f \sqrt{c+d x} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^6 f^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{f (c+d x)^2 (d e-c f) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{(c+d x) (d e-c f)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (c+d x)^{3/2} (d e-c f) \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b \sqrt{c+d x} (d e-c f)^2 \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int (e+f x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right ) \, dx &=-\frac{2 \operatorname{Subst}\left (\int \left (\frac{f^2 \sin (a+b x)}{d^2 x^7}+\frac{2 f (d e-c f) \sin (a+b x)}{d^2 x^5}+\frac{(d e-c f)^2 \sin (a+b x)}{d^2 x^3}\right ) \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d}\\ &=-\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^7} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}-\frac{(4 f (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^6} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{(b f (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (b (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{\left (b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{\left (b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{\left (b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{\left (b^3 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{\left (b^2 (d e-c f)^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}+\frac{\left (b^2 (d e-c f)^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{d^3}\\ &=-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{180 d^3}-\frac{\left (b^4 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}\\ &=-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{\left (b^5 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{\left (b^4 f (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}-\frac{\left (b^4 f (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{6 d^3}\\ &=\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}-\frac{b^4 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{6 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{b^4 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{\left (b^6 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}\\ &=\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}-\frac{b^4 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{6 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}-\frac{b^4 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{\left (b^6 f^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}+\frac{\left (b^6 f^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt{c+d x}}\right )}{360 d^3}\\ &=\frac{b^5 f^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^3 f (d e-c f) \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b (d e-c f)^2 \sqrt{c+d x} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{180 d^3}+\frac{b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b f^2 (c+d x)^{5/2} \cos \left (a+\frac{b}{\sqrt{c+d x}}\right )}{15 d^3}+\frac{b^6 f^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{360 d^3}-\frac{b^4 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{6 d^3}+\frac{b^2 (d e-c f)^2 \text{Ci}\left (\frac{b}{\sqrt{c+d x}}\right ) \sin (a)}{d^3}+\frac{b^4 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^2 f (d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{60 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt{c+d x}}\right )}{3 d^3}+\frac{b^6 f^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{360 d^3}-\frac{b^4 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{6 d^3}+\frac{b^2 (d e-c f)^2 \cos (a) \text{Si}\left (\frac{b}{\sqrt{c+d x}}\right )}{d^3}\\ \end{align*}
Mathematica [C] time = 2.20019, size = 557, normalized size = 0.91 \[ \frac{i e^{-i a} \left (-e^{2 i a} b^2 \left (f^2 \left (60 b^2 c+b^4+360 c^2\right )-60 d e f \left (b^2+12 c\right )+360 d^2 e^2\right ) \text{Ei}\left (\frac{i b}{\sqrt{c+d x}}\right )-\sqrt{c+d x} e^{i \left (2 a+\frac{b}{\sqrt{c+d x}}\right )} \left (-2 i b^3 f (-29 c f+30 d e+d f x)-6 b^2 f \sqrt{c+d x} (-9 c f+10 d e+d f x)+b^4 f^2 \sqrt{c+d x}+i b^5 f^2+24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )+120 \sqrt{c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+b^2 \left (f^2 \left (60 b^2 c+b^4+360 c^2\right )-60 d e f \left (b^2+12 c\right )+360 d^2 e^2\right ) \text{Ei}\left (-\frac{i b}{\sqrt{c+d x}}\right )+\sqrt{c+d x} e^{-\frac{i b}{\sqrt{c+d x}}} \left (2 i b^3 f (-29 c f+30 d e+d f x)-6 b^2 f \sqrt{c+d x} (-9 c f+10 d e+d f x)+b^4 f^2 \sqrt{c+d x}-i b^5 f^2-24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )+120 \sqrt{c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )\right )}{720 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 696, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.04818, size = 1184, normalized size = 1.94 \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 = 2.38604, size = 1119, normalized size = 1.83 \begin{align*} \frac{{\left (360 \, b^{2} d^{2} e^{2} - 60 \,{\left (b^{4} + 12 \, b^{2} c\right )} d e f +{\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \operatorname{Ci}\left (\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) +{\left (360 \, b^{2} d^{2} e^{2} - 60 \,{\left (b^{4} + 12 \, b^{2} c\right )} d e f +{\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \operatorname{Ci}\left (-\frac{b}{\sqrt{d x + c}}\right ) \sin \left (a\right ) + 2 \,{\left (360 \, b^{2} d^{2} e^{2} - 60 \,{\left (b^{4} + 12 \, b^{2} c\right )} d e f +{\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \cos \left (a\right ) \operatorname{Si}\left (\frac{b}{\sqrt{d x + c}}\right ) + 2 \,{\left (24 \, b d^{2} f^{2} x^{2} + 360 \, b d^{2} e^{2} - 60 \,{\left (b^{3} + 10 \, b c\right )} d e f +{\left (b^{5} + 58 \, b^{3} c + 264 \, b c^{2}\right )} f^{2} + 2 \,{\left (60 \, b d^{2} e f -{\left (b^{3} + 36 \, b c\right )} d f^{2}\right )} x\right )} \sqrt{d x + c} \cos \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\right ) + 2 \,{\left (120 \, d^{3} f^{2} x^{3} + 360 \, c d^{2} e^{2} - 60 \,{\left (b^{2} c + 6 \, c^{2}\right )} d e f +{\left (b^{4} c + 54 \, b^{2} c^{2} + 120 \, c^{3}\right )} f^{2} - 6 \,{\left (b^{2} d^{2} f^{2} - 60 \, d^{3} e f\right )} x^{2} -{\left (60 \, b^{2} d^{2} e f - 360 \, d^{3} e^{2} -{\left (b^{4} + 48 \, b^{2} c\right )} d f^{2}\right )} x\right )} \sin \left (\frac{a d x + a c + \sqrt{d x + c} b}{d x + c}\right )}{720 \, d^{3}} \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 (e + f x\right )^{2} \sin{\left (a + \frac{b}{\sqrt{c + d 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 (f x + e\right )}^{2} \sin \left (a + \frac{b}{\sqrt{d x + c}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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